Classification of all alternatives to the Born rule in
terms of informational properties
Thomas D. Galley and Lluis Masanes
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
June 14, 2017
The standard postulates of quantum the-
ory can be divided into two groups: the
first one characterizes the structure and
dynamics of pure states, while the sec-
ond one specifies the structure of measure-
ments and the corresponding probabilities.
In this work we keep the first group of pos-
tulates and characterize all alternatives to
the second group that give rise to finite-
dimensional sets of mixed states. We prove
a correspondence between all these alter-
natives and a class of representations of
the unitary group. Some features of these
probabilistic theories are identical to quan-
tum theory, but there are important dif-
ferences in others. For example, some the-
ories have three perfectly distinguishable
states in a two-dimensional Hilbert space.
Others have exotic properties such as lack
of “bit symmetry”, the violation of “no si-
multaneous encoding” (a property similar
to information causality) and the existence
of maximal measurements without phase
groups. We also analyze which of these
properties single out the Born rule.
1 Introduction
There is a long-standing debate on whether the
structure and dynamics of pure states already en-
codes the structure of measurements and proba-
bilities. This discussion arises within the dynam-
ical description of a quantum measurement, de-
coherence theory, and the many-worlds interpre-
tation. There have been attempts to derive the
Born rule [13], however these are often deemed
controversial. In this work we suggest a more
neutral approach where we consider all possible
Thomas D. Galley: thomas.galley.14@ucl.ac.uk
alternatives to the Born rule and explore their
physical and informational properties.
More precisely, we provide a complete classifi-
cation of all the alternatives to the structure of
measurements and the formula for outcome prob-
abilities with the following property: in any sys-
tem with a finite-dimensional Hilbert space, the
number of parameters that is required to specify
a mixed state is finite. This property is neces-
sary if state estimation can be performed with a
finite number of measurements and no additional
assumptions. We also show that, among all these
alternative theories, those which have no restric-
tion on the allowed effects violate a physical prin-
ciple called “bit symmetry” [4], which is satisfied
by quantum theory. Bit symmetry states that
any pair of perfectly distinguishable pure states
can be reversibly mapped to any other. Thus the
Born rule is the only possible probability assign-
ment to measurement outcomes under these re-
quirements. We hope that these results may help
to settle the above mentioned debate.
Most other known non-quantum general prob-
abilistic theories seem somewhat pathological [5
7] with some exceptions such as quantum the-
ory over the field of real numbers [8] or quater-
nions [9] and theories based on euclidean Jordan
algebras [10].
For example they often violate most of the ax-
ioms used to reconstruct quantum theory, rather
than just one or two per reconstruction. In par-
ticular, modifying the Born rule without creating
inconsistencies or absurdities has been argued to
be difficult (see, e.g. [11]).
For example, if we keep the usual association of
the outcomes of a measurement with the elements
of an orthonormal basis {|ki}, a straightforward
alternative for the probability of |ki given sate
|ψi is
P (k|ψ) =
|hk|ψi|
α
P
k
0
|hk
0
|ψi|
α
, (1)
1
arXiv:1610.04859v3 [quant-ph] 13 Jun 2017
with α 6= 2. However, this violates the finiteness
assumption stated above.
In this work, we provide many non-quantum
probabilistic theories with a modified Born rule
which still inherit many of quantum theory’s
properties. These theories can be used as foils to
understand quantum theory “from the outside”.
In particular, we analyse in full detail all finite-
dimensional state spaces where pure states are
rays in a 2-dimensional Hilbert space. We discuss
properties such as the number of distinguishable
states, bit symmetry, no-simultaneous encoding
and phase groups for maximal measurements. We
also analyze a class of alternative theories which
have a restriction on the allowed effects such that
they share all the above properties with quantum
theory.
In section 2 we introduce the necessary frame-
work and outline the main result showing the
equivalence between alternative sets of measure-
ment postulates and certain representations of
the dynamical group. We then find these repre-
sentations. In section 3 we study these alternative
theories starting with a classification of theories
with pure states given by rays on a 2-dimensional
Hilbert space. We then show that all unrestricted
non-quantum theories with pure states given by
rays on a d-dimensional Hilbert space violate bit
symmetry. In section 4 we discuss the results and
their relation to Gleason’s theorem. Proofs of all
the results are in the appendix.
2 All alternatives to the Measurement
Postulates
2.1 Outcome probability functions
The standard formulation of quantum theory can
be divided in two parts. The first part postu-
lates that the pure states of a system are the
rays of a complex Hilbert space
1
ψ PC
d
, and
that the evolution of an isolated quantum sys-
tem is unitary ψ 7→ U ψ, where U PU(d) is
an element of the projective unitary group
2
. The
second part postulates that each measurement is
1
The set of rays of a Hilbert space H is called the pro-
jective Hilbert space, denoted PH.
2
PU(d) is defined by taking SU(d) and identifying the
matrices that differ by a global phase U
=
e
U. For
example, the two matrices ± SU(2) correspond to the
same element in PU(2).
characterized by a list of positive semi-definite
operators (Q
1
, Q
2
, . . . , Q
k
) adding up the iden-
tity
P
i
Q
i
= , and that the probability of out-
come Q
i
when the system is in state ψ is given
by P (Q
i
|ψ) = hψ|Q
i
|ψi (Born’s rule).
In this article we consider all possible alterna-
tives to this second part. For this, we denote by
P (F |ψ) the probability of outcome F given the
pure state ψ. And we recall that, from an oper-
ational perspective, the mathematical character-
ization of an outcome F is given by the prob-
ability of its occurrence for all states. Hence,
each outcome F is represented by the function
F : PC
d
[0, 1] defined as F (ψ) = P (F |ψ).
Note that, unlike in [12], we do not require out-
comes F to correspond to those of quantum me-
chanics (POVM elements). And even more, a pri-
ori, these outcome probability functions (OPFs)
F are arbitrary; for example, they can be non-
linear in |ψihψ| or even discontinuous. How-
ever, we show below that, if mixed states are
required to have finitely-many parameters then
OPFs must be polynomial in |ψihψ|.
In this generalized setup, each measure-
ment is characterized by a list of OPFs
(F
1
, F
2
, . . . , F
k
) satisfying the normalization con-
dition
P
i
F
i
(ψ) = 1 for all ψ PC
d
. There-
fore, each alternative to the Measurement Postu-
lates can be characterized by the list of all “al-
lowed” measurements in each dimension d, e.g.
{(F
1
, . . . , F
k
), (F
0
1
, . . . , F
0
k
0
), . . .}. For each alter-
native, we denote the set of OPFs on PC
d
by
F
d
= {F
1
, F
2
, . . . , F
0
1
, F
0
2
, . . .}. From an opera-
tional perspective, the set F
d
is not completely
arbitrary: the composition of a measurement and
a unitary generates another measurement. That
is, if F F
d
and U PU(d) then F U F
d
.
In this generalized setup, a mixed state is an
equivalence class of indistinguishable ensembles.
Let (p
i
, ψ
i
) be the ensemble where state ψ
i
PC
d
is prepared with probability p
i
. Two ensembles
(p
i
, ψ
i
) and (p
0
i
, ψ
0
i
) are indistinguishable in F
d
if
X
i
p
i
F (ψ
i
) =
X
i
p
0
i
F (ψ
0
i
) (2)
for all F F
d
. This equation defines the equiva-
lence relation with which the equivalence classes
of ensembles (mixed states) are defined.
2
ψ
U
//
Uψ
ψ
Γ
U
//
Uψ
Figure 1: This diagram expresses the commutation of
equation (6).
2.2 The convex representation
In what follows we introduce a different represen-
tation for pure states, unitaries and OPFs, which
naturally incorporates mixed states. This repre-
sentation is the one used in general probabilistic
theories [1316], which is sometimes called the
convex operational framework [1720]. Before
starting let us introduce some notation. Given
a real vector space V , the general linear group
GL(V ) is the set of invertible linear maps T :
V V , and E(V ) is the set of affine
3
functions
E : V R.
Result 1. Given a set F
d
of OPFs for PC
d
(en-
coding an alternative to the measurement pos-
tulates) there is a (possibly infinite-dimensional)
real vector space V and the maps
Ω : PC
d
V , (3)
Γ : PU(d) GL(V ) , (4)
Λ : F
d
E(V ) , (5)
satisfying the following properties:
Preservation of dynamical structure (see
Figure 1):
Γ
U
ψ
= Ω
Uψ
, (6)
Γ
U
1
Γ
U
2
= Γ
U
1
U
2
. (7)
Preservation of probabilistic structure:
Λ
F
(Ω
ψ
) = F (ψ) . (8)
3
A function E : V R is affine if it satisfies
E (
1
+ (1 x)ω
2
) = xE(ω
1
)+(1x)E(ω
2
) for all x R
and ω
1
, ω
2
V . When V is finite-dimensional, each
E E (V ) can be written in terms of a scalar product
as E(ω) = e · ω + c, where e V and c R.
Minimality of V
4
:
Aff (Ω
PC
d
) = V . (9)
Uniqueness: for any other maps
0
, Γ
0
, Λ
0
satisfying all of the above, there is an in-
vertible linear map L : V V such that
0
ψ
= L(Ω
ψ
) , (10)
Γ
0
U
= LΓ
U
L
1
, (11)
Λ
0
F
= Λ
F
L
1
. (12)
In this new representation, the affinity of the
OPFs Λ
F
implies the following. All the predic-
tions for an ensemble (p
i
, ψ
i
) can be computed
from the vector ω =
P
i
p
i
ψ
i
V . In more de-
tail, if a source prepares state ψ
i
with probability
p
i
then the probability of outcome F is
X
i
p
i
P (F |ψ
i
) =
X
i
p
i
Λ
F
(Ω
ψ
i
) = Λ
F
(ω) .
Hence, the “mixed state” ω contains all the physi-
cal information of the ensemble, in the same sense
as the density matrix ρ =
P
i
p
i
|ψ
i
ihψ
i
| does in
quantum theory. More precisely, ω is the ana-
logue of the traceless part of the density ma-
trix ρ
trρ
d
, which in the d = 2 case is the
Bloch vector. So V is the analogue of the space
of traceless Hermitian matrices or Bloch vectors,
not the full space of Hermitian matrices. This
is why probabilities are affine instead of linear
functions. For example,in this representation, the
maximally mixed state is always the zero vector
R
ψ
= 0 V . Also, the linearity of the ac-
tion Γ
U
: V V implies that, transforming the
ensemble as (p
i
, ψ
i
) (p
i
, Uψ
i
), is equivalent to
transforming the mixed state as
X
i
p
i
Uψ
i
=
X
i
p
i
Γ
U
ψ
i
= Γ
U
ω .
When we include all these mixed states the (full)
state space S becomes convex:
S = conv
PC
d
V . (13)
An effect on S is an affine function E : S [0, 1].
In the representation introduced in Result 1,
OPFs Λ
F
are effects. We say that a given set
4
Here Aff(S) refers to the affine span of vectors in S.
The affine span is defined as all linear combinations of
vectors in S where the coefficients add to 1.
3
of OPFs F
d
is unrestricted [21] if it contains all
effects of the corresponding state space S. That
is, for every effect E on S there is an OPF F F
d
such that Λ
F
= E. When F
d
is unrestricted the
map Λ is redundant, and the theory is character-
ized by and Γ, which provide the state space S.
When F
d
is not unrestricted, the theory can be
understood as an unrestricted one with an extra
restriction on the allowed effects. Hence, unre-
stricted theories play a special role.
The dimension of S (equal to that of V ) cor-
responds to the number of parameters that are
necessary to specify a general mixed state. In
quantum theory this number equal to d
2
1. De-
spite the finite dimensionality of PC
d
the state
space S = conv
PC
d
corresponding to a gen-
eral F
d
, can be infinite-dimensional. When this
is the case, general state estimation requires col-
lecting an infinite amount of experimental data,
which makes this task impossible. Recall that,
when performing state estimation in infinite-
dimensional quantum mechanics, additional as-
sumptions are required; like, for example, an up-
per bound on the energy of the state. For this rea-
son, in the rest of the paper, we assume that when
the set of pure states PC
d
is finite-dimensional
then so is the set of mixed states S.
Let us also assume that any mathematical
transformation T : S S which can be
arbitrarily-well approximated by physical trans-
formations Γ
PU(d)
should be considered a physical
transformation T Γ
PU(d)
. This means that the
group of matrices Γ
PU(d)
is topologically closed.
Together with the homomorphism property (7)
and the finite-dimensionality of V this implies
that Γ : PU(d) GL(V ) is a continuous group
homomorphism (this is shown in appendix A.1.1).
In other words, Γ is a representation of the Lie
group PU(d).
2.3 Classification results
In quantum mechanics all pure states ψ can be re-
lated to a fixed state ψ
0
via a unitary ψ = Uψ
0
.
This and equation (6) imply that, given the repre-
sentation Γ and the image of the fixed pure state
ψ
0
ψ
0
, we can obtain the image of any pure
state as
ψ
= Ω
Uψ
0
= Γ
U
ψ
0
. (14)
Now note that, for any OPF F , we can write
F (ψ) = F (Uψ
0
) = Λ
F
U
ψ
0
) . (15)
This, the affinity of Λ
F
and the continuity of Γ
imply that F : PC
d
[0, 1] is continuous.
We also see that, an unrestricted theory is char-
acterized by a representation Γ and a reference
vector
ψ
0
V . But not all pairs ,
ψ
0
) sat-
isfy (6). Below we prove that only a small class
of representations Γ allow for (6), and for each of
these, there is a unique (in the sense of (10)-(12))
vector
ψ
0
satisfying (6).
Any representation of PU(d) is also a represen-
tation of SU(d), and these are all well classified
in the finite-dimensional case. We now consider
a family of irreducible representations of SU(d)
which we call D
d
j
. There are many other repre-
sentations of SU(d) but these are not compat-
ible with the dynamical structure of quantum
theory (6). For any positive integer j, let D
d
j
:
SU(d) GL(R
D
d
j
) be the highest-dimensional ir-
reducible representation inside the reducible one
Sym
j
U Sym
j
U
, where Sym
j
U is the projec-
tion of U
j
into the symmetric subspace [22,
Appendix 2]. Here U is the fundamental rep-
resentation of SU(d) acting on C
d
. Note that
any global phase e
U disappears in the prod-
uct Sym
j
U Sym
j
U
, hence D
d
j
is also an irre-
ducible representation of PU(d). We recall that
these irreducible representations are real, in the
sense that there exists a basis in which all ma-
trix elements of D
d
j
(U) are real, for all U. The
dimension D
d
j
of the real vector space acted upon
by D
d
j
(U) is
D
d
j
=
2j
d 1
+ 1
d2
Y
k=1
1 +
j
k
2
, (16)
(see [22, p.224]).
Note that quantum theory corresponds to j =
1. In figure 2 the weight diagrams of the quantum
(D
3
1
) and lowest dimensional non-quantum (D
3
2
)
representations of su(3) (the Lie algebra of SU(3))
are shown. For d = 2, D
2
j
are the SU(2) irre-
ducible representations with integer spin j, which
are also irreducible representations of SO(3)
=
PU(2). For d 3, these irreducible represen-
tations are also denoted with the Dynkin label
(j, 0, . . . , 0
| {z }
d3
, j).
Result 2. Let ω
d
j
R
D
d
j
be the unique (up to
proportionality) invariant vector D
d
j
(U)ω
d
j
= ω
d
j
4
for all elements of the subgroup
U =
e
0 ··· 0
0
.
.
. e
iα/(d1)
u
0
, u SU(d 1) .
(17)
Each finite-dimensional representation :
PC
d
R
n
and Γ : PU(d) GL(R
n
) satisfy-
ing (6), (7), (9) is of the form
Γ
U
=
M
j∈J
D
d
j
(U) , (18)
ψ
0
=
M
j∈J
ω
d
j
, (19)
where J is any finite set of positive integers.
Each set J corresponds to an unrestricted the-
ory, and clearly the dimension of the state space
n =
P
j∈J
D
d
j
. Quantum theory corresponds to
J = {1} and n = d
2
1. Now, let us analyze a
simple example: quantum theory with d = 2. In
this case, the subgroup SU(d1) = { } is trivial,
and the subgroup (17) is e
Zt
where
Z =
i 0
0 i
!
. (20)
The action of this subgroup on the Bloch vec-
tor has two invariant states (0, 0, ±1). But, as
the result says, both are the same vector up to
a proportionality factor. Also note that the two
state spaces generated by the two reference vec-
tors ω
2
1
= (0, 0, ±1) are related as in (10)-(12).
A consequence of Result 2 is that all finite-
dimensional group actions Γ are polynomials of
the fundamental action U. Alternative (1) is
not polynomial, and hence needs an infinite-
dimensional representation V . This shows our
previous claim that an infinite number of param-
eters is necessary to specify a mixed state in this
alternative theory.
A desirable feature of any alternative to the
measurement postulates F
d
is faithfulness: for
any pair pure states ψ, ψ
0
PC
d
there is a
measurement F F
d
which distinguish them
F (ψ) 6= F (ψ
0
). When this does not happen, the
two states ψ, ψ
0
become operationally equivalent.
Faithfulness translates to the injectivity of the
map : PC
d
V . The following result tells us
which of the representations (18)-(19) are faith-
ful.
Result 3 (Faithfulness). When d 3 the map
is always injective. When d = 2 the map
is injective if and only if J contains at least one
odd number.
In this work we are specially interested in faithful
state spaces. Because if two pure states in PC
d
are indistinguishable, then, from an operational
point of view, they become the same state.
2.4 A simpler characterization
In this section we present a different representa-
tion for all these alternative theories that makes
them look closer to quantum theory. For this,
we have to relax the “minimality of V property
(given by equation (9)).
For any ψ PC
d
and U PU(d) let
ψ
= |ψihψ|
N
, (21)
Γ
U
ω = U
N
ω U
†⊗N
, (22)
where N is a positive integer and ω is a Hermi-
tian matrix acting on the symmetric subspace of
(C
d
)
N
. It can be seen that this representation
decomposes as
Γ
U
=
N
M
j=0
P
d
N,j
D
d
j
(U) , (23)
where P
d
N,j
is a projector whose rank depends on
N, j, d. The rank of projector P
d
N,j
counts the
number of copies of representation D
d
j
. For ex-
ample rankP
d
N,0
= 1 and rankP
d
N,N
= 1. Also, we
define D
d
0
to be the trivial irreducible representa-
tion.
The fact that representation (21)-(22) contains
the trivial irreducible representation, allows for
effects to be linear instead of affine function.
Hence, all effects can be written as
P (E|ψ) = tr(E|ψihψ|
N
) , (24)
where E is a Hermitian matrix. However, this
matrix E does not need to be positive semi-
definite. For example, any N-party entanglement
witness W has negative eigenvalues but gives a
positive value tr(W |ψihψ|
N
) 0 for all ψ. An
example of this is given in Section 3.1.
If we want to use notation (21)-(22) to repre-
sent the alternative theory characterized by J,
then we have to set N max J and constrain
5
(a)
(b)
Figure 2: (a) The weight diagram of the irreducible rep-
resentation of su(3) with j = 1. This is the adjoint
representation and corresponds to quantum theory. (b)
The weight diagram of the irreducible representation of
su(3) with j = 2. This corresponds to the simplest
non-quantum state space for PC
3
.
the allowed effects to only have support on the
subspaces of (23) with j J. The fact that for
some j J the representations D
d
j
may be re-
peated is irrelevant. We can restrict the effects
to act on a single copy or not. In the second case
the action of the effect becomes the average of its
action in each copy.
3 Phenomenology of the alternative
theories
In this section we explore the properties of the
alternative theories classified in the previous sec-
tion. We first consider the case d = 2 for both ir-
reducible and reducible representations, and later
we generalize to d 3. Before considering these
families of theories we provide an example of a
specific theory of PC
2
which differs significantly
from quantum theory.
3.1 Three distinguishable states in C
2
Result 4. The d = 2 state space J = {1, 2}
has at least three perfectly distinguishable states
when all effects are allowed.
(Result independently obtained in [23].) The irre-
ducible representations D
2
j
are given by the sym-
metric product D
2
j
(U) = Sym
2j
U [22, p.150],
where U is the fundamental representation on C
2
.
According to Result 3, these are faithful when j is
odd ( injective). To prove Result 4 we first show
that the unfaithful PC
2
theory J
0
= {2} corre-
sponds to a quantum PC
3
system restricted to the
real numbers (i.e. PR
3
), and hence has 3 distin-
guishable states. By including the trivial repre-
sentation, the representation J
0
can be expressed
as: Sym
4
USym
0
U = Sym
2
(Sym
2
U) [22, p.152].
Using the fact that Sym
2
U is the quantum repre-
sentation, the states generated by applying J
0
to
the reference state can be expressed as the rank-
one projectors
|φihφ| = D
2
1
(U)ω
2
1
[D
2
1
(U)ω
2
1
]
T
, (25)
where ω
2
1
R
3
is a unit Bloch vector and hence
|φi R
3
with hφ|φi = 1. This is the state space
of a 3-dimensional quantum system restricted to
the real numbers PR
3
, and hence, has three dis-
tinguishable states.
Coming back to J = {1, 2}, this theory
has transformations given by the representation
Sym
4
U Sym
2
U Sym
0
U and hence it is faith-
ful. It has at least three distinguishable states
since one can make measurements with support
on the blocks Sym
4
U Sym
0
U to distinguish the
three states. It may be the case that by mak-
ing measurements with support in both blocks
one could distinguish more states. We now con-
sider families of alternative theories with dynam-
ics SU(2) and find various properties which dis-
tinguish them from the qubit.
3.2 Irreducible d = 2 theories
In this section we define a class of d = 2 theories
and explore which properties they have in com-
mon with the qubit, and which properties distin-
guish them from quantum theory. We denote by
T
I
2
the class of non-quantum d = 2 theories which
are faithful, unrestricted (all effects allowed) and
have irreducible Γ. According to Result 3, each
6
of these theories is characterized by an odd in-
teger j 3, such that J = {j}, or equivalently
Γ = D
2
j
.
3.2.1 Number of distinguishable states
An important property of a system is the max-
imal number of perfectly distinguishable states
it has. This quantity determines the amount of
information that can be reliably encoded in one
system. For instance a classical bit has two dis-
tinguishable states, as does a qubit. Distinguish-
able states of a qubit are orthogonal rays in the
Hilbert space, or equivalently, antipodal states on
the Bloch sphere.
The following result tells us about the similar-
ities of the theories in T
I
2
with respect to quan-
tum mechanics. It also applies to other theories,
because it does not require the assumption of un-
restricted effects.
Result 5. Any d = 2 theory with irreducible Γ
has a maximum of two perfectly distinguishable
states.
However all these theories have an important dif-
ference with quantum theory.
Result 6. All theories in T
I
2
have pairs of non-
orthogonal (in the underlying Hilbert space) rays
which are perfectly distinguishable.
We also prove that two rays ψ, φ PC
2
are or-
thogonal if and only if they are represented by
antipodal states
ψ
=
φ
. The existence of
distinguishable non-antipodal states entails that
theories in T
I
2
have exotic properties not shared
by qubits. We discuss a few of these in what fol-
lows.
3.2.2 Bit symmetry
Bit symmetry, as defined in [4], is a property of
theories whereby any pair of pure distinguishable
states (ω
1
, ω
2
) can be mapped to any other pair
pure of distinguishable states (ω
0
1
, ω
0
2
) with a re-
versible transformation U belonging to the dy-
namical group, i.e. Γ
U
ω
1
= ω
0
1
and Γ
U
ω
2
= ω
0
2
.
The qubit is bit symmetric since distinguishable
states are orthogonal rays, and any pair of or-
thogonal rays can be mapped to any other pair
of orthogonal rays via a unitary transformation.
Result 7. All theories in T
I
2
violate bit symme-
try.
This result follows directly from Result 6 and
from the fact that there does not exist any re-
versible transformation which maps a pair of
antipodal states (representations of orthogonal
rays) to a pair of non-antipodal states (represen-
tations of non-orthogonal rays).
3.2.3 Phase invariance of measurements
We now consider the concept of phase groups
of measurements following [24, 25]. The phase
group T of a measurement (E
1
, ..., E
n
) is the max-
imal subgroup of the transformation group of the
state space S which leaves all outcome probabil-
ities unchanged:
E
i
Γ
U
= E
i
U T . (26)
For example, in the case of the qubit and a mea-
surement in the Z basis, the phase group associ-
ated to this measurement is T = {e
Zt
: t R},
where (20). We maintain the historical name
“phase group”, although in general, it need not
be abelian.
A measurement which distinguishes the max-
imum number of pure states in a state space is
often called a maximal measurement [24]. For
a qubit, these are the projective measurements
(since they perfectly distinguish two states) which
have a phase group U(1).
Result 8. All theories in T
I
2
have maximal mea-
surements with trivial phase groups T = { }.
These maximal measurements with trivial phase
groups are those used to distinguish non-
antipodal states. We note that the antipodal
states can be distinguished by measurements with
U(1) phase groups (as in quantum theory). We
observe that contrary to [24] we find maximal
measurements with trivial phase groups in a non-
classical theory. This is because unlike [24] we
do not consider all allowed transformations which
map states to states, but only the transformations
Γ
U
.
3.2.4 No simultaneous encoding
No simultaneous encoding [26] is an information-
theoretic principle which states that if a system is
used to perfectly encode a bit it cannot simulta-
neously encode any other information (similarly
for a trit and higher dimensions). More precisely,
7
consider a communication task involving two dis-
tant parties, Alice and Bob. Similarly as in the
scenario for information causality [27], suppose
that Alice is given two bits a, a
0
{0, 1}, and
Bob is asked to guess only one of them. He will
base his guess on information sent to him by Al-
ice, encoded in one T
I
2
system. Alice prepares
the system with no knowledge of which of the two
bits, a or a
0
, Bob will try to guess. No simultane-
ous encoding imposes that, in a coding/decoding
strategy in which Bob can guess a with probabil-
ity one, he knows nothing about a
0
. That is, if
b, b
0
are Bob’s guesses for a, a
0
then
P (b|a, a
0
) = δ
a
b
P (b
0
|a, a
0
= 0) = P (b
0
|a, a
0
= 1)
where δ
a
b
is the Kronecker tensor.
As an example consider a qubit. Alice decides
to perfectly encode bit a, which she can only do
by encoding a = 0 and a = 1 in two perfectly
distinguishable states. Without loss of generality
she can choose to encode a = 0 in |0i and a = 1 in
|1i, with h0|1i = 0. She now also needs to encode
a
0
whilst keeping a perfectly encoded. Since, |0i is
the only state which is perfectly distinguishable
from |1i, we have that both a = 0, a
0
= 0 and
a = 0, a
0
= 1 combinations must be assigned to
|0i. Similarly a = 1, a
0
= 0 and a = 1, a
0
= 1
combinations must be assigned to |1i. In this
case we see that whilst Bob can perfectly guess
the value of a if he chooses to, if he chooses to
guess the value of a
0
he cannot do so. Hence this
property is met by qubits, however
Result 9. All theories in T
I
2
violate no simulta-
neous encoding.
We see that the above three properties: bit sym-
metry, existence of phase groups for maximal
measurements and no-simultaneous encoding sin-
gle out the qubit amongst all T
I
2
theories.
3.2.5 Restriction of effects
The study of theories in T
I
2
has shown that they
differ from quantum theory in many ways. We
now consider a new family of theories
˜
T
I
2
, which
is constructed by restricting the effects of the the-
ories in T
I
2
. These theories turn out to be closer
to quantum theory in that they obey all the above
properties. This approach is similar to the self-
dualization procedure outlined in [21] (which also
recovers bit symmetry).
Qubit T
I
2
˜
T
I
2
T
R
2
Distinguishable states 2 2 2 2
Bit symmetry X X X X
Phase invariant group X X X ?
No simultaneous enco ding X X X ?
Figure 3: Summary of results for d = 2 theories.
We call a theory pure-state dual if the only
allowed effects are “proportional” to pure states.
That is, for every allowed effect, there is a pure
state ψ and a pair of normalization constants α, β
such that
E(ω) = α(Ω
T
ψ
· ω) + β . (27)
All theories in
˜
T
I
2
have a maximum of two distin-
guishable states, and all pairs of distinguishable
states are antipodal.
Result 10. All theories in
˜
T
I
2
are bit symmetric,
have phase groups for all maximal measurements
and obey no-simultaneous encoding.
3.3 Reducible d = 2 theories
We consider the set of all unrestricted faithful
non-quantum state spaces generated by reducible
representations of SU(2). These representations
are given by equation (18) with d = 2 and |J| > 1
containing at least one odd number. We denote
the set of all these theories T
R
2
.
For theories in T
R
2
the number of distinguish-
able states can be more than two (as shown for
J = {1, 2} in Section 3.1). It is in general a dif-
ficult task to find the number of distinguishable
states. However we find:
Result 11. All theories in T
R
2
violate bit sym-
metry.
That is, there are pairs of distinguishable pure
states (ψ
1
, ψ
2
) which cannot be mapped to an-
other pair of distinguishable states (ψ
0
1
, ψ
0
2
) with
a reversible transformation U belonging to the
dynamical group. This implies that
Bit symmetry singles out quantum theory
amongst all d = 2 unrestricted faithful state
spaces (both irreducible and reducible).
8
3.4 Arbitrary d
Having studied in detail different families of d = 2
theories, we now consider properties of arbitrary-
d unrestricted theories. We also show that the
property of bit symmetry singles out quantum
theory in all dimensions. This is proven by show-
ing that all faithful PC
d
state spaces have embed-
ded within them faithful (reducible) PC
d1
state
spaces. We then show that if a state space PC
d1
violates bit symmetry, then any state space PC
d
it is embedded in also does. Using a proof by
induction we find:
Result 12. All unrestricted non-quantum the-
ories with pure states PC
d
and transformations
D
d
j
are not bit symmetric.
This result shows that the Born rule can be sin-
gled out amongst all possible probability assign-
ments from the following set of assumptions.
The Born rule is the unique probability as-
signment satisfying: (i) no restriction on the
allowed effects, and (ii) bit symmetry.
Although many works in GPT’s only consider un-
restricted theories no-restriction remains an as-
sumption of convenience with a less direct phys-
ical or operational meaning. Bit symmetry how-
ever is a property which has computational and
physical significance as discussed in [4]. It is gen-
erally linked to the possibility of reversible com-
putation, since bit symmetric theories allow any
logical bit (pair of distinguishable states) of the
theory to be reversibly transformed into any other
logical bit.
4 Conclusion
In this work we have considered theories with the
same structure and dynamics for pure states as
quantum theory but different sets of OPF’s (not
necessarily following the Born rule). These alter-
native theories were shown to be in correspon-
dence with certain class of representations of the
unitary group. Moreover these theories (assum-
ing no restriction on the effects) were shown to
differ from quantum theory in that they are not
bit symmetric. In the case of d = 2, the num-
ber of distinguishable states was bounded for ir-
reducible representations, and further properties
which single out quantum theory were found.
In order to fully develop these alternative the-
ories it is necessary to determine how the state
spaces studied compose. This would then allow
us to study important properties related to mul-
tipartite systems.
Gleason’s theorem [28] can be understood as
a derivation of the Born rule. Gleason assumes
that measurements are associated with orthogo-
nal bases and shows that states must be given
by density matrices and measurement outcome
probabilities by the Born rule. This is a differ-
ent approach to the one presented in this paper.
In this work we show that, assuming that pure
states are rays on a Hilbert space and time evo-
lution is unitary, the Born rule can be derived (or
singled out) from the requirement of bit symme-
try. We allow for measurements to take any form,
not just being associated with orthogonal bases
in the corresponding Hilbert space. Unlike Glea-
son’s theorem this applies in the case d = 2 and
may arguably provide a more physical justifica-
tion for the Born rule. There exists a derivation
of Gleason’s theorem for POVM’s [29] which is
closer in spirit to our approach than the original.
However we emphasise that in our work we begin
from the structure of pure states and dynamics to
derive that of measurements. In Gleason’s theo-
rem (and its extension) it is the structure of mea-
surements which are assumed and that of states
derived from it.
Future work will be focused on multipartite
systems and post-measurement state update rules
for the alternative theories analysed in this work.
This will allow us to consider more general in-
formation processing tasks for these novel proba-
bilistic theories.
5 Acknowledgements
We are grateful to Howard Barnum, Jonathan
Barrett, Markus Mueller, Jonathan Oppenheim,
Matthew Pusey, Jonathan Richens and Tony
Short for valuable discussions. We acknowledge
the helpful referee comments which allowed us to
correct the proofs of Lemma 2 and Result 7 as
well as clarify some technical points about the
continuity of the OPF’s. TG is supported by the
9
EPSRC Centre for Doctoral Training in Deliver-
ing Quantum Technologies. LM is supported by
the EPSRC.
References
[1] David Deutsch. Quantum theory of proba-
bility and decisions. Proceedings of the Royal
Society of London A: Mathematical, Physical
and Engineering Sciences. 455, 3129-3137.
1999.
[2] Wojciech Hubert Zurek. Probabilities from
entanglement, born’s rule p
k
= | ψ
k
|
2
from
envariance. Phys. Rev. A. 71, 052105. May
2005.
[3] David Wallace. How to Prove the Born Rule.
In Adrian Kent Simon Saunders, Jon Barrett
and David Wallace, editors, Many Worlds?
Everett, Quantum Theory, and Reality. Ox-
ford University Press. 2010.
[4] Markus P. Müller and Cozmin Ududec.
Structure of reversible computation deter-
mines the self-duality of quantum theory.
Phys. Rev. Lett.. 108, 130401. Mar 2012.
[5] David Gross, Markus Müller, Roger Colbeck,
and Oscar C. O. Dahlsten. All reversible dy-
namics in maximally nonlocal theories are
trivial. Phys. Rev. Lett.. 104, 080402. Feb
2010.
[6] Sabri W Al-Safi and Anthony J Short. Re-
versible dynamics in strongly non-local box-
world systems. Journal of Physics A: Math-
ematical and Theoretical. 47, 325303. 2014.
[7] Sabri W Al-Safi and Jonathan Richens. Re-
versibility and the structure of the local state
space. New Journal of Physics. 17, 123001.
2015.
[8] L. Hardy and W. K. Wootters. Lim-
ited Holism and Real-Vector-Space Quan-
tum Theory. Foundations of Physics. 42,
454-473. March 2012.
[9] D. Finkelstein, J.M. Jauch, and D. Speiser.
Notes on Quaternion Quantum Mechanics.
CERN publications: European Organization
for Nuclear Research. CERN. 1959.
[10] H. Barnum, M. Graydon, and A. Wilce.
Composites and Categories of Euclidean
Jordan Algebras. eprint ArXiv:1606.09331
quant-ph. June 2016.
[11] Scott Aaronson. Is Quantum Mechanics An
Island In Theoryspace? eprint arXiv:quant-
ph/0401062. 2004.
[12] Y. D. Han and T. Choi. Quantum Probabil-
ity assignment limited by relativistic causal-
ity. Sci. Rep.. 6, 22986. July 2016.
[13] L. Hardy. Quantum theory from five
reasonable axioms. eprint arXiv:quant-
ph/0101012. January 2001.
[14] Jonathan Barrett. Information processing in
generalized probabilistic theories. Phys. Rev.
A. 75, 032304. March 2007.
[15] Lluís Masanes and Markus P. Muller. A
derivation of quantum theory from physical
requirements. New Journal of Physics. 13,
063001. 2011.
[16] C. Brukner B. Dakic. Quantum Theory
and Beyond: Is Entanglement Special? In
H. Halvorson, editor, Deep Beauty - Under-
standing the Quantum World through Math-
ematical Innovation. pages 365–392. Cam-
bridge University Press. 2011.
[17] Robin Giles. Foundations for Quantum Me-
chanics. Journal of Mathematical Physics.
11, 2139. 1970.
[18] Bogdan Mielnik. Generalized quantum me-
chanics. Communications in Mathematical
Physics. 37, 221–256. 1974.
[19] G. Chiribella, G. M. D’Ariano, and
P. Perinotti. Probabilistic theories with pu-
rification. Physical Review A. 81, 062348.
jun 2010.
[20] H. Barnum, J. Barrett, L. Orloff Clark,
M. Leifer, R. Spekkens, N. Stepanik,
A. Wilce, and R. Wilke. Entropy and infor-
mation causality in general probabilistic the-
ories. New Journal of Physics. 12, 033024.
mar 2010.
[21] Peter Janotta and Raymond Lal. General-
ized probabilistic theories without the no-
restriction hypothesis. Physical Review A.
87, 052131. 2013.
[22] William Fulton and Joe Harris. Representa-
tion theory : a first course. Graduate texts
in mathematics. Springer-Verlag. New York,
Berlin, Paris. 1991.
[23] Jonathan Barrett. private communication.
[24] Andrew J P Garner, Oscar C O Dahlsten,
Yoshifumi Nakata, Mio Murao, and Vlatko
Vedral. A framework for phase and inter-
ference in generalized probabilistic theories.
New Journal of Physics. 15, 093044. 2013.
10
[25] Ciarán M Lee and John H Selby. Generalised
phase kick-back: the structure of computa-
tional algorithms from physical principles.
New Journal of Physics. 18, 033023. 2016.
[26] L. Masanes, M. P. Muller, R. Augusiak, and
D. Perez-Garcia. Existence of an informa-
tion unit as a postulate of quantum theory.
Proceedings of the National Academy of Sci-
ences. 110, 16373-16377. 2013.
[27] M. Pawłowski, T. Paterek, D. Kaszlikowski,
V. Scarani, A. Winter, and M. Żukowski. In-
formation causality as a physical principle.
nature. 461, 1101-1104. October 2009.
[28] Andrew M. Gleason. Measures on the Closed
Subspaces of a Hilbert Space. pages 123–133.
Springer Netherlands. Dordrecht. 1975.
[29] Carlton M Caves, Christopher A Fuchs, Ki-
ran K Manne, and Joseph M Renes. Gleason-
type derivations of the quantum probability
rule for generalized measurements. Founda-
tions of Physics. 34, 193–209. 2004.
[30] Karl Heinrich. Hofmann and Sidney A. Mor-
ris. The structure of compact groups : a
primer for students, a handbook for the ex-
pert. Berlin, Boston: De Gruyter. third edi-
tion, revised and augmented edition. 2013.
[31] Ingemar Bengtsson and Karol Zyczkowski.
Geometry of Quantum States. Cambridge
University Press. 2006. Cambridge Books
Online.
[32] M. L. Whippman. Branching Rules for Sim-
ple Lie Groups. Journal of Mathematical
Physics. 6, 1534-1539. October 1965.
[33] W.J. Holman and L.C. Biedenharm. The
Representations and Tensor Operators of the
Unitary Groups U(n). In Ernest M. Loebl,
editor, Group Theory and its Applications.
pages 1 73. Academic Press. 1971.
[34] Roe Goodman and Nolan R. Wallachs.
Symmetry, Representations, and Invariants.
Springer-Verlag New York. New York. 2009.
[35] Fernando Antoneli, Michael Forger,
and Paola Gaviria. Maximal sub-
groups of compact Lie groups. eprint
ArXiv:math/0605784. 2006.
A Proofs
A.1 The convex representation (proof of Re-
sult 1)
Let us fix the value of d and the set of OPFs
F
d
. Also, we assume that all OPFs of the form
F U are contained in F
d
, for all U PU(d) and
F F
d
. And recall that, if this is not the case,
we can always re-define F
d
to include them. Be-
fore defining the representation (Ω, Γ, Λ) of Re-
sult 1, it is convenient to define a temporary
one (
˜
,
˜
Γ,
˜
Λ) in which states
˜
ψ
are represented
by a list of outcome probabilities. For this, we
take a minimal subset {F
(1)
, F
(2)
, F
(3)
, . . .} F
d
that affinely generates F
d
. That is, for any
F F
d
there are constants c
(0)
, c
(1)
, c
(2)
, . . . R
such that F = c
(0)
+
P
r
c
(r)
F
(r)
. This con-
struction requires the minimal generating subset
{F
(1)
, F
(2)
, . . .} to be countable, but it can still
be infinite. This does not pose any loss of gener-
ality, because, from an operational point of view,
the characterization of arbitrary states would be
impossible.
If we represent the pure state ψ by the list of
probabilities
˜
ψ
=
P (F
(1)
|ψ)
P (F
(2)
|ψ)
P (F
(3)
|ψ)
.
.
.
=
F
(1)
(ψ)
F
(2)
(ψ)
F
(3)
(ψ)
.
.
.
V ,
(28)
then we can use the rules of probability to extend
this definition to mixed states. For example, the
probability of outcome F when the system is pre-
pared in the ensemble (p
i
, ψ
i
) is
P (F |(p
i
, ψ
i
)
i
) =
X
i
p
i
P (F |ψ
i
) =
X
i
p
i
F (ψ
i
) .
(29)
Therefore, the probabilistic representation of en-
semble (p
i
, ψ
i
) is
P
i
p
i
˜
ψ
i
. And the set of all
states, pure and mixed, is
˜
S = conv
˜
PC
d
. (30)
In the representation
˜
, any probability F (ψ)
is either a component of the vector
˜
ψ
or an affine
function of its components. Hence, for any F ,
there is an affine function
˜
Λ
F
: V R such that
˜
Λ
F
(
˜
ψ
) = F (ψ) ψ . (31)
Any U can be seen as a relabeling of the pure
states PC
d
, hence, {(F
(1)
U), (F
(2)
U), . . .}
11
F
d
is also a minimal generating subset. There-
fore, there is an affine map
˜
Γ
U
: V V such
that
˜
Uψ
=
˜
Γ
U
(
˜
ψ
) U, ψ . (32)
Next, we define a new representation , Γ, Λ in
which Γ
U
: V V is linear. This new represen-
tation is that of Result 1. For this, we need to
define the maximally mixed state
˜ω
mm
=
Z
PU(d)
dU
˜
Γ
U
(
˜
ψ
) , (33)
where dU is the Haar measure and ψ is any pure
state. We also note that the maximally mixed
state is invariant under any unitary:
˜
Γ
U
(˜ω
mm
) =
˜ω
mm
for any U. Now, we define the new repre-
sentation as
ψ
=
˜
ψ
˜ω
mm
, (34)
Γ
U
(ω) =
˜
Γ
U
(˜ω) ˜ω
mm
, (35)
Λ
F
(ω) =
˜
Λ
F
(˜ω) , (36)
which extends to general mixed states as ω =
˜ω ˜ω
mm
. Using (31) and (32) we obtain (6),
(7). In this representation, the maximally mixed
state is the zero vector ω
mm
= 0 V . Now,
recalling that ω
mm
is invariant under unitaries,
we have that Γ
U
(0) = 0, which together with the
affinity of Γ
U
implies that Γ
U
: V V is linear,
for all U .
To continue with the proof of Result 1, we note
that the affinity of
˜
Λ
F
implies that of Λ
F
; and
that (30) implies (9).
Now, it only remains to prove uniqueness. Let
us suppose that there are other maps
0
, Γ
0
, Λ
0
with the properties stated in Result 1. These
properties imply the existence of an affine func-
tion
˜
L such that
˜
ψ
=
F
(1)
(ψ)
F
(2)
(ψ)
.
.
.
=
Λ
0
F
(1)
(Ω
0
ψ
)
Λ
0
F
(2)
(Ω
0
ψ
)
.
.
.
=
˜
L(Ω
0
ψ
) .
(37)
Which in turn implies the existence of another
affine function L such that
ψ
= L(Ω
0
ψ
). Accord-
ing to the properties stated in Result 1, both Γ
U
and Γ
0
U
are linear; which implies that the maxi-
mally mixed state in both representations is the
zero vector ω
mm
= ω
0
mm
= 0 V . The affine map
L also changes the representation of the maxi-
mally mixed state ω
mm
= L(ω
0
mm
). But 0 = L(0)
is only possible if L is not just affine, but linear.
All the results of this section are valid when V
is finite- and infinite-dimensional. In the rest of
the appendices, only the finite-dimensional case
is considered.
A.1.1 Continuity of homomorphisms Γ
The number of parameters that define a mixed
state in S is equal to the dimension of V =
conv S. Hence, the requirement that general
mixed states in S can be estimated with finite
means implies that V is finite-dimensional. Now,
we recall that in the probabilistic representation
˜
, the entries of states ˜ω
˜
S are bounded.
Hence, due to (34), the same is true in S. This
implies that the absolute value of all matrix ele-
ments of the group Γ
PU(d)
are also bounded.
Now, we argue that, from a physical point of
view, the group of transformations Γ
PU(d)
must
be topologically closed. This follows from the
fact that any mathematical transformation that
can be approximated arbitrarily well by physi-
cal transformations should be a physical trans-
formation too. In summary, the fact that the set
Γ
PU(d)
R
n
is bounded and closed implies that
it is compact. And this is the premise of the fol-
lowing theorem.
It is proven in Theorem 5.64 of [30, p.167] that
any group homomorphism Γ : PU(d) G in
which G is compact must be continuous.
A.2 Preservation of dynamical structure (Re-
sult 2)
The stabilizer subgroup G
ψ
PU(d) of a pure
state ψ PC
d
is the set of unitaries U leaving the
state invariant Uψ = ψ. If the state ψ
0
is related
to ψ via ψ
0
= Uψ, then their corresponding sta-
bilizer subgroups are related via G
ψ
0
= UG
ψ
U
.
Therefore, all stabilizers are isomorphic.
Equation (7) implies that Γ is a representation
of SU(d). Following the premises of Result 2,
this representation is finite-dimensional. Hence,
we can decompose Γ into real irreducible repre-
sentations
Γ =
M
r
Γ
r
. (38)
Using the same partition into real linear sub-
spaces, we also decompose the map
Ω =
M
r
r
. (39)
12
Equation (6) independently holds for each sum-
mand Γ
r
U
r
ψ
=
r
Uψ
. In particular Γ
r
U
r
ψ
=
r
ψ
for all U G
ψ
, which implies that each Γ
r
has a
G
ψ
-invariant subspace.
Now, let us concentrate on the d 3 case. To
figure out the structure of the stabilizer subgroup,
we consider the state ψ
0
= (1, 0, . . . , 0), and note
that the group G
ψ
0
is the set of unitaries of the
form (17). Hence, when d 3, all stabilizers are
isomorphic to U(1) × SU(d 1).
Lemma 1. The finite-dimensional irreducible
representations of SU(d) that have U(1)×SU(d
1)-invariant vectors are the D
d
j
introduced above.
Additionally, the vector is always unique (up to
a constant).
This lemma, proven in appendix A.12, tells us
that most of the irreducible representations of
SU(d) do not have a vector which is invariant
under U(1) × SU(d 1), and hence violate (6).
As mentioned above, the SU(d) representations
D
d
j
are also PU(d) representations, which is what
is required (see Result 1). A technical point
(fully addressed in appendix A.12.6) arises. The
representations we are looking for act on real
vector spaces, however the representation theory
tools used to find the D
d
j
deal with representa-
tions acting on complex vector spaces. A repre-
sentation which is irreducible on a real space is
not necessarily irreducible on a complex space.
We have found all complex irreducible repre-
sentations (which are also real irreducible) with
the correct properties, however it may be the
case that there are real irreducible representa-
tions which are complex reducible which were not
found with the methods used. It is shown in ap-
pendix A.12.6 that the D
d
j
are the only real irre-
ducible representations with the required proper-
ties.
A very convenient fact is that there is a single
one-dimensional invariant subspace. Hence, the
maps
r
are completely determined up to a fac-
tor. However, this factor does not play any role,
since it can be modified via an equivalence trans-
formation L of the type (10)-(12). If there were
multiple trivial invariant subspace, one could con-
struct different alternatives to the measurement
postulates having the same representation Γ.
In the case d = 2, the stabilizer subgroup G
ψ
is isomorphic to U(1). This subgroup is gener-
ated by a single Lie algebra element (e.g. Z).
The irreducible representations D in which, the
subgroup G
ψ
has an invariant vector, are those
in which D(Z) has at least one zero eigenvalue.
These are the integer spin representations. Also,
the multiplicity of this eigenvalue is always one,
implying that, as in the d 3 case, the map is
completely determined up to equivalence trans-
formations of the type (10)-(12).
A.3 Faithfulness (Result 3)
Let us prove that when d 3 the map specified
in Result 2 is injective. We start by assuming the
opposite: there are two different pure states ψ 6=
ψ
0
which are mapped to the same vector
ψ
=
ψ
0
. This vector must be invariant under the
action of the two stabilizer subgroups: Γ
U
ψ
=
ψ
for all U G
ψ
and all U G
ψ
0
. Now, note
that if Γ
U
ψ
= Γ
U
0
ψ
= Ω
ψ
then also Γ
UU
0
ψ
=
ψ
. Hence, the stabilizer group of the vector
ψ
contains the group {UU
0
|∀U G
ψ
, U
0
G
ψ
0
}.
Lemma 2. When d 3 the group SU(d) can
be generated by two stabilizer subgroups SU(d
1) × U(1) of two distinct rays on PC
d
.
This lemma, proven in Appendix A.13, shows
that the group generated by any two stabilizer
subgroups is the full group SU(d). Therefore, the
vector
ψ
is invariant under any transformation
Γ
U
. This implies that all states ψ
0
are mapped
to the same vector
ψ
0
=
ψ
and have exactly
the same outcome probabilities. Equivalently, all
functions F F
d
are constant.
Now, let us analyze the d = 2 case. In what
follows we prove that, the map
r
associated to
an irreducible representation Γ
r
= D
2
j
is injec-
tive when j is odd. Hence, the global map is
injective if it contains at least one summand
r
that is injective. An irreducible representation of
SU(2) can be expressed as a symmetric power of
the fundamental representation [22, p.150]:
D
2
j
= Sym
(2j)
D
2
1
2
, (40)
where D
2
1
2
is the fundamental representation of
SU(2) acting on C
2
, with basis {ψ
0
, ψ
1
}. The
action of the Lie algebra element Z on this basis
is:
Zψ
0
=
0
, (41)
Zψ
1
=
1
. (42)
13
Given D
2
j
, we take as reference state
j
ψ
0
, the 0
eigenstate of D
2
j
(Z) (since ψ
0
is invariant under
all transformations generated by Z).
e
D
2
j
(Z)t
j
ψ
0
= Ω
j
ψ
0
. (43)
The 0 eigenstate is given by
j
ψ
0
= ψ
j
0
ψ
j
1
(where the product is the symmetric product) [22,
p.150]. All states can be obtained by applying a
unitary to the reference state:
j
ψ
= D
2
j
(U)Ω
j
ψ
0
.
We call U
Z
the set of transformations generated
by Z. All states ψ 6= ψ
0
are of the form ψ =
Uψ
0
, U / U
Z
.
We show that for j odd there are no states
Uψ
0
, U / U
Z
such that D
2
j
(U)Ω
j
ψ
0
=
j
ψ
0
hence
ψ
0
and ψ are mapped to distinct states. For j
even we show that there is a U / U
Z
such that
j
ψ
= D
2
j
(U)Ω
j
ψ
0
=
j
ψ
0
and hence the represen-
tation is not faithful. A generic U SU(2) acting
on C
2
has the following action:
ψ
0
αψ
0
+ βψ
1
,
ψ
1
α
ψ
1
β
ψ
0
, (44)
where |α|
2
+ |β|
2
= 1. The action D
2
j
(U) on
j
ψ
0
is the same as that of U
2j
since
j
ψ
0
belongs to
the symmetric subspace.
ψ
j
0
(αψ
0
+ βψ
1
)
j
,
ψ
j
1
(α
ψ
1
β
ψ
0
)
j
. (45)
We now determine which unitaries U preserve the
state:
ψ
j
0
ψ
j
1
= (αψ
0
+βψ
1
)
j
(α
ψ
1
β
ψ
0
)
j
. (46)
This only holds when either α or β is 0. When
β = 0 this corresponds to a U U
Z
. When α = 0
we have:
ψ
j
0
ψ
j
1
= (1)
j
ψ
j
0
ψ
j
1
, (47)
since by unitarity requirement |β| = 1. For even j
there is a unitary U / U
Z
, such that D
2
j
(U)Ω
j
ψ
0
=
j
ψ
0
hence the map is not injective. It maps
orthogonal rays to the same state. For odd j any
unitary U / U
Z
maps
j
ψ
0
to a different state
and so the map is injective. Moreover we see
that Uψ
0
= ψ
1
and hence orthogonal rays ψ
0
and ψ
1
are mapped to antipodal states
ψ
0
and
ψ
1
=
ψ
0
for odd j and to the same state
ψ
0
= Ω
ψ
1
for even j.
A.4 Distinguishable states in T
I
2
(Result 5)
The proof is by contradiction: we assume the ex-
istence of 3 distinguishable states, and show that
any distinguishing measurement has an outcome
with a negative probability for certain states.
Hence the maximum number of perfectly distin-
guishable states is two. Since effects are affine
functions we can write an effect E as a pair (e, c).
E(ω) = e · ω + c . (48)
A measurement is a set of effects {E
i
} such that
P
E
i
(ω) = 1 for all states ω; this entails
P
e
i
= 0
and
P
c
i
= 1. If three states ω
1
, ω
2
and ω
3
are
distinguishable then there exists a measurement
which distinguishes them of the following form:
{E
1
= (c
1
, e
1
), E
2
= (c
2
, e
2
), E
3
= (c
3
, e
3
)}, with
E
i
(ω
j
) = δ
ij
. This entails
e
i
· ω
i
= 1 c
i
. (49)
From Result 3 there exists an antipodal state ω
i
for every ω
i
in theories in T
I
2
. We compute the
outcome probabilities for these antipodal states:
E
i
(ω
i
) = e
i
· (ω
i
) + c
i
= 1 + 2c
i
(50)
The sum of these three measurement outcomes
probabilities is:
X
i
E
i
(ω
i
) = 3 + 2(c
1
+ c
2
+ c
3
) = 1 (51)
Therefore at least one of the outcome probabili-
ties is negative and therefore not legitimate. By
contradiction this proves there is no measurement
which perfectly distinguishes three states in these
theories.
A.5 Distinguishable non-antipodal states in T
I
2
(Result 6)
In this section we show that for each non-
quantum theory in T
I
2
there exists a measurement
of the form M = {(e, c), (e, 1 c)} with e
ω
0
· D
2
j
(X)
which perfectly distinguishes pairs
of non antipodal states lying on the D(U
X
(t))ω
0
orbit.
Theories in T
I
2
are generated by representa-
tions D
2
j
with j odd. The dimension of such repre-
sentations is n = 2j + 1. We write U
K
(t) = e
Kt/2
for an arbitrary element of the Lie Algebra K.
We will use the following elements of su(2):
Z =
i 0
0 i
!
, (52)
14
X =
0 i
i 0
!
, (53)
H =
1
2
i i
i i
!
, (54)
We note that D
2
j
(e
Kt/2
) = e
D
2
j
(K)t
. The reference
state is chosen to be invariant under all U
Z
(t):
D
2
j
(U
Z
(t))ω
2
j
= ω
2
j
, (55)
which implies D
2
j
(Z)ω
2
j
= 0. The 0-weight sub-
space is one dimensional, the other possible (nor-
malised) state is ω
2
j
. This will be the antipodal
state. The manifold of pure states can be gener-
ated in the following manner:
ω(s, t) = D
2
j
(U
Z
(s))D
2
j
(U
X
(t))ω
2
j
. (56)
This is just a 2-sphere (embedded in a space of
dimension n) parametrised by polar angle s and
azimuthal angle t. The generators of the irre-
ducible (unitary) representations of SU(2) can be
written as [31, p.387]:
D
2
j
(X) = i
0
2j
2
0 0 0 0 0
2j
2
0
2(2j1)
2
0 0 0 0
0
2(2j1)
2
0
.
.
.
0 0 0
0 0
.
.
.
0
.
.
.
0 0
0 0 0
.
.
.
0
2(2j1)
2
0
0 0 0 0
2(2j1)
2
0
2j
2
0 0 0 0 0
2j
2
0
(57)
D
2
j
(Z) = i
j 0 0 0 0 0 0
0 j 1 0 0 0 0 0
0 0
.
.
.
0 0 0 0
0 0 0 0 0 0 0
0 0 0 0
.
.
.
0 0
0 0 0 0 0 j + 1 0
0 0 0 0 0 0 j
. (58)
We define k = j + 1 where D(K)
(k,k)
is the
central element of the matrices D(K) for some
Lie algebra element K. It is clear that in this
basis the reference state (0 eigenstate of D
2
j
(Z))
is:
(ω
2
j
)
i
= δ
ik
. (59)
In the following we drop the explicit reference to
the representation, and use D(U ) for D
2
j
(U). We
now use the notation ω
K
k+i
for the i eigenstate of
the representation matrix D(K) of the Lie Alge-
bra element K. Here i runs from j to j. In the
case of the diagonal operator D(Z) we observe
that an eigenvector with eigenvalue i has a sin-
gle entry at position k + i. The zero eigenstate
(ω
Z
k
)
m
= δ
km
which is just a vector with a sin-
gle entry at position k (the central entry). The
0 eigenstate of the operator D(X) for example
is ω
X
k
. Hence we use ω
Z
k
for the reference state
ω
2
j
(since it is the 0 eigenstate of the Z operator).
The 0 eigenstate of D(X) corresponds to the state
ω
X
k
which is equal to D(U
Y
(
π
2
))ω
Z
k
(and is in-
variant under D(U
X
(t))). These 0 eigenstates of
Lie Algebra matrices D(K) correspond to states
since they have the correct invariance properties,
that is, invariant under e
iD(K)t
.
As shown in the proof of Result 3 these state
spaces have antipodal states for all states. Con-
sider an effect which has as maximum ω
m
will
have as minimum ω
m
. This effect may be noisy
(i.e. not have values 0 and 1 for the minimum
and maximum) but will be proportional to an ef-
fect which is tight (gives values 0 and 1 on the
minimum and maximum). Hence any tight ef-
fect which has as maximum ω
m
will distinguish
15
it from ω
m
. In the following we show that
there exist effects which have 2 global maxima
and minima when applied to the manifold of pure
states. These effects are proportional to tight ef-
fects which distinguish ω
0
and ω
0
and also dis-
tinguish ω
0
and ω
1
(for ω
1
6= ω
0
).
Since effects (e, c) are of the form e ·ω(s, t) + c,
we need only consider e · ω(s, t) to establish the
number of global extrema and their locations. We
now prove the effect (e, c) with e = (ω
Z
k
)
D(X)
(up to normalisation) perfectly distinguishes non-
antipodal states in all representations. Here we
use the since the specific representation of the
states have complex entries (even though the vec-
tor space spanned by the states is a real vector
space). We first show that e gives real values
when applied to the state space (whose states
span a real vector space). There is a basis for the
representation where all entries are real which is
related to this representation by AD(U)A
1
. It
acts on states as and effects as eA
1
. Hence
(ω
Z
j
)
A
1
·AD(U)A
1
is real for all ω (since it
is an inner product of vectors with real entries).
This is equal to e · ω for all omega. Hence e ap-
plied to the state space is real valued.
We study the maxima and minima of f(s, t) =
e ·ω(s, t) which corresponds to applying e to the
entire manifold of pure states:
f(s, t) = e · D(U
Z
(s))D(U
X
(t))ω
Z
k
. (60)
We consider the first part (a row vector)
e
s
= e e
D(Z)s
= (ω
z
k
)
D(X)
e
D(Z)s
, (61)
for which entries are 0 apart from entries k ± 1:
(e
s
)
k±1
= i
p
j(j + 1)
2
e
is
. (62)
We can write this as:
e
s
=
(0, ..., i
p
j(j + 1)
2
e
is
, 0, i
p
j(j + 1)
2
e
is
, 0, ..., 0) .
(63)
We now determine the second part of the func-
tion f(s, t): D(U
X
(t))ω
Z
k
. Since ω
Z
k
has a single
entry equal to 1 at position k this just selects
the k
th
column of D(U
X
(t)), we call these states
ω
X
(t); they form an orbit. The inner product
e
s
·ω
X
(t) will only have two terms corresponding
to the k 1 and k + 1 entries of each vector.
e
s
· ω
t
=
i
p
j(j + 1)
2
(e
is
ω
X
(t)
k1
+ e
is
ω
X
(t)
k+1
) .
(64)
This is real valued for all values of s, hence it
must be the case that ω
X
(t)
k1
= ω
X
(t)
k+1
and
both are imaginary (or both real and ω
X
(t)
k1
=
ω
X
(t)
k+1
). However since the ω
Y
k
state be-
longs to the orbit ω
X
(t) and has the property
(ω
Y
k
)
k1
= (ω
Y
k
)
k+1
it must be the former.
e
s
· ω
t
=
q
j(j + 1)ω
X
(t)
k+1
cos(s) . (65)
The maxima and minima of this function occur
for s = 0 (or π but this corresponds to the same
orbit). Hence we need to find the maxima and
minima of the function:
f(s, t = 0) = g(t) = e · D(U
X
(t))ω
Z
k
= (ω
Z
k
)
D(X)
· D(U
X
(t))ω
Z
k
= (ω
X
k
)
D(Z)
· D(U
Z
(t))ω
X
k
.
(66)
Where we have used H the Hadamard trans-
formation defined above which changes the ba-
sis from Z to X: HZH
= X. Moreover
HU
Z
(t)H
= U
X
(t). Therefore:
D(U
X
(t))ω
Z
k
= D(H)D(U
Z
(t))D(H)
ω
Z
k
, (67)
we note that D(H)
ω
Z
k
is a 0 eigenstate of D(X)
and is therefore equal to ω
X
k
.
We now evaluate the second part of g(t) :
D(U
Z
(t))ω
X
k
where ω
X
k
is a normalised zero eigen-
vector of D(X) and has the following form:
ω
X
k
=
a
j
0
a
(j2)
0
.
.
.
0
a
(j2)
0
a
j
, (68)
where |a
n
| > |a
m
| for n > m.
16
D(e
Zt
)ω
X
k
=
a
j
e
ijt
0
a
(j2)
e
i(j2)t
0
.
.
.
0
a
(j2)
e
i(j+2)t
0
a
j
e
ijt
. (69)
We can also determine (ω
X
k
)
D(Z)
:
(ω
X
k
)
D(Z)
=
(ja
j
, 0, (j 2)a
j2
, 0, ...., 0, (j 2)a
j2
, 0, ja
j
) .
(70)
We can now compute g(t):
g(t) = (ω
X
k
)
D(Z)
· D(U
Z
(t))ω
X
k
= 2
j
X
l=1, l o dd
l |a
l
|
2
sin(kt) .
(71)
|a
n
| > |a
m
| for n > m hence |a
n
|
2
> |a
m
|
2
for
n > m. We restrict ourselves to the interval
[0, 2π). We have been considering non-quantum
theories with j > 1, however we briefly describe
what this effect would correspond to for the quan-
tum case. When j = 1 (i.e k = 2) there is a sin-
gle maximum occurring for t
m
= π/2. Hence the
global minimum occurs for t = 3π/2. The two
states distinguished by this effect are antipodal
and correspond to the D(X) eigenstates. Indeed
we observe that this effect is just given by the
tangent to ω
Z
2
(image of the ray |0i) in the D(X)
direction which is proportional to ω
X
2
. We know
that this effect gives the outcome probability of
being in the ω
X
2
state (image of the |+i ray) which
is maximal for ω
X
2
and minimal for ω
X
2
(image
of the |−i ray).
We now show that g(t) has two global maxima
and two global minima for j > 1. g(t) = g(π t),
therefore given a maximum/minimum we can find
another (unless t
m
= π/2). Moreover since g(t) =
g(t + π) given a maximum/minimum we can
find a minimum/maximum. To prove our claim
we need to find a global maximum/minimum in
the interval [0, π). If this extremum occurs for a
value which is not π/2 then we can find the other
maximum/minimum in the same interval and the
two minima/maxima in [π, 2π). We now show
that for j > 1 the global maximum/minimum
does not occur for t
m
= π/2.
We first note that g(0) = 0, hence a global
maximum is positive and a global minimum is
negative. We first compute g(π/2) and show that:
g(π/2) > 0 g(π/(2j)) > g(π/2). This implies
that g(π/2) cannot be a global maximum, since
if it is positive there is a τ = π/(2j) such that
g(τ) > g(π/2). Similarly we show that g(π/2) <
0 g(π/(2j)) < g(π/2) which entails that when
g(π/2) is negative it cannot be a global minimum.
Hence the global extrema of g(t) do not occur for
t = π/2 when j > 1.
g(π/2) =
j
X
l=1,l odd
(1)
l1
2
l |a
l
|
2
. (72)
We note that:
l|a
l
|
2
> (l 2)|a
l2
|
2
, l odd, l > 1 . (73)
Since each term in g(π/2) alternates sign and the
absolute value of each term increases for each
l the sign of g(π/2) is determined by that of
its highest term (1)
j1
2
l |a
j
|
2
. This implies
g(π/2) > 0 for j = 5, 9, 13, .... We consider
these cases. The last term (which is the largest)
(1)
j1
2
j |a
j
|
2
is always positive. We observe that
the sum of the remaining terms is negative. This
implies:
g(π/2) j|a
j
|
2
< 0 , (74)
We now determine
g(
π
2j
) =
j
X
l=1,l odd
l |a
l
|
2
sin(
kπ
2j
) . (75)
The arguments in each of the sin functions in
g(
π
2j
) are always between 0 and
π
2
, hence each
term is positive. Therefore:
g(
π
2j
) j|a
l
|
2
> 0 . (76)
Combining equations 74 and 76 we obtain
g(
π
2j
) > g(
π
2
) when g(
π
2
) > 0 . This shows that
the maximum of the function does not occur for
t = π/2. The function therefore has at least two
maxima in the interval [0, π].
For j = 3, 7, 11, ... the same argument can be
applied to show that the function has two minima
within the interval [0, π] and hence two maxima
within [π, 2π].
17
This entails that there exists a global maximum
which is separated from a global minimum by a
value which is not π. Hence there exist states
which are distinguishable but are not antipodal.
A.6 Bit symmetry (Result 7)
In theories belonging to T
I
2
images of the orthog-
onal rays ψ
0
and ψ
1
are antipodal states
ψ
0
and
ψ
1
with
ψ
0
=
ψ
1
. These states can
be perfectly distinguished using the measurement
(e,
1
2
), (e
T
,
1
2
) where e = Ω
T
ψ
0
/2. From Result 6
there exists a state
ψ
2
which is distinguishable
from
ψ
0
and not antipodal to it. Due to the
faithfulness of for j odd we have ψ
2
6= ψ
1
. Since
ψ
1
is the unique ray orthogonal to ψ
0
in PC
2
ψ
2
is
not orthogonal to ψ
0
. There is no unitary which
maps the pair of orthogonal rays (ψ
0
, ψ
1
) to the
pair of non-orthogonal rays (ψ
0
, ψ
2
). Hence there
exist pairs of distinguishable states which are not
related by a reversible transformation belonging
to the dynamical group.
A.7 Phase groups in T
I
2
(Result 8)
We show that the maximal measurement (from
the proof of Result 6) which distinguishes two
non-antipodal states
ψ
and
φ
in theories T
I
2
cannot have a phase group. Without loss of gen-
erality we choose ψ to be invariant under trans-
formations U
Z
(t). The phase group of a measure-
ment is a subgroup of the transformation group
which leaves all outcome probabilities of the mea-
surement unchanged. This measurement has ex-
actly two maxima and two minima on the state
space (as shown in the proof of Result 6). We call
E one of the effects which compose this measure-
ment:
E(Ω
ψ
) = E(
φ
) = 1 ,
E(Ω
φ
) = E(
ψ
) = 0 .
The states
ψ
and
φ
are the images of non-
orthogonal rays on C
2
. We now show that there
is no subgroup of the transformation group which
leaves the outcome probabilities of the 4 states
ψ
,
φ
,
ψ
and
φ
invariant (which are the
images of ψ, φ , ψ
and φ
). This is sufficient
that there is no phase group for this measure-
ment. Such a phase group must either preserve
the states, or map them to states which give the
same outcome probabilities. For example it must
either preserve ψ or map it to φ
. First consider
transformations which map
φ
to itself. These
correspond to the U(1) subgroup U
Z
(t) (or a dis-
crete subgroup of this subgroup). These transfor-
mations do not map
φ
to itself (nor to
ψ
) and
hence do not preserve the outcome probabilities
of the effect. The only other possible transforma-
tions which preserve the outcome probabilities for
these 4 states are of the form:
Uψ = φ
,
Uφ = ψ
.
In order for a phase group to exist U
must also
preserve the outcome probabilities:
U
ψ = φ
,
U
φ = ψ
,
entailing that U
= U. The only U PU(2)
which has this property is the identity. Therefore
the only phase group of this maximal measure-
ment is the trivial one.
A.8 No-simultaneous encoding (Result 9)
All theories in T
I
2
have pairs of non-antipodal
states ω
0
and ω
1
which are perfectly distinguish-
able. The measurement which distinguishes them
perfectly, also distinguishes ω
0
and ω
1
.
E
i
(ω
j
) = δ
ij
, (77)
E
i
(ω
j
) = δ
ij
+ 1 . (78)
A first bit a can be perfectly encoded as follows:
a = 0 ω
0
or ω
1
,
a = 1 ω
1
or ω
0
.
The second bit a
0
can be encoded as:
a
0
= 0 ω
0
or ω
1
,
a
0
= 1 ω
1
or ω
0
.
For example if Alice needed to encode the bits
a = 0 , a
0
= 0 she would choose the state ω
0
. Ac-
cording to the scenario Alice encodes her bits a
and a
0
in a single system and sends it to Bob. He
then tries to guess one of the bits. If he chooses
to guess the value of bit a he can do so with
certainty (using the measurement which perfectly
distinguishes ω
0
and ω
1
from ω
0
and ω
1
). If
Bob chooses to guess the value of bit a
0
he can use
another effect to obtain partial information about
18
whether the state is ω
0
or ω
1
or whether the state
is ω
1
or ω
0
. This could be any effect which
partially distinguishes
ω
0
+ω
1
2
from
ω
0
+ω
1
2
. Since
neither of these is the maximally mixed state such
an effect exists.
A.9 Pure-state dual theories (Result 10)
A two outcome measurement is a pair of effects
E
1
= (e, c) and E
2
= (e, 1 c). In theories
with irreducible Γ (for d = 2) for each pure state
ψ
S (where S is the state space) the state
ψ
also exists. For a pure state dual theory we
impose that effects are proportional to states and
tight. Hence the effects are given by:
{e =
T
ψ
2
|ψ PC
2
} . (79)
The set of effects is such that for every e the lin-
ear functional e also exists. Two outcome mea-
surements are of the form M = {(e,
1
2
), (e,
1
2
)}.
The linear functional e has a single maximum
for state
T
ψ
and a single minimum for
T
ψ
.
Hence the only states which are distinguishable
are antipodal. This entails that the state spaces
are bit symmetric. The fact that for all effects
there is a single maximum/minimum entails that
no-simultaneous encoding holds. Given a func-
tional e there is a U(1) subgroup which leaves
ψ
and hence e invariant. Hence every measurement
M = {(e,
1
2
), (e,
1
2
)} has a U(1) phase group.
A.10 Bit symmetry in T
R
2
(Result 11)
Consider a theory in T
R
2
; its state space S is a
direct sum of state spaces of theories in T
I
2
:
S =
M
j∈J
S
j
. (80)
Moreover S has at least one faithful block in the
decomposition. In the case where this block is
non-quantum we denote it by k (k > 1, k odd).
We assume all effects are allowed. Now consider
effects with support solely on subspace k. The
state space restricted to this subspace is just a
state space corresponding to a theory with j = k
( and hence in T
I
2
). There exists a measurement
(with support just in this subspace) which can
distinguish pairs of non-antipodal states (by Re-
sult 6). Moreover there also exists a measure-
ment (with support only in this subspace) which
distinguishes pairs of antipodal states. Hence the
entire state space S has pairs of distinguishable
states which are images of orthogonal rays and
pairs which are not images of orthogonal rays.
If the generators contain a single faithful block
j = 1 (with all others unfaithful) then a mea-
surement on that block can distinguish a pair
of states (which are images of orthogonal rays).
The unfaithful blocks have pairs of distinguish-
able states which are not images of orthogonal
rays. This follows from the fact that orthogonal
rays are mapped to the same state in unfaithful
representations and that the unfaithful block are
also state spaces when considered alone. Since
all state spaces have at least two distinguish-
able states it follows that pairs of states which
are not images of orthogonal rays can be distin-
guished using effects with support in the unfaith-
ful blocks.
Therefore all theories in T
R
2
have pairs of dis-
tinguishable states which are images of orthogo-
nal rays and pairs which are not; these theories
violate bit symmetry.
A.11 Proof of Result 12
In order to show that all unrestricted state spaces
are not bit symmetric we first need to establish
how a PC
d1
state space is embedded in a PC
d
state space. In the following we assume that all
state spaces are unrestricted.
Lemma 3. Given a theory of PC
d
correspond-
ing to maps and Γ, where the highest weight
representation in Γ is D
d
j
, the image of a PC
d1
subspace of PC
d
under is equivalent to a state
space of a PU(d 1) theory with representation
Γ
0
, whose highest weight component is D
d1
j
.
This lemma is proven in appendix A.14. It fol-
lows from the lemma that any non-quantum PC
3
(i.e. which has a block D
3
j
with j > 1) state space
when restricted to a PC
2
subspace is equivalent to
a non-quantum state space with a representation
which has a block D
2
j
(if j is even it must also be
the case that the representation has a block with j
odd since the representation of states is faithful).
Hence the restricted state space belong to T
R
2
. By
Result 11 this PC
2
state space is not bit symmet-
ric, that is to say it has pairs of distinguishable
states which have different (Hilbert space) inner
products. Moreover the PC
3
state space is not
bit symmetric since there are no transformations
19
in PU(3) mapping pairs of states with different
Hilbert space inner products.
Any non-quantum PC
d
state space (i.e. which
has a block D
d
j
with j > 1) has a PC
d1
subspace
which is equivalent to a state space associated
to a reducible representation which has a block
D
d1
j
. If the PC
d1
subspace is not bit symmet-
ric (i.e. the Hilbert space inner product between
two pairs of distinguishable state is not the same)
then the PC
d
state space is not bit symmetric
either (since even considering the whole PU(d)
transformation group its elements will still only
map between pairs of states with the same inner
product).
Hence by induction any non-quantum PC
d
(d 2) unrestricted state space is not bit sym-
metric.
A.12 Proof of Lemma 1
A.12.1 Partitions, U(d) and SU(d)
A partition λ is a set of d integers such that λ
1
λ
2
... λ
d1
λ
d
0. We define the size
of the partition |λ| =
P
n
i=1
λ
i
= k and say that
λ is a partition of k into d parts. We use square
brackets to denote λ = [λ
1
, ..., λ
d
].
Each partition λ is associated to an irreducible
representation π
λ
d
of U(d) [32]. Moreover any ir-
reducible representation π
λ
d
of U(d) is also an irre-
ducible representation of SU(d) [32] . By this we
mean that if we consider a subgroup π
λ
d
(U), U
SU(d) this representation is not reducible. Two
irreducible representations λ
(1)
and λ
(2)
of U(d),
when restricted to SU(d), are equivalent repre-
sentations of SU(d) if and only if λ
(1)
i
λ
(2)
i
is
constant for all i [32].
A representation of SU(d) corresponding to a
partition λ and a representation of SU(d) corre-
sponding to λ + c are therefore equivalent. Hence
an irreducible representation of SU(d) is charac-
terised by the differences between elements of a
partition λ. We can map λ to (λ
1
λ
2
, λ
2
λ
3
, ..., λ
d1
λ
d
) which is a vector of positive
integers of size d 1 which is the same for λ
and λ + c for all c . This vector (λ
1
λ
2
, λ
2
λ
3
, ..., λ
d1
λ
d
) is the previously introduced
Dynkin index (which is in one-to-one correspon-
dence with irreducible representations of SU(d)).
We use square brackets to differentiate a partition
from a Dynkin index. In the following it will be
useful to describe representations of SU(d) using
their partition (even if the map from partitions to
representations of SU(d) is many to one). Since
partitions differing by a constant give the same
representation we fix λ
d
= 0. This requirement
fixes a representation of SU(d) for each λ.
A.12.2 U(1) × SU(d 1) invariant subspaces
For a representation λ of SU(d) acting on V to
have an U(1)×SU(d1)-invariant vector v means
the following. If we take the representation of a
U(1) × SU(d 1) subgroup (which is reducible)
and consider its action on v, it leaves v invariant.
The reducible representation of U(1) ×SU(d 1)
is composed of irreducible blocks. If we decom-
pose v according to these blocks, we see that
the components of v in these subspaces (for non-
trivial representations of U(1) × SU(d 1)) will
not be left invariant by all of the transformations
U(1)×SU(d1) (since each block is irreducible, so
cannot have invariant subspaces within it). How-
ever if one of the blocks corresponds to a trivial
representation of U(1) × SU(d 1) then it will
leave that component of v invariant. Thus we
can choose v to have support just in those sub-
spaces which correspond to trivial representations
in the decomposition of λ when restricting to a
U(1) × SU(d 1) subgroup.
We now find the representations λ such that
there exists at least one trivial representation of
U(1)×SU(d1) when λ is restricted to this U(1)×
SU(d 1) subgroup.
A.12.3 Branching rule for U(d) U(d 1)
Restricting representations of groups to sub-
groups leads to a reducible representation of the
subgroup given by a branching rule. We intro-
duce the necessary tools.
We consider a partition λ = [λ
1
, ..., λ
n
]. A
partition µ = [µ
1
, ..., µ
n1
] is said to interlace λ
when:
λ
1
µ
1
... λ
n1
µ
n1
λ
n
(81)
The branching rule from U(d) to U(d 1) is as
follows. Let U(d) have irreducible representation
π
λ
d
acting on a space V
λ
. Then there is a unique
decomposition of V
λ
into subspaces under the ac-
tion of U(d 1) [33, p.19]:
V
λ
=
M
µ
V
µ
, (82)
20
where the sum is over every µ which interlaces λ
and V
µ
is a carrier space for an irreducible repre-
sentation of U(d 1) labelled by µ.
A.12.4 Branching rule for SU(d) SU(d 1) ×
U(1)
The restriction of π
λ
d
(a representation of U(d))
to an SU(d) subgroup is an irreducible represen-
tation of SU(d) with partition λ (which for SU(d)
is defined up to a constant). This representation
acts irreducibly on V
λ
. The subgroup U(d 1) is
given by:
1 0 ··· 0
0
.
.
. U
(d1)×(d1)
0
, U
(d1)
U
(d1)
= . (83)
It’s action on V
λ
decomposes as a direct sum of
V
µ
. We wish to consider the action of SU(d 1)
on V
λ
. This subgroup is given by:
1 0 ··· 0
0
.
.
. U
(d1)×(d1)
0
, U
(d1)×(d1)
SU(d1) .
(84)
U(d 1) acts on each V
µ
irreducibly and restrict-
ing an irreducible representation of U(d 1) to
SU(d1) gives an irreducible representation with
the same partition. Hence the SU(d 1) acts ir-
reducibly on each V
µ
in the decomposition. The
branching rule SU(d) SU(d 1) is the same as
U(d) U(d 1) [32].
We now establish the branching U(d)
SU(d 1) × U(1) subgroup. The U(1) part cor-
responds to all matrices C of the following form:
C =
e
it
e
i
d1
t
.
.
.
e
i
d1
t
. (85)
We note that this U(1) subgroup commutes with
the SU(d1) subgroup hence its action will leave
the subspaces V
µ
invariant. Therefore the space
V
λ
decomposes into V
µ
invariant subspaces under
the action of SU(d 1) ×U(1). We now establish
the action of the U(1) subgroup on these sub-
spaces. We use a similar technique as that found
in proof of theorem 8.1.2 of [34, p.364]. We con-
sider a matrix A U(d):
A =
e
it
e
it
.
.
.
e
it
, (86)
And a matrix B U(d 1):
B =
1
e
i
d
d1
t
.
.
.
e
i
d
d1
t
. (87)
We note that AB = C for the above defined U(1)
subgroup. We determine the action of A and B
on V
λ
which will allow us to know that of C. The
action of A on the whole carrier space V
λ
is mul-
tiplication by a scalar e
it|λ|
. Similarly the action
of B (which belongs to the U(d1) subgroup and
commutes with all elements of U(d 1)) on each
subspace V
µ
is multiplication by e
i
d
d1
t|µ|
[34,
theorem 8.1.2. (proof) , p.364]. Hence we can
compute the action of the U(1) subgroup on each
subspace which is multiplication by :
e
it|λ|
e
i
d
d1
t|µ|
= e
it(|λ|−
d
d1
|µ|)
, (88)
Hence the action of this U(1) subgroup on the
carrier space V
λ
acts by scalar multiplication on
each V
µ
(where the scalar can be the same for dif-
ferent µ). We can now summarise: given a rep-
resentation π
λ
d
of SU(d) acting on V
λ
there exists
a decomposition (into invariant subspaces) under
the action of SU(d 1) × U(1) given by:
V
λ
=
M
µ
V
µ
, (89)
where the sum is over every µ which intertwines
λ. The µ determine irreducible representations
of SU(d 1) acting on each subspace. The U(1)
parts acts like e
it(|λ|−
d
d1
|µ|)
on each subspace.
A.12.5 Representation of SU(d) with trivial rep-
resentations under the action of SU(d 1) × U(1)
We can now address the problem of finding which
representations of SU(d) are such that their re-
striction to SU(d 1) × U(1) contains a trivial
representation of SU(d 1) × U(1).
21
A trivial representation of SU(d 1) has
Dynkin coefficients (0, ...0) and hence partition
µ = [µ
1
, ...µ
d1
] where µ
1
= µ
2
= ... = µ
d1
.
Hence representations of SU(d) which contain
trivial representations of SU(d1) in this decom-
position will have partitions λ where λ
2
= λ
3
=
... = λ
n1
= µ
1
, following the requirement that
µ intertwine λ.
We now consider the requirement that the U(1)
action is trivial. Since the action of U(1) on V
µ
is given by e
it(|λ|−
d
d1
|µ|)
, it is trivial when |λ|
d
d1
|µ| = 0. We note that some authors multiply
the U(1) charge by a constant so that it is an
integer value. Since we are considering only 0
U(1) charge this will not concern us.
We can now add this requirement on λ to the
preceding one: λ
2
= λ
3
= ... = λ
d1
= µ
1
. As
stated above we are considering λ
d
= 0 in order
to identify one representation of SU(d 1) with
each partition λ. We have
|µ| = (d 1)µ
1
, (90)
and
|λ| = λ
1
+ (d 2)µ
1
, (91)
We substitute this into the requirement |λ|
d
d1
|µ| = 0:
λ
1
+ (d 2)µ
1
1
= 0
λ
1
= 2µ
1
(92)
Hence the representation π
λ
n
of SU(d) with par-
tition [2µ
1
, µ
1
, ..., µ
1
, 0] which corresponds to
Dynkin label (µ
1
, 0, ..., 0, µ
1
) meets the require-
ments. This shows that any representation of
SU(d) with Dynkin label (j, 0, ..., 0, j) (for any
positive integer j) has a trivial subspace under
the action of SU(d) × U(1). We now show that
these representations have a single trivial repre-
sentation in the decomposition. These are repre-
sentations with partition λ = [2j, j, ..., j, 0]. Un-
der the branching rule into SU(d1)×U(1) trivial
representations correspond to partitions µ have
components µ
i
which are all equal (in order for
the SU(d 1) component to be trivial) and these
must be equal to j. There is a single such µ of
this form. This shows that there is a unique triv-
ial representation in the decomposition.
The representations of SU(d) of the form
(j, 0, ..., 0, j) correspond to the representations
D
d
j
. These are the only representations of SU(d)
(and PU(d)) which leave a vector invariant under
SU(d) × U(1) (moreover this vector is unique up
to normalisation).
A.12.6 A comment on real and complex irre-
ducibility
In this work we are interested in classifying
certain representations of the dynamical group
SU(d). These representations act on state spaces
which span real vector spaces.
All the representation theory results used
above concern representations acting on complex
vectors spaces. A representation which acts on a
complex vector space is real if it can be expressed
in a basis where all the matrix elements are real.
This is the case of the irreducible representa-
tions D
d
j
. These representations are irreducible
when acting on both complex and real vector
spaces. However not all real irreducible represen-
tations correspond to complex irreducible repre-
sentations. There are reducible representations
acting on a complex vector space whose action is
irreducible when acting on a real vector space.
We have found all real irreducible representa-
tions which are also complex irreducible which
meet our criteria. However it may be the case
that there are representations which are real ir-
reducible but complex reducible which meet our
criteria. If this is the case our approach would
not have found them. We now show that there
are no such representations satisfying the invari-
ance properties we require.
A real representation Γ which is irreducible on
a real vector space but reducible on a complex one
can be block diagonalised in the following form
Γ = ρ ¯ρ [22, Exercise 3.39, p. 41]. For example
consider the following representation of SO(2)
=
U(1):
cos(θ) sin(θ)
sin(θ) cos(θ)
!
. (93)
This is irreducible when acting on R
2
. However
its action on C
2
can be diagonalised as follows:
e
0
0 e
!
. (94)
Hence there may be representations which are
real irreducible which have SU(d 1) × U(1) in-
variant vectors but are complex reducible. All
real irreducible representations are either: com-
plex irreducible or complex reducible of the form
ρ + ¯ρ.
22
Let us consider a real irreducible representa-
tion Γ which is complex reducible. There exists
a transformation L such that Γ = L(ρ ¯ρ)L
1
.
We now ask the same question: does there exist
a vector which is invariant under the subgroup
H = SU(d 1) × U(1). Let us call this vector v.
We have:
Γ
|H
v = v . (95)
We observe that if such a vector exists then
the (complex) vector L
1
v is invariant under
L
1
Γ
|H
L = ρ
|H
¯ρ
|H
. Moreover if there exists a
vector w which is left invariant by ρ
|H
¯ρ
|H
then
the vector Lw is left invariant under Γ
|H
. Hence
if we show that there are no vectors left invari-
ant under ρ
|H
¯ρ
|H
then there are no vectors left
invariant under Γ
|H
.
There exists a vector w invariant under ρ
|H
¯ρ
|H
if and only if this representation (obtained by
restricting ρ ¯ρ) has at least one trivial compo-
nent. This is only possible if at least one of the
representations ρ
|H
or ¯ρ
|H
has one or more trivial
components.
Here ρ and ¯ρ are complex irreducible repre-
sentations of SU(d). The only such representa-
tions with a SU(d 1) × U(1) invariant vector
are of the form D
d
j
which is real (and we observe
D
d
j
=
¯
D
d
j
). Hence there are no representations
which are complex reducible but real irreducible
which have these invariant vectors.
A.13 Proof of Lemma 2
We consider two SU(d 1) ×U(1) stabilizer sub-
groups of two distinct rays on PC
d
(d 3). Con-
sider a ray ψ with stabilizer group G
ψ
.The Lie al-
gebra g
ψ
which generates this group has elements
X
ψ
(corresponding to U(1))
X
ψ
=
i 0 ··· 0
0
.
.
.
i
d1
I
d1
0
, (96)
and elements Y
ψ
(corresponding to the SU(d 1)
group)
Y
ψ
=
0 0 ··· 0
0
.
.
. A
0
, A = A
, Tr(A) = 0 .
(97)
As can be seen by the generators ψ is the unique
ray stabilized by this subgroup. Hence distinct
rays ψ and ψ
0
have distinct stabilizer groups
(which are equivalent up to conjugation). This
is not the case for SU(2) for example; the U(1)
stabilizer of |0i also stabilizes |1i.
The two subgroups G
ψ
and G
ψ
0
(which stabi-
lizes ψ
0
) are maximal subgroups of SU(d) [35]. A
maximal subgroup H of G is a proper subgroup
(i.e. H 6= G) such that if H K G then
H = K or K = G. The group H generated by
these two groups G
ψ
and G
ψ
0
is not equal to ei-
ther G
ψ
and G
ψ
0
. Hence it is equal to the full
group SU(d).
A.14 Proof of Lemma 3
A.14.1 Embeddedness of simple theories
In this section we look at theories which do not
obey the constraint given by (9) and study how
PC
d1
subspaces are embedded in PC
d
state
space. We consider a PC
d
state space, which can
be generated by applying all transformations Γ
U
to a reference state
0
= |0ih0|
N
:
Γ
U
0
= U
N
|0ih0|
N
U
†⊗N
= Sym
N
U|0ih0|
N
Sym
N
U
,
(98)
The action of the product Sym
N
U Sym
N
U
can be decomposed (using known rules for Young
tableaux) as:
(N, 0, ..., 0) (0, ..., 0, N)
= (0, ..., 0) (1, 0, ..., 0, 1) ... (N, 0, ..., 0, N) .
(99)
In our previous notation this is just:
Γ
U
=
N
M
i=0
D
d
i
. (100)
The state space is given by Hermitian matri-
ces acting on the symmetric subspace of (C
d
)
n
.
However it is not immediately clear that every
summand in (100) acts on the space of Hermitian
matrices (which is the state space). For example
when we generate the state space by acting with
a reducible representation on a reference state, if
the reference state does not have support in ev-
ery block of the representation then there are cer-
tain components of the representation which are
superfluous (since they do not act on the state
23
space). The dimension of the space of Hermitian
matrices is:
D
ω
=
d + n 1
n
!
2
(101)
Using the dimension formula for the representa-
tions D
d
j
given by (16) and requiring that the rep-
resentation of the transformations acts on a space
of dimension D
ω
we see that every summand in
(100) acts on the state space. Hence we see that
these theories are equivalent to reducible theo-
ries of PC
d
with J = {1, 2, ..., N}. We consider
a PC
d1
subspace of the Hilbert space and see
which subspace of the state space it corresponds
to. It can be generated from the state |1ih1|
N
by applying an SU(d 1) subgroup:
U
N
|1ih1|
N
U
†⊗N
, (102)
With:
U =
1 0 ··· 0
0
.
.
. U
(d1)×(d1)
0
(103)
We notice that this state space is just a PC
d1
state space of the type described above with rep-
resentation:
Γ
0
=
N
M
i=0
D
d1
i
. (104)
A.14.2 Embeddedness of irreducible theories
We consider a PC
d
theory with representation
D
d
j
. We want to know the state space correspond-
ing to a PC
d1
subspace. We consider a basis
{|0i, |1i, ..., |d 1i} and determine the image of
all states of the form α
1
|1i + ... + α
d1
|d 1i.
These are all the states (apart from |0i) which are
invariant under the U(1) group of the SU(d1)×
U(1) subgroup specified in the proof of Lemma 2.
From Lemma 4 (proven in appendix A.15) we
know that restricting the representation D
d
j
to a
SU(d 1) ×U(1) subgroup gives a reducible rep-
resentation of SU(d 1) × U(1). Moreover the
U(1) action is trivial for the following blocks:
j
M
i=0
D
d1
i
. (105)
The image of α
1
|1i + ... + α
d1
|d 1i therefore
lies in this subspace (since it is invariant under
this U(1) action) which we call the α = 0 sub-
space. Moreover the image of |0i is uniquely de-
termined as the trivial subspace D
d1
0
(which is
invariant under the whole subgroup). However
we do not know if the image of the PC
d1
sub-
space spans the whole α = 0 subspace. We show
that it must have support in the subspace D
d1
j
(which is part of the α = 0 subspace).
We first consider a reducible theory described
with representation:
Γ =
j
M
i=0
D
d
i
. (106)
This corresponds to a reducible theory as char-
acterised above. We call its state space S. Con-
sidering a PC
d1
subspace gives a state space S
0
with representation:
Γ
0
=
j
M
i=0
D
d1
i
. (107)
We observe that the state space with representa-
tion Γ is equivalent to a direct sum of state spaces
with representation D
d
k
for k = 1, ..., j. We label
these state spaces S
k
. We write:
S =
j
M
k=0
S
k
. (108)
Moreover reducing a state space S
k
gives a PC
d1
state space with a representation inside the direct
sum (i.e. which may or may not have support in
each representation):
k
M
i=0
D
d1
i
. (109)
We have seen that the restriction of S gives a
state space S
0
with support in block D
d1
j
. More-
over since S is a direct sum of state spaces S
k
it
must be the case that (at least) one of these S
k
reduces to a state space with support in D
d1
j
.
The state space S
j
is the only state space in the
direct sum in (108) which can have support in
D
d1
j
when restricted. That is to say the reduc-
tion of S
j
(with representation D
d
j
) gives a PC
d1
state space with support in block D
d1
j
.
A.14.3 Embeddedness of arbitrary reducible theo-
ries
From the above considerations we see that an ir-
reducible theory J = j of PC
d
when restricted to
24
a PC
d1
subspace must give a state space which
has support in the subspace D
d1
j
, and hence cor-
responds to a theory of PC
d1
which has a repre-
sentation which contains at least a block D
d1
j
. A
reducible theory of PC
d
J = j
1
, ..., j
n
has a state
space S which is a direct sum of state spaces S
j
i
.
The restriction of S to a PC
d1
subspace is equiv-
alent to restricting each of the S
j
i
state spaces.
Hence the restriction of S to PC
d1
will have sup-
port in at least subspaces D
d1
k
for k = j
1
, ..., j
n
.
The Lemma follows directly from this.
A.15 Proof of Lemma 4
Lemma 4. Consider a representation D
d
j
and its
decomposition under a SU(d1)×U(1) subgroup.
The reducible representation of SU(d 1) ×U(1)
has blocks with various U(1) charges. The sub-
space with 0 U(1) charge is acted upon by the
representation of SU(d 1):
j
M
i=0
D
i
d1
. (110)
Similarly to Lemma 1 we label D
d
j
with partition
λ = [2j, j, j...., j, 0]. The restriction to a SU(d
1) × U(1) subgroup acts reducibly on the carrier
space V
λ
as:
V
λ
=
M
µ
V
µ
, (111)
where the sum is over every µ which intertwines
λ and V
µ
is a carrier space for an irreducible rep-
resentation of SU(d 1) with partition µ . The
µ which intertwine λ are of the form:
µ = [a, j, j, ..., b], 2j a j , j b 0 . (112)
We have |λ| = dj and |µ| = (d 3)j + a + b.
From Lemma 1 we have the condition that the
U(1) charge is 0 when |λ|
d
d1
|µ| = 0. We now
substitute in the relevant expressions:
|λ|
d
d 1
|µ| = 0
dj
d
d 1
((d 3)j + a + b) = 0
(d 1)j (d 3)j a b = 0
a = 2j b .
(113)
This entails that when b = 0, a = 2j, when
b = 1, a = 2j 1... and when b = j, a = j. Every
possible µ (intertwining λ) is in the direct sum
and hence every µ which meets the above condi-
tion for having 0 U(1) charge is in the decompo-
sition. Hence there are j + 1 subspaces V
µ
where
the action of U(1) is trivial. If we express the rep-
resentations acting on these in terms of Dynkin
notation we observe that µ = [2jb, j, ..., j, 2jb]
becomes (j b, 0, 0..., j b) for b = 0, ..., j. The
terms in the direct sum are therefore just the rep-
resentations D
d1
i
for i = 0, ..., j.
25