On the classiﬁcation of two-qubit group orbits and the use
of coarse-grained ‘shape’ as a superselection property
Thomas Hebdige
1
and David Jennings
1,2,3
1
Controlled Quantum Dynamics Theory Group, Imp erial College London, Prince Consort Road, London SW7 2BW, UK
2
Department of Physics, University of Oxford, Oxford, OX1 3PU, UK
3
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK.
January 30, 2019
Recently a complete set of entropic con-
ditions has been derived for the intercon-
version structure of states under quan-
tum operations that respect a speciﬁed
symmetry action, however the core struc-
ture of these conditions is still only par-
tially understood. Here we develop a
coarse-grained description with the aim
of shedding light on both the structure
and the complexity of this general prob-
lem. Speciﬁcally, we consider the degree
to which one can associate a basic ‘shape’
property to a quantum state or channel
that captures coarse-grained data either
for state interconversion or for the use of
a state within a simulation protocol. We
provide a complete solution for the two-
qubit case under the rotation group, give
analysis for the more general case and dis-
cuss possible extensions of the approach.
1 Introduction
Symmetry principles are ubiquitous in both the
classical and quantum realms. They constrain
dynamics, connect with conservation laws and
simplify computations of physical properties. In
crystallography the crystal structure is described
by a symmetry group and largely determines the
properties of the material [1].
Symmetry considerations have been extremely
successful in the ﬁeld of quantum information
theory [24]. Moreover a substantial component
of this work has addressed the regime in which
symmetry principles disconnect from conserva-
tion laws [5], and which has found application in
the study of quantum features of thermodynam-
ics [68]. Quantum states that break the symme-
try can also be used to circumvent limitations on
Figure 1: Residual symmetry-types in two-qubit sys-
tems. We associate to each state a ‘shape’ that de-
scribes the residual symmetries of that state. As an ex-
ample, in the above ﬁgure we show the symmetry proper-
ties of the set of all mixtures of Bell states under SU(2).
This is determined solely by the triplet component of the
state. The only possibilities are for the state to have a
residual symmetry of (a) a cylinder, (b) a double-sided
rectangle, or (c) a sphere. This ‘shape’ can be used
to provide superselection rules when using the two-qubit
state to simulate quantum operations. Details are given
in section 4.
the precision of measurements imposed by conser-
vation laws, as described by the WAY theorem
[914], while metrology can be viewed as using
broken symmetries to distinguish diﬀerent group
transformations [15]. Symmetry principles have
also provided new insights into quantum speed
limits and how quickly quantum operations can
be performed [16]. Finally, symmetry groups pro-
vide an invaluable toolkit in the context of quan-
tum computing [1720].
In this work we look at how general quantum
states ρ can be classiﬁed by groups that break a
Accepted in Quantum 2019-01-23, click title to verify 1
arXiv:1804.09967v2 [quant-ph] 29 Jan 2019
given symmetry constraint to an equal degree. A
concrete example makes this clearer: both a chair
and an arrow break rotational symmetry, however
an arrow has a residual (axial) symmetry group
to it, and so breaks rotations to a weaker degree
than the chair. For a ﬁxed quantum system of
dimension d, and symmetry action G, we can ask
what residual symmetries does the system admit,
and how do these residual symmetries transform
under symmetric dynamics. We also address the
degree to which this aspect of a quantum system
constitutes a property in the ‘resource-theoretic’
sense.
The structure of the paper is as follows. In
the next section we provide notation and back-
ground motivation. In section 3 we provide the
coarse-graining scheme and some basic relations
involved. This demonstrates the complexity of
the topic, and highlights some subtleties if one
wishes to associate the scheme to a quantum re-
source. In section 4 we provide a complete classi-
ﬁcation for two-qubit systems (partly pictured in
Fig. 1 and more fully in Fig. 5), and which illus-
trates how quantum state spaces admit a coarse-
grained partial order.
2 Background and Motivation
In this paper we shall use the following nota-
tion. For any quantum system A we denote
by H
A
its associated Hilbert space, and B(H
A
)
the set of linear operators on H
A
. We assume
that a symmetry action of G is deﬁned on H
A
through the unitary representation g 7→ U (g)
B(H
A
). The group action on a quantum state
ρ B(H
A
) is given by the adjoint action U
g
(ρ) :=
U(g) ρ U
(g).
For any quantum channel E : B(H
A
)
B(H
A
0
), the action of the group on E is U
g
(E) :=
U
g
E U
g
1
(where denotes the concatenation
of channels). This can be interpreted as moving
to a diﬀerent frame, performing the channel and
returning to the original frame.
If U
g
(E) = E for all g G then E is called
a symmetric operation. This includes certain
preparation procedures, and the resulting states
are known as symmetric states because they are
invariant under the group action: U
g
(ρ) = ρ for
all g G.
2.1 General state transformations under a
symmetry constraint
A primary concern here is the study of general
quantum states or quantum operations under a
symmetry action. In particular we would like
to study the degree to which any given quan-
tum state breaks the symmetry (a ‘resource state’
[21, 22]). One way of comparing states in this
manner is to say that ρ σ if and only if
ρ σ = E(ρ) for some symmetric quantum op-
eration E, which deﬁnes a partial order on quan-
tum states [23, 24]. A measure of how much a
state breaks the symmetry is then a function M
that respects the partial order namely, if ρ σ
then M(ρ) M(σ) [21, 22]. A central ques-
tion is whether a complete set of measures ex-
ist that fully capture this ordering of states, and
thus asks when two quantum states can or can-
not be interconverted under the symmetry. Very
recently [7] an answer was given for this general
problem in terms of single-shot entropies of the
form H
min
(R|A) that quantify the amount of in-
formation in a system A about an external quan-
tum reference frame R. Thus it is the case that
ρ σ under the symmetry constraint if and only
if H
min
(R|A) decreases with respect to all refer-
ence frames R.
However this set of measures for the resource
theory is over-complete and highly complex to
use. Applying it to general systems requires so-
phisticated techniques in single-shot information
theory. Here we wish to tackle a more mod-
est goal, and instead of describing the complete
symmetry properties of a quantum state, wish
to coarse-grain states into groups that break the
symmetry in the same way. Notably, this itself
turns out to be a highly non-trivial problem and
so sheds light on the structure of the more general
set of measures given in [7].
2.2 Eﬃcient simulation of quantum channels
One very grounded way of probing the degree
to which a quantum state breaks a symmetry
is to see how useful it is when we wish to per-
form tasks, such as induce a channel on another
system. More generally, if we have some target
quantum operation E on a system, we would like
to know what kind of resource states and interac-
tions are required to realise that operation. The
degree to which a resource σ can do this under
Accepted in Quantum 2019-01-23, click title to verify 2
symmetric dynamics therefore constitutes a mea-
sure of its properties.
In [25], the symmetry properties of quantum
channels were analysed, with a focus on the orbit
of a channel E, deﬁned as
M(E) := {U
g
(E) : g G}. (1)
A similar deﬁnition applies to the orbit of states
under the group, so M(ρ) = {U
g
(ρ) : g G}.
Now the type of orbit one obtains provides a
natural context in which to understand both how
the channel breaks a symmetry constraint, and
moreover how one might simulate such a channel
using an auxiliary quantum system B prepared
in a non-symmetric state σ
B
.
It is clear that an auxiliary resource state may
be used with varying eﬃciency in the simulation
of a channel on a system S, however in [25] the
question of when a state σ
B
can be used to in-
duce a channel E such that its ability to simulate
the very same channel on subsequent systems is
undiminished. This was motivated by the dis-
covery of a “catalytic coherence” protocol [26] in
which a quantum state σ
B
is used as a phase ref-
erence. The state undergoes non-trivial changes
σ
B
σ
(1)
B
σ
(2)
b
· · · σ
(n)
B
yet for all n 1
its abilities as a phase reference never diminish.
In [25] the structure of such arbitrarily repeatable
protocols was spelled out using harmonic anal-
ysis. It was shown that if a channel E on S is
simulated using a quantum state σ
B
under an ar-
bitrarily repeatable protocol then there exists a
POVM measurement {M
k
} on B such that
E(ρ) =
X
k
tr(M
k
σ
k
(ρ) (2)
where {Φ
k
} are completely positive maps on the
primary system S. The POVM {M
k
} on the
auxiliary system depends explicitly on the tar-
get channel E. Crucially however, it can be un-
derstood as resolving the location x of E on its
unitary orbit M(E) to some scale that depends
on the dimension d of S. For example, in the case
of catalytic coherence, the POVM measurement
on B corresponds simply to resolving a point x
on a circle (the orbit of the channel E under phase
rotations) down to an angular scale
2π
d
.
The degree to which x is resolved on M(E)
clearly depends on the resource state σ
B
, and
corresponds to the fact that σ
B
is a quantum
reference frame. This in turn can be viewed as
inducing non-classical geometry on M(E) for
example, for G = SU(2) a channel can have
orbit M(E) that is a 2-sphere. The use of a
bounded reference system in this case means that
only ﬁnite resolution of points on M(E) is pos-
sible, however since the 2-sphere carries a phase
space structure the bounded reference case cor-
responds to the so-called “fuzzy sphere” model
from non-commutative geometry [27]. This dif-
fers from catalytic coherence in that we now have
complementarity in resolving the coordinates on
the sphere, and which ultimately arises because
the two orbits have quite diﬀerent structure.
Thus the general eﬃcient use of a quantum
state σ
B
to simulate a channel E on S can be
analysed in terms of two aspects:
1. (‘Shape’) The type of orbit M(σ) the state
has under the group action compared to
M(E), independent of metrical aspects.
2. (‘Geometry’) The ability of σ to encode clas-
sical coordinate data for x in M(E).
What is a necessary relation between M(E) and
M(σ)? It is clear that we need σ to break the
symmetry to a ‘larger’ degree than E, however in
this paper we would like to make this statement
more precise. Given the complexity of the general
problem, one motivation in this work is to start
with the above separation into ‘shape’ and ‘ge-
ometry’, and develop an organisational setting in
which we provide a coarse-graining over states of
the same ‘shape’, with the remaining task being
to analyse the ‘geometry’ aspect of the state.
2.3 Related topics in quantum state tomogra-
phy and quantum computation
Beyond the abstract problem of simulating an ar-
bitrary quantum channel using a resource state,
there are speciﬁc contexts of importance where
such a coarse-grained division naturally arises. In
[28], quantum state tomography was studied in
the context of prior information that restricts the
state to a lower-dimensional submanifold of the
state space. The analysis shows that the topolog-
ical genus of the manifold can be used to bound
the number of measurements needed to discrim-
inate states on the submanifold. The orbit of a
state (or channel) is one such submanifold, and
so the study of what kinds of orbits can exist and
Accepted in Quantum 2019-01-23, click title to verify 3
how they transform among themselves is of po-
tential relevance to quantum tomography under
symmetry constraints.
A wholly practical direction where such analy-
sis might be of relevance is in the computational
power of gate-sets in quantum computation, and
how such gate-sets interact with states that in-
crease the computational power of the gate-set
(such as magic states for the stabilizers [29]). For
example, a recent work provides a classiﬁcation
of all Cliﬀord gate-sets [17] in a lattice hierar-
chy, and so aspects of the present work might be
applicable in studying how noisy quantum states
increase the computational power of easily realis-
able gate-sets.
2.4 Core questions of the present work
In subsection 2.1 we highlighted that state inter-
conversion ρ σ under a symmetry constraint
is highly non-trivial, and while progress has been
made on this fundamental problem, it is not clear
how complex it is in general. Also, as discussed
in subsection 2.2 recent results on optimal proto-
cols show that the structure of this theory natu-
rally decomposes into a geometric component and
the particular kind of group orbit the quantum
channel has. In light of these considerations one
expects that the group orbit level of description
might provide a form of coarse-grained descrip-
tion that sheds light on the more complex parent
resource theory. By studying this coarse-grained
theory we can shed light on the parent theory, and
moreover extend the notion of a resource theory
from being about pre-orders on quantum states to
considering pre-orders on sub-sets of states. How-
ever, in order for this to work and be meaningful,
it is crucial that the coarse-grained theory be (a)
physically sensible and (b) relate naturally to the
parent theory. Therefore the core questions we
address in this work are the following:
1. Does a coarse-grained resource-theory frame-
work exist?
2. Is this theory consistent with its parent re-
source theory? Is it physically sensible?
3. How complex is the general structure of
symmetry-based resource theories?
In the next section we tackle the ﬁrst two of
these questions, by showing how a natural par-
tition of the state space arises that is consistent
with the ﬁner-grained resource theory, and de-
scribe its limitations and physical robustness. In
the process, and through exhaustive analysis of
the G = SU(2) case for two qubits, we shed light
on the crucial third question posed.
3 Coarse-graining states and channels
under a Symmetry Action
3.1 Simulating Operations Under Symmetry
Constraints
If one wishes to realise a quantum channel E via
a symmetric interaction with a state σ, the most
natural way to model this is as
E(ρ) = tr
B
V (ρ
A
σ
B
)V
(3)
where V is a covariant unitary, namely
[V, U
A
(g) U
B
(g)] = 0 for all g G. If such
a relation holds, we say that E can be simulated
using the state σ
B
.
A basic classiﬁcation of states and channels
under a symmetry can be done by studying the
subgroup of residual symmetries for the state or
channel, called the isotropy subgroup or stabilizer
of E and deﬁned as
Iso(E) := {g G : U
g
(E) = E}. (4)
It speciﬁes the residual symmetry that the chan-
nel has under the group action. Symmetric chan-
nels are invariant under all group transforma-
tions, and simply have Iso(E) = G. This notion
also applies to states, where Iso(ρ) := {g G :
U
g
(ρ) = ρ}.
In many cases the quantum operation E has
some residual symmetry, which is characterised
an isotropy subgroup H of G. The possible
isotropy subgroups form an abstract structure
called a lattice [30]. The partial ordering of the
subgroups is deﬁned by subset inclusion, namely
H
1
H
2
if H
1
H
2
for H
1
, H
2
every pair of elements have a unique supremum
and unique inﬁmum, which deﬁne binary opera-
tions called the meet and join [31]. For two sub-
groups H
1
and H
2
, their meet is denoted H
1
H
2
and deﬁned as H
1
H
2
, while their join is denoted
H
1
H
2
, deﬁned as the subgroup generated by
H
1
H
2
[30]. This structure is illustrated by a
Hasse diagram of the subgroups, as shown in Fig.
2.
Accepted in Quantum 2019-01-23, click title to verify 4
Figure 2: Coarse-grained classiﬁcation of quantum states under a symmetry. The set of all density operators
is partitioned into sets C(H) with deﬁnite symmetries (seen on the left). The C(H) correspond to the subgroups
shown in the Hasse diagram in the middle. The Hasse diagram represents the subgroup lattice, with lines indicating
subgroup inclusion, i.e. H
5
H
4
H
3
etc. Note that for ﬁnite-dimensional systems many of the C(H) will be
empty. The subsets C(H) can be associated, up to diﬀeomorphisms, with group orbits (indicated on the right). The
left ﬁgure shows the action of group-averaging (over H
2
), going from σ to P
H
2
(σ) as described in section 3.4.
Since quantum operations can be combined in
a number of ways, we present some basic state-
ments that constrain the residual symmetry of
the resultant quantum operation. The proofs for
this section are provided in Appendix A.
Lemma 1. Given any two quantum channels E :
B(H
A
) B(H
A
0
) and F : B(H
B
) B(H
B
0
),
with unitary representations of a group G deﬁned
on all input and output spaces. Then we have the
following:
1. Iso(EF) Iso(E)Iso(F) under the tensor
product group action U
g
(E F) = U
g
(E)
U
g
(F).
2. If B
0
= A then Iso(E F) Iso(E) Iso(F).
3. If A = B and A
0
= B
0
, and p some proba-
bility 0 p 1, then Iso(pE + (1 p)F)
Iso(E) Iso(F).
4. Iso(U
g
E U
g
) = g[Iso(E)]g
1
.
These results also apply for quantum states, once
we view the state as the state preparation map
1 ρ. We can now make precise a coarse-grained
(necessary, but far from suﬃcient) requirement
that a state σ allow the simulation of a channel
E under symmetric dynamics and show that this
bound is tight in general. The proof is provided
in Appendix A.
Theorem 1. If a system B in a state σ
B
can be
used to simulate a CPTP map E under symmetric
dynamics, then Iso(σ
B
) Iso(E). Moreover, if
E has isotropy group Iso(E) then there exists a
quantum system B and quantum state σ
B
with
Iso(σ
B
) = Iso(E) that allows the simulation of E.
This result gives the basic relation, in the coarse-
grained picture, for when a quantum state σ
B
can
be used to simulate a quantum channel and also
shows that this relation is a tight one. This is
necessary in order for the isotropy subgroup ap-
proach to be consistent with the parent resource
theory, where one is concerned with the minimal
resources needed to realise a particular quantum
channel.
A similar result can be given in the case of ap-
proximate simulation. For any target operation
E we can deﬁne the noisy version given by
E
= (1 ) E + D (5)
where D is the complete depolarization map ρ 7→
1
d
for all ρ. From Lemma 1 we see that Iso(E
) =
Iso(E) G = Iso(E). If σ allows the simulation
of E
for some 0 < < 1 then this corresponds
Accepted in Quantum 2019-01-23, click title to verify 5
to an approximate, ‘isotropic’ simulation of the
original map with noise parameter . Theorem 1
extends to this case in an obvious way.
3.2 Is ‘shape’ a resource-theoretic property?
While we normally associate measurable proper-
ties with either projective measurements or more
generally POVMs, there is an alternative way
that is more general again. Speciﬁcally, a prop-
erty is associated with a pre-order deﬁned on
the set of all quantum states. This recent ap-
proach is called the resource-theory method, and
has found success in areas such as entanglement,
coherence, thermodynamics, and many other sce-
narios [4, 3234]. Quantum maps that respect the
pre-order are called ‘free operations’ and any real-
valued function on quantum states that respects
the pre-order is a measure of the property.
Given an ability to order the isotropy sub-
groups of quantum states and channels under the
group action, we can ask if a meaningful notion
of ‘shape’ can be deﬁned for quantum systems,
along such a resource-theoretic line. This would
essentially characterise the asymmetry of states
and channels without reference to a measure.
For continuous Lie groups, we ﬁnd that the or-
bit of a state or channel is always a homogeneous
space [35, 36], speciﬁed by both the group G and
the particular isotropy subgroup H of the chan-
nel. More precisely the orbit of the channel is
M(E) and coincides with the quotient G/H up to
diﬀeomorphisms, which we write M(E)
=
G/H
(and likewise for states).
1
This is what we call
the ‘shape’ of a state or channel.
A simple example can be given for a single
qubit state under an SU(2) symmetry, where the
possible types of group orbit are
M
1
2
[1 + r · σ]
=
(
SU(2)/U(1)
=
S
2
r 6= 0
SU(2)/SU (2)
=
{e} r = 0,
(6)
and these are illustrated in Fig. 3.
The lattice of isotropy subgroups gives us a way
of comparing the ‘shapes’ of group orbits under a
group action G, with M(E) M(F) iﬀ Iso(E)
Iso(F) (up to isomorphism of the isotropy sub-
groups). Likewise for states, M(ρ) M(σ) iﬀ
1
Technically we deﬁne the ‘shape’ of the group orbit
with respect to the conjugacy class of the isotropy sub-
group H. This ‘shape’ is also called the Orbit-Type in the
literature. See [37] for further details.
Figure 3: Basic orbits of states. Group orbits for a
single qubit under the action of SU(2) shown in the
Bloch sphere. There are only two possibilities in this
case: a sphere for non-zero Bloch vectors (left), or a
point for the symmetric maximally mixed state (right).
Iso(ρ) Iso(σ) (up to isomorphism). The con-
vention here is to match with resource-theoretic
measures of asymmetry [21, 23], so that one
‘shape’ is ‘bigger’ than another if it has a smaller
isotropy group. For example, a single-qubit state
with non-zero Bloch vector (M(ρ)
=
S
2
) has a
‘bigger’ group orbit than the maximally mixed
state (M(1/2)
=
{e}). This gives valuable infor-
mation regarding the resources required to per-
form a quantum channel in the presence of sym-
metry constraints.
The isotropy subgroup gives us a natural equiv-
alence relation between states that behave in
the same way under the group action. We can
then say that ρ
1
and ρ
2
are related ρ
1
ρ
2
if
we have that Iso(ρ
1
) = Iso(ρ
2
). This equiva-
lence relation partitions the state space into sets
C(H) := {ρ : Iso(ρ) = H}, and collects together
all states that break the symmetry in the same
manner. Taking the union of C(H) corresponding
to isomorphic H partitions the states and chan-
nels according to their ‘shape’.
Viewed as a mapping, C maps from the sub-
group lattice onto a partition of the state space,
and so we can simply allow the partition to in-
herit the lattice structure, and order subsets of
states as C(H
1
) C(H
2
) whenever H
1
H
2
.
Note however that the sets {C(H)} are not
convex in general. For example a qubit system
under G = SU(2), both |0i h0| and |1i h1| have
the same U(1) isotropy subgroup, but
1
2
(|0i h0| +
|1i h1|) =
1
2
1, which is symmetric. However,
C(G) is always a convex set. This follows be-
cause if ρ
1
, ρ
2
C(G) then Lemma 1 implies that
G Iso(p
1
ρ
1
+p
2
ρ
2
) Iso(ρ
1
)Iso(ρ
2
) = G, and
so any mixture has the same isotropy subgroup.
The SU(2) qubit example also highlights that
if C(H) 6= it does not imply that C(H
0
) 6=
Accepted in Quantum 2019-01-23, click title to verify 6
for all H
0
H, since there are subgroups of
SU(2) containing U(1) which do not appear as
the isotropy subgroup of a single qubit state. Not
all possible isotropy subgroups will be seen in a
given system, and for a ﬁnite dimensional system
many of the sets C(H) will be empty.
However in order to have a meaning-
ful resource-theory interpretation, the coarse-
grained ordering must be compatible with the set
of free operations namely the symmetric oper-
ations. It is easy to show that this is in fact the
case at the level individual quantum states.
Lemma 2. Under a symmetric operation E,
Iso(E(ρ)) Iso(ρ).
This shows that the coarse-grained ordering of
states is consistent with the set of free (symmet-
ric) operations, as expected. However it does not
imply that a functional mapping is deﬁned on the
sets C(H); it is possible to have ρ
1
and ρ
2
in the
same set C(H), but each get sent to diﬀerent sets
C(H
0
1
) and C(H
0
2
). This aspect means that inter-
preting the coarse-graining as deﬁning a quantum
resource needs care.
3.3 The speck of dust argument tiny pertur-
bations to symmetric states & channels
Having outlined how the set of quantum states or
the set of quantum channels of a quantum system
is partitioned according to the symmetry action
one might raise the following concern: what hap-
pens if one perturbs a channel via some small
perturbation? How does this aﬀect its isotropy
subgroup?
It is readily seen that one can make an arbitrar-
ily small perturbation to any quantum channel E
so as to break the symmetry in a “maximal” way.
Informally this amounts to the “speck of dust”
argument where, for example, a perfectly rota-
tionally symmetric ball can be made completely
asymmetric by sticking arbitrarily small pieces of
dust onto it (Fig. 4).
In more abstract terms, one can use the princi-
pal orbit-type theorem [37] for the space of quan-
tum channels of a system to see that there is
always a principal isotropy subset that forms a
dense subset in the set of all quantum channels.
Thus “most” quantum channels on a system will
break a symmetry to a maximal degree (the prin-
cipal isotropy).
Figure 4: The speck of dust argument. If we start
with a perfectly rotationally symmetric object (left im-
age) we can always add arbitrarily small perturbations
that completely break the rotational symmetry (right
image). However physically the fact that we have ﬁ-
nite resolution means that quantum states and channels
should always be understood as being deﬁned up to some
–smoothing scale. This does not aﬀect the resource-
theoretic account and therefore the ‘shape’ is a robust
feature in the presence of such tiny symmetry-breaking
perturbations.
This argument of course applies equally well to
the partitioning of the quantum state space into
diﬀerent subsets with ﬁxed isotropies, however in
both cases this does not pose a problem for our
formalism for the following reasons. Firstly, in
resource theories we are fundamentally interested
in the most eﬃcient use of resources and therefore
our task is to look for those quantum resource
states that break the symmetry not in a maximal
way, but in a minimal way.
Secondly, at the level of channels, the most
interesting and commonly addressed channels in
quantum information science are not of princi-
pal isotropy type namely they have non-trivial
isotropies. For example, consider quantum chan-
nels on a single qubit, and the irreducible group
action of G = SU(2). A typical quantum chan-
nel on the qubit will only have a small resid-
ual isotropy group Z
2
= {1, 1} (the principal
isotropy subgroup of the set of all qubit chan-
nels). However consider the following standard
single qubit channels:
1. The depolarizing channel: E(ρ) = + (1
p)(
1
2
), with 0 p 1.
2. A rotation about the ˆn axis: U
ˆn
(θ) =
exp[iˆn · σ ].
3. A projective measurement along the ˆn axis:
E(ρ) =
P
k
tr(Π
k
ρ
k
, where Π
0
= |ˆni hˆn|
and Π
1
= 1 Π
0
= |−ˆni h−ˆn|.
Accepted in Quantum 2019-01-23, click title to verify 7
4. Partial dephasing along the axis ˆn: E(ρ) =
+(1p)
P
k
tr(Π
k
ρ
k
, where Π
0
= |ˆni hˆn|
and Π
1
= 1 Π
0
= |−ˆni h−ˆn|.
5. A qubit state preparation channel: E(ρ) =
1
2
(1 + p ˆn · σ).
It is readily seen that the isotropy subgroup for
channel (1) is SU(2), while for channels (2–5) it
is a U (1) subgroup of SU(2). The orbits will have
‘shapes’ SU(2)/SU(2)
=
{e} and SU(2)/U (1)
=
S
2
respectively. None of these channels have prin-
cipal isotropy type { , }, however they are
clearly key quantum operations in quantum in-
formation science for which one might wish to
determine the minimal quantum resources neces-
sary to realise them.
Thirdly, the fact that something is “measure
zero” in some space does not imply that it is
physically irrelevant. For example, a 2-d plane
is a measure zero set in three spatial dimensions,
however this does not mean that anyonic physics
or the quantum hall eﬀect is irrelevant. Or of
closer bearing on our present work, we can con-
sider the case of Noether’s theorem: clearly the
set of dynamics with rotational symmetry is again
“measure zero”, however this does not mean that
Noether’s theorem is irrelevant for physics. In
both cases while the restrictions are strictly un-
stable under small perturbations, the important
thing is that the features of interest are opera-
tionally robust under small perturbations.
The speck of dust subtlety makes itself appar-
ent in our setting if one tries to naively exploit
some metric on quantum states to quantify how
far apart the sets {C(H)} are from each other.
It is readily seen that the distance been any two
such sets is in fact zero.
Lemma 3. Let d(·, ·) be any metric on the space
of quantum states. In terms of this metric we
deﬁne
d(C(H
1
), C(H
2
)) := inf
σ
1
C(H
1
)
σ
2
C(H
2
)
d(σ
1
, σ
2
). (7)
Then d(C(H
1
), C(H
2
)) = 0 for all H
1
, H
2
G.
This also shows that any set C(H) is arbitrar-
ily close to the set C(G). However, this simply
means that any symmetry properties must always
be considered up to some ﬁnite resolution scale
arbitrarily small perturbations can always elimi-
nate residual symmetries, even though this state
is practically indistinguishable from the unper-
turbed one.
Therefore any membership of a quantum state
ρ to a subset should only be considered up to
some smoothing scale based on a distance mea-
sure d (such as from the L
1
norm). For each state
there is an -ball of nearby states, B
(ρ) = {σ :
d(ρ, σ) }. Rather than Iso(ρ), we should con-
sider Iso
(ρ) := max
σ∈B
(ρ)
Iso(σ). Intuitively,
if a state ρ is within a distance of another
state with more residual symmetries, we asso-
ciate those additional residual symmetries with ρ.
The smoothing scale selects the size of symmetry-
breaking perturbations that we wish to consider,
and will depend on the particular physical con-
text.
This reasoning applies equally to quantum
channels. For example, the depolarisation chan-
nel on a qubit is fully invariant under the G =
SU(2) and so no non-trivial resource state is
needed to simulate it. If one perturbs this channel
to E E
by some > 0 perturbation in the dia-
mond norm one can clearly break this isotropy to
the principal isotropy { , }, however it is also
clear (e.g. from the continuity of the Stinespring
dilation [38]) that E
can either be simulated ex-
actly with a resource state on B that -close to
be being symmetric, or simulated approximately
to the same threshold with a perfectly symmet-
ric state on B. Thus the use of isotropy groups
within the resource theory context is robust under
such small perturbations and the speck of dust
argument is moot. We illustrate this point ex-
plicitly in Section 4, where we demonstrate that
for the case of two qubits under SU(2) the notion
of diﬀerent ‘shapes’ is perfectly robust under the
particular case of perturbations that allow a 4%
error rate.
3.4 Projecting out residual symmetries via
quantum operations
Given the structure of states under the above
partition, we can ask how easy it is to move
from one set C(H) to another. As discussed
one does not in general have quantum operations
that map any C(H) neatly into some other sub-
set. Instead it makes sense to consider the sets
ˆ
C(H) :=
S
W H
C(W ). These sets are quite nat-
ural to consider because Lemma 2 implies that
each
ˆ
C(H) is closed under symmetric operations,
and any symmetric operation provides a well-
Accepted in Quantum 2019-01-23, click title to verify 8
deﬁned mapping of
ˆ
C(H)
ˆ
C(f (H)).
The quantum operation
P
H
(ρ) :=
Z
H
dh U
h
(ρ) (8)
is the average of the state ρ over a ﬁxed subgroup
H weighted by the invariant Haar measure dh.
In the case of a ﬁnite subgroup of size |H|, the
integral
R
H
dh is replaced by the sum
1
|H|
P
hH
.
Lemma 4. The map P
H
has the following prop-
erties:
1. P
H
is the (orthogonal) projector onto
ˆ
C(H).
2. P
H
(ρ) = arg min
σ
ˆ
C(H)
S(ρ||σ), where
S(ρ||σ) is the relative entropy.
Therefore P
H
projects onto
ˆ
C(H), as illus-
trated in Fig 2. Dephasing [20], E(ρ) =
1
2π
R
2π
0
e
iθZ
ρe
iθZ
, can be viewed as P
U(1)
,
sending ρ to the nearest incoherent state while
preserving the diagonal terms of a density ma-
trix.
The mapping P
H
moves down chains of the
subgroup Hasse diagram, however this is not al-
ways a symmetric operation. The following tells
us when such a transformation can be performed
freely in the resource theory.
Lemma 5. Given a group action for G,
Iso(P
H
) = N
G
(H), where N
G
(H) = {g G :
gHg
1
= H} is the normalizer [39] of H in G,
and therefore if H is a normal subgroup of G
(H G) then P
H
is a symmetric operation.
This constrains the kind of resources needed to
move from one C(H) to another using P
H
, since
this projects onto a given
ˆ
C(H), although less
resource-hungry ways may exist to perform the
same transformation.
3.5 Discussion of the section results
In this section we have given an analysis of the
what happens when one classiﬁes quantum states
or channels in terms of their residual symmetries.
We discussed how composition and mixing aﬀect
the ordering, and how one can relate these fea-
tures to the issue of simulation (exact or approx-
imate) of a target quantum operation with some
resource state. We also saw that the ordering
has subtleties if one wishes to interpret it in a re-
source theoretic sense. The statements one makes
are also quite blunt, and a good example of this
is the fact that the sets {C(H)} are all arbitrarily
close to one another. However the relations still
carry non-trivial content, and for example Theo-
rem 1 can be viewed as a form of superselection
rule on quantum operations that tells us which
states are ruled out and which are not.
While this is conceptually neat and provides
high-level insight into the complexities of the full
classiﬁcation of states (as described in [7]), it is
less clear how computationally useful or simple
these structures are in practice. To address this
point, in the next section we consider the impor-
tant case of a two-qubit system under an SU (2)
action. We ﬁnd that already in just this sim-
ple scenario the hierarchy of states is quite com-
plex, however the example does provide insight
into what to expect in the more general case.
4 Illustrative Example: Two Qubits
Under SU(2) Symmetry Constraints
The goal of this section is to illustrate the clas-
siﬁcation of quantum states in a simple quan-
tum system that has suﬃcient structure, yet is
tractable to the point of being fully solvable.
The state space of a two-qubit system is 15 di-
mensional and is suﬃciently non-trivial, more-
over there is a very natural group action to con-
sider, namely the tensor product representation
of SU(2). The orbits of 2-qubit states have been
studied in relation to thermodynamics and corre-
lations within these states [40, 41].
Any two qubit state ρ
AB
can be written
ρ
AB
=
1
4
1 1 + a · σ 1 + 1 b · σ
+
3
X
i,j=1
T
ij
σ
i
σ
j
, (9)
with local Bloch vectors a and b and correlation
terms determined by the correlation matrix T
ij
[42]. In general |a| 1 and |b| 1. Moreover,
we can put all 2-qubit states into diagonal form,
ρ
AB
=
1
4
1 1 + a · σ 1 + 1 b · σ
+
3
X
i=1
τ
i
c
i
· σ d
i
· σ
!
(10)
Accepted in Quantum 2019-01-23, click title to verify 9