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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
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).
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-
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
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.
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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