Fermat's Library | On the classification of two-qubit group orbits and the use of coarse-grained ‘shape’ as a superselection property annotated/explained version.

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 [2–4]. 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 [6–8]. 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

[9–14], 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 [17–20].

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)Φ

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

⊆ G. In addition,

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(E⊗F)  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, 32–34]. 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) = pρ + (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)Π

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) =

pρ+(1−p)

P

k

tr(Π

k

ρ)Π

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

h∈H

.

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

dθ 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

where {c

i

} and {d

i

} are orthonormal bases

of R

3

. The coeﬃcients τ

i

specify a posi-

tion (τ

1

, τ

2

, τ

3

) in a tetrahedron with vertices

(−1, −1, −1), (1, 1, −1), (1, −1, 1) and (−1, 1, 1).

When a = b = 0, then ρ

AB

is a quantum state

with maximally mixed marginals represented by

a point in the tetrahedron. In the case a or b

being non-zero, every quantum state ρ

AB

corre-

sponds to a point in the tetrahedron, however the

converse is not true: not all triples (a, b, t) cor-

respond to a valid quantum states.

Via local unitaries U

1

⊗ U

2

, these can be trans-

formed into the canonical form

˜ρ

AB

=

1

4



1 ⊗ 1 + a

0

· σ ⊗ 1 + 1 ⊗ b

0

· σ

+

3

X

i=1

τ

i

σ

i

⊗ σ

i

!

(11)

where a

0

·σ = U

1

(a·σ)U

†

1

and b

0

·σ = U

2

(b·σ)U

†

2

.

The states in this canonical form with maximally

mixed marginals are called T-states [42],

ρ

T

=

1

4





1 ⊗ 1 +

X

j

τ

j

σ

j

⊗ σ

j





. (12)

The set of T-states is the convex hull of the four

Bell states. The corners of the tetrahedron de-

ﬁned by the coeﬃcients {τ

i

} correspond to the

Bell states.

We now consider the symmetry properties of 2-

qubit states under SU (2) symmetry constraints,

assuming an SU(2) tensor product representa-

tion for the group action, i.e. U

g

(ρ) = U(g) ⊗

U(g) ρ U

†

(g) ⊗ U

†

(g), where U (g) is a two di-

mensional irrep of SU(2).

Our description of the group orbit as the quo-

tient space G/Iso(ρ) starts with the group man-

ifold of G and eliminates the redundancy caused

by the symmetries of ρ. The group transfor-

mations in Iso(ρ) have trivial group action on

ρ, so we quotient out the equivalence classes

g Iso(ρ) := {gh

i

: h

i

∈ Iso(ρ)}.

Certain groups have structure which permits

us to further simplify our description of the group

orbit manifold. Here we consider SU(2) symme-

try constraints, however this SU(2) action has a

subgroup Z

2

= {±1} that is always a symmetry

of a quantum state, and which is associated with

SU(2) being the double cover of SO(3). More-

over this sub-group has a key property which al-

lows us to simplify our description of the SU(2)

isotropy groups to those of SO(3). We can there-

fore think in terms of SO(3) subgroups, which are

the familiar chiral point groups one encounters in

for example crystallography [1].

A subgroup N of G is a normal subgroup, de-

noted N  G, if it satisﬁes gNg

−1

= N for all

g ∈ G [1]. These partition G into equivalence

classes gN := {gn

i

: n

i

∈ N}, which themselves

constitute elements of the quotient group G/N.

Every element of G is speciﬁed by the pairing of

the equivalence class gN and an element of the

normal subgroup N. Likewise, elements of H are

speciﬁed by the pairing of an element of H/N and

an element of N because N H. This means that

the group orbit manifold G/H can be described

by the quotient groups G/N and H/N, and we

can write M(ρ)

∼

=

G/H

∼

=

(G/N )/(H/N ). In

our present analysis Z

2

is a normal sub-group

of SU(2) common to all quantum states and so

M(ρ)

∼

=

SO(3)/(H/Z

2

) where H is the isotropy

group of ρ in SU(2). We can therefore use the

more intuitive SO(3) subgroups, listed in Table

1.

Group Description

C

n

Cyclic group of order n

C

∞

Symmetries of Cone

D

n

Symmetries of a regular n-sided Polygon

D

∞

Symmetries of Cylinder

T Symmetries of Tetrahedron

O Symmetries of Cube/Octahedron

I Symmetries of Dodecahedron/Icosahedron

Table 1: Chiral point groups: subgroups of SO(3),

as detailed in [1]. All quantum states have isotropy sub-

groups under SU(2) that can be related to particular

chiral point group.

In the next section we look at the isotropy sub-

groups of the Bell states, and then go on to

classify the symmetry properties of the T-states.

Then we continue onto the wider class of 2-qubit

states with maximally mixed marginals and ﬁ-

nally we complete the classiﬁcation for general 2-

qubit states by considering non-zero local Bloch

vectors.

4.1 Bell States

The Bell states correspond to the extremal points

of the tetrahedron of T-states. Their isotropy

subgroups under the SU (2) group action can be

Accepted in Quantum 2019-01-23, click title to verify 10

calculated directly, as described in Appendix B.

The singlet Bell state ψ

−

is symmetric un-

der SU(2), therefore Iso(ψ

−

) = SU(2) and so

M(ψ

−

) = {e} as expected. The triplet Bell

states do break the SU(2) symmetry, with

Iso(φ

+

) = {(iZ)

α

e

iθY

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}.

(13)

We will call this subgroup K

∞

∼

=

U(1) o Z

4

, and

it provides an example of a semi-direct product

group

2

. Similarly,

Iso(φ

−

) = {(iY)

α

e

iθX

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

(14)

and

Iso(ψ

+

) = {(iX)

α

e

iθZ

: 0 ≤ θ < 2π, α = 0, 1, 2, 3},

(15)

both of which are isomorphic to K

∞

.

Given the triplet Bell states have isomorphic

isotropy subgroups, their group orbits will have

the same shape, M(φ

+

)

∼

=

M(φ

−

)

∼

=

M(ψ

+

)

∼

=

SU(2)/K

∞

. In the previous section we described

how normal subgroups simplify our description

of the group orbit, and we can use this here

to put the description in terms of SO(3) sub-

groups. There is a 2-to-1 homomorphism from

the quotient G/{1, −1}. Elimination of the Z

2

normal subgroup from the group orbit descrip-

tion gives SU (2)/K

∞

∼

=

SO(3)/D

∞

, where D

∞

is the isotropy subgroup of the cylinder.

4.2 T-States

From the Bell states, we can begin to map out the

possible shapes of group orbit in the tetrahedron

2

The semi-direct product [1] is constructed from two

groups, in this case Z

4

and U (1). Each element of the

semidirect product group can be thought of as a pairing

(h, n) where h ∈ H = Z

4

and n ∈ N = U(1). However the

multiplication law of the semidirect product group does

not treat the elements of this pairing independently, with

(h

1

, n

1

)(h

2

, n

2

) = (h

1

h

2

, f

h

2

(n

1

)n

2

) where f

h

: N → N

is a automorphism on the second group speciﬁed by an

element of the ﬁrst group. The semi-direct product o

is deﬁned by the choice of automorphism f

h

. Note that

N / (N o H). If a trivial (identity) automorphism is cho-

sen, where f

h

(n) = n for any h and n, we have a direct

product group [1]. In the U (1) o Z

4

example, the group

multiplication law is



(iZ)

α

1

e

iθ

1

Y



(iZ)

α

2

e

iθ

2

Y



= (iZ)

α

1

+α

2

e

i[(−1)

α

2

θ

1

+θ

2

]Y

.

of T-states. Consider the convex hull of triplet

Bell states to be the base of the tetrahedron, and

ψ

−

to be the peak. The vertical height above the

base of this tetrahedron indicates the proportion

of the singlet state in the convex mixture p

1

φ

+

p

2

φ

−

+ p

3

ψ

+

+ p

4

ψ

−

, where p

1

+ p

2

+ p

3

+ p

4

= 1.

Since ψ

−

is symmetric under the SU (2) group

action,

M(p

1

φ

+

+ p

2

φ

−

+ p

3

ψ

+

+ p

4

ψ

−

)

= M(p

1

φ

+

+ p

2

φ

−

+ p

3

ψ

+

), (16)

therefore the possible types of group orbit for T-

states will all be exhibited in the convex hull of

the triplet states. These can be visualized as a

triangle of states, as shown in Fig. 5.

The vertices of this triangle are the triplet Bell

states, and each have a K

∞

isotropy subgroup.

Appendix B also shows that the midpoints of

the edges of this triangle, corresponding to states

1

4

(1 ⊗ 1 + σ

i

⊗ σ

i

), also have K

∞

isotropy sub-

groups. Therefore the diagonals of this triangle,

corresponding to states

ρ = pφ

+

+ (1 − p)

1

4

(1 ⊗ 1 + Y ⊗ Y ) (17)

ρ = pφ

−

+ (1 − p)

1

4

(1 ⊗ 1 + X ⊗ X) (18)

ρ = pψ

+

+ (1 − p)

1

4

(1 ⊗ 1 + Z ⊗ Z) (19)

for 0 ≤ p < 1, have K

∞

≺ Iso(ρ). The states

not on these diagonals remain invariant under the

subgroup K

2

= {±1, ±iX, ±iY, ±iZ}.

The possible isotropy subgroups exhibited by

the T-states can be seen with the projectors P

H

,

as detailed in Appendix C. These are:

• Iso(ρ

T

) = SU(2) ⇒ M(ρ)

∼

=

{e}

Werner states along the central axis of the

T-state tetrahedron, where τ

1

= τ

2

= τ

3

.

These are the states

ρ = (1 − p) ψ

−

+

p

3

(ψ

+

+ φ

+

+ φ

−

) (20)

for 0 ≤ p ≤ 1.

• Iso(ρ

T

) = K

∞

⇒ M(ρ)

∼

=

SO(3)/D

∞

T-states with τ

i

= τ

j

6= τ

k

. These states lie

on or above the diagonals of the triangle of

triplet Bell states, as seen in Fig.5, excluding

the Werner states on the central axis of the

tetrahedron.

Accepted in Quantum 2019-01-23, click title to verify 11

Figure 5: The partitioning of the 2-qubit state space. a) Tetrahedron of T-states coloured to indicate diﬀerent

group orbit shapes: grey – SU(2)/K

2

∼

=

SO(3)/D

2

, light blue – SO(3)/D

∞

, black – {e}. Slices through the

tetrahedron that are parallel to the plane containing the triplet Bell states exhibit the same structure, exhibited

in the triangle to the right. b) Tetrahedron mapping out the smoothed isotropy subgroups with smoothing scale

 = 0.04. The bands along the diagonals indicate Iso



(ρ)

∼

=

K

∞

, while states in the central hexagon are assigned

Iso



(ρ)

∼

=

SU(2). The remaining states retain their K

2

isotropy subgroup. c) Hasse diagram of the observed isotropy

subgroups (up to isomorphism) for two qubits. Arrows indicate subset inclusion. This is a sublattice of the full SU(2)

subgroup lattice.

• Iso(ρ

T

) = K

2

⇒ M(ρ)

∼

=

SO(3)/D

2

T-states with τ

1

6= τ

2

6= τ

3

. These are states

that do not lie on or above the diagonals of

the triangle formed by the triplet Bell states.

The smoothed isotropy subgroups are illustrated

in the lower part of Fig. 5, with a smoothing

scale of  = 0.04 under the trace distance. Under

this smoothing, the horizontal slices through the

tetrahedron are no longer equivalent; near ψ

−

all

states are assigned SU(2) as their isotropy sub-

group, while at the base there remain many states

with Iso



(ρ) = K

2

.

4.3 Maximally Mixed Marginals

The set of 2-qubit states with maximally mixed

marginals is a wider class of states reached

from the T-states through local SU(2) unitaries

U(g

1

) ⊗ U(g

2

). Appendix C details how the

isotropy subgroups are enumerated.

If these local unitaries are the same for each

qubit, we reach states of the form

˜ρ

T

= U(g) ⊗ U(g) ρ

T

U

†

(g) ⊗ U

†

(g) = U

g

(ρ

T

)

=

1

4

1 ⊗ 1 +

3

X

i=1

τ

i

c

i

· σ ⊗ c

i

· σ

!

, (21)

where the vectors {c

i

} form an orthonormal basis

of R

3

.

These states behave similarly to the T-states,

but with isomorphic isotropy subgroups. This

can be seen from the ﬁnal part of Lemma 1, where

Iso(U

g

◦ ρ

T

◦ U

g

−1

) = g[Iso(ρ

T

)]g

−1

. Explicitly,

instead of {±1, ±iX, ±iY, ±iZ}, these rotated

T-states will be invariant under {±1, ±i(c

1

·

σ), ±i(c

2

· σ), ±i(c

3

· σ)}

∼

=

K

2

.

The possible types of group orbit for these

states are the same as for the T-states:

• Iso(˜ρ

T

) = SU(2) ⇒ M(˜ρ

T

)

∼

=

{e}

The same symmetric states as the T-states,

unchanged by the unitary transformation

Accepted in Quantum 2019-01-23, click title to verify 12

U(g) ⊗ U(g). These are states ˜ρ

T

with

τ

1

= τ

2

= τ

3

.

• Iso(˜ρ

T

) = K

∞

⇒ M(˜ρ

T

)

∼

=

SO(3)/D

∞

States ˜ρ

T

with τ

i

= τ

j

6= τ

k

. These are

reached via unitary transformations U(g) ⊗

U(g) from the T-states with K

∞

isotropy

subgroups.

• Iso(˜ρ

T

) = K

2

⇒ M(˜ρ

T

)

∼

=

SO(3)/D

2

States ˜ρ

T

with τ

1

6= τ

2

6= τ

3

. These are

reached via unitary transformations U(g) ⊗

U(g) from the T-states with K

2

isotropy sub-

groups.

Further states with maximally mixed marginals

are reached from the T-states by diﬀerent local

unitaries on each qubit. Suppose that these two

local unitaries have the same generator r · σ, such

that U(g

1

) ⊗ U(g

2

) = e

iθ

1

r·σ

⊗ e

iθ

2

r·σ

. This gives

states of the form

ρ

M

= e

iθ

1

r·σ

⊗ e

iθ

2

r·σ

ρ

T

e

−iθ

1

r·σ

⊗ e

−iθ

2

r·σ

=

1

4

1 ⊗ 1 + τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

!

(22)

where {r, c

2

, c

3

} and {r, d

2

, d

3

} are two sets of

orthonormal bases for R

3

. If U(g

1

) = U(g

2

) in the

local unitaries, these sets of bases will be the same

and we retrieve ˜ρ

T

. However, in the case that

θ

1

6= θ

2

in the local unitaries, new possibilities

for the isotropy subgroup are introduced:

• Iso(ρ

M

) = U(1) ⇒ M(ρ

M

)

∼

=

S

2

States ρ

M

with τ

2

= τ

3

. These are reached

from T-states with K

∞

isotropy subgroups

via local unitaries e

iθ

1

r·σ

⊗e

iθ

2

r·σ

where θ

1

6=

θ

2

.

• Iso(ρ

M

) = Z

4

⇒ M(ρ

M

)

∼

=

SO(3)/C

2

States ρ

M

with τ

2

6= τ

3

. These are reached

from T-states with K

2

isotropy subgroups

via local unitaries e

iθ

1

r·σ

⊗e

iθ

2

r·σ

where θ

1

6=

θ

2

.

The states ˜ρ

T

are only invariant under a K

2

sub-

group because the correlation terms are put into

diagonal form by the same orthonormal bases

{c

i

} and {d

i

} on each qubit. For example, the

correlation terms of the T-states are put into

the canonical form by the standard cartesian or-

thonormal basis {

ˆ

x,

ˆ

y,

ˆ

z} on each qubit. When

the local unitaries are applied, this can break the

K

2

symmetry, and the local bases become ‘mis-

aligned’. However, when c

1

= d

1

= r there is

still a subgroup Z

4

= {±1, ±i(r · σ)} that leaves

these states invariant.

The remaining 2-qubit states with maximally

mixed marginals are reached from the T-states

through local SU(2) unitaries that do not share

a generator, such that U(g

1

) ⊗ U(g

2

) = e

iθ

1

r

1

·σ

⊗

e

iθ

2

r

2

·σ

where |r

1

| = |r

2

| = 1 and r

1

6= r

2

. These

states can be expressed in the form

˜ρ

M

= e

iθ

1

r

1

·σ

⊗ e

iθ

2

r

2

·σ

ρ

T

e

−iθ

1

r

1

·σ

⊗ e

−iθ

2

r

2

·σ

=

1

4

1 ⊗ 1 +

3

X

i=1

τ

i

c

i

· σ ⊗ d

i

· σ

!

, (23)

where {c

i

} and {d

i

} are orthonormal bases of R

3

that do not share any elements, i.e. c

i

6= d

j

for

all i and j. These states exhibit another type of

group orbit:

• Iso(ρ) = Z

2

⇒ M(ρ)

∼

=

SO(3)

States ˜ρ

M

. The orthonormal bases {c

i

} and

{d

i

} do not share any elements, therefore the

only subgroup that leaves these states invari-

ant is Z

2

= {±1}. These are in a sense max-

imally asymmetric states, as they have no

non-trivial residual symmetries.

4.4 Local Bloch Vectors

Finally, the introduction of local Bloch vectors

completes the set of 2-qubit states. In the states

with maximally mixed marginals, the alignment

of the orthonormal bases describing the corre-

lation terms played a key role in determining

the symmetry properties. The eﬀect of the lo-

cal Bloch vectors depends on the how they relate

to each other, as well as how they relate to the

bases for the correlation terms.

There are three possible isotropy subgroups for

these states:

• Iso(ρ) = Z

2

⇒ M(ρ)

∼

=

SO(3)

This will occur for any states

ρ =

1

4



1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ

+

3

X

i=1

τ

i

c

i

· σ ⊗ d

i

· σ

!

(24)

Accepted in Quantum 2019-01-23, click title to verify 13

whose local Bloch vectors a · σ and b · σ are

not parallel or anti-parallel, such that a 6=

γb for any γ ∈ R.

This isotropy subgroup will also occur in

states where the basis in which the correla-

tion terms are diagonalised does not align

with the local Bloch vectors. These are

states of the form

ρ =

1

4



1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ

+τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

!

(25)

where {r, c

2

, c

3

} and {r, d

2

, d

3

} are two sets

of orthonormal bases for R

3

and either a

or b 6= γr for any γ ∈ R. In this case,

the correlation terms would be invariant un-

der Z

4

= {±1, ±ir · σ}, however the lo-

cal Bloch vectors prevent this. Similarly if

τ

2

= τ

3

in states of this form, then the lo-

cal Bloch vectors break a U(1) symmetry,

while if {c

i

} = {d

i

} then the local Bloch

vectors break a K

2

symmetry. Likewise if

both {c

i

} = {d

i

} and τ

2

= τ

3

, the corre-

lation terms are invariant under a K

∞

sub-

group that is again broken by the local Bloch

vectors.

• Iso(ρ) = Z

4

⇒ M(ρ)

∼

=

SO(3)/C

2

States of the form

ρ =

1

4



1 ⊗ 1 + ar · σ ⊗ 1 + b1 ⊗ r · σ

+τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

!

(26)

where {r, c

2

, c

3

} and {r, d

2

, d

3

} are two sets

of orthonormal bases for R

3

, a, b ∈ R and

τ

2

6= τ

3

. The local Bloch vectors are invari-

ant under the same Z

4

subgroup that leaves

the correlation terms invariant. This also in-

cludes the case where c

i

= d

i

for i = 2, 3.

The correlation terms are invariant under a

K

2

subgroup, but the local Bloch vectors

break this to a Z

4

subgroup.

Similarly, states of the form

ρ =

1

4



1 ⊗ 1 + ac

1

· σ ⊗ 1 + b1 ⊗ c

1

· σ

+

3

X

i=1

τ

i

c

i

· σ ⊗ c

i

· σ

!

(27)

where {c

1

, c

2

, c

3

} is an orthonormal basis for

R

3

and τ

1

= τ

2

6= τ

3

. The correlation terms

are invariant under a K

∞

subgroup, but the

local Bloch vectors break the residual sym-

metry down to a Z

4

subgroup as the local

Bloch vectors are orthogonal to c

3

· σ, the

generator of the relevant U(1) subgroup.

• Iso(ρ) = U(1) ⇒ M(ρ)

∼

=

S

2

These are states

ρ =

1

4



1 ⊗ 1 + ar · σ ⊗ 1 + b1 ⊗ r · σ

τ

1

r · σ ⊗ r · σ + τ

3

X

i=2

c

i

· σ ⊗ d

i

· σ

!

(28)

where {r, c

2

, c

3

} and {r, d

2

, d

3

} are two sets

of orthonormal bases for R

3

, a, b ∈ R. The

local Bloch vectors are invariant under the

same U (1) subgroup that leaves the correla-

tion terms invariant. This also includes the

cases when {c

i

} = {d

i

} and when τ

1

= τ,

where the correlation terms are invariant un-

der K

∞

and SU(2) respectively.

This completes the classiﬁcation of the isotropy

subgroups of 2-qubit states.

4.5 Some simple consequences of the classiﬁ-

cation

This classiﬁcation allows us to infer constraints

on the resources required to simulate asymmetric

channels under symmetry constraints. If we wish

to simulate an axial channel (i.e. M(E) = S

2

)

under SU(2) symmetry constraints using the T-

states, the resource state must have M(σ)  S

2

.

T-states on the diagonals of the tetrahedron, with

M(ρ) ≺ SO(3)/D

∞

, will not be able to per-

form such simulations; only the T-states with

τ

1

6= τ

2

6= τ

3

will achieve this task up to some

approximation as discussed earlier.

However, convex combinations of diﬀerent T-

states can create more states that break the sym-

metry to a larger degree. For example, Iso(pφ

+

Accepted in Quantum 2019-01-23, click title to verify 14

(1−p)φ

−

) = K

2

when p 6= 1/2, despite Iso(φ

+

)

∼

=

Iso(φ

−

)

∼

=

K

∞

.

Group orbits also constrain the dynamics of

the T-states. For instance, Lemma 2 implies that

symmetric operations move states situated on a

diagonal of the tetrahedron only along that diag-

onal. The T-states are closed under symmetric

dynamics – leaving would entail further symme-

tries being broken.

Also note that H

1

⊆ H

2

does not imply

there exists a symmetric channel from C(H

1

) into

C(H

2

). T-states on the diagonals of the base

of the tetrahedron cannot be reached from those

that lie oﬀ these diagonals under symmetric op-

erations [25], despite K

2

≺ K

∞

. Group orbits

impose the highest level constraints; further tech-

niques such as those of [25] then impose more de-

tailed constraints.

5 Outlook

Our analysis has shown that the basic prob-

lem of state interconversion under a symmetry

constraint has a rich and non-trivial structure.

We have established a high-level description of

the problem that is consistent with the resource-

theoretic picture, however the question whether

the framework can ﬁnd practical use in the same

way that superselection rules simplify computa-

tion remains to be explored.

We have classiﬁed the set of all two-qubit states

under the SU(2) tensor product representation,

which may be of use in its own right for the study

of quantum correlations. However, beyond two-

qubits, the general problem can be extremely dif-

ﬁcult – for example even the ﬁnite subgroups of

SU(5) remain unclassiﬁed [17]. Despite this com-

plexity, we believe that the tools we have devel-

oped here provide useful insight into the struc-

ture of information processing under symmetry

constraints.

Other techniques for studying the eﬀect of sym-

metry constraints on quantum operations con-

nect with the present analysis. As already men-

tioned, recent work on the harmonic analysis of

quantum channels suggests that the geometry of

group orbits provides an insight into the use of

ﬁnite quantum resources [25]. The present paper

has focused purely on the ‘shape’ of this orbit,

but the role of the induced geometry on the or-

bits is only partially understood. To extend such

an analysis would require adapting techniques in

metrology [15] and diﬀerential geometry [43]. It

would also be of interest to draw connections with

recent work [44] that extends classical coarse-

graining into the quantum regime with symme-

tries present.

Finally, these ideas may inform us about other

resource theories, using the symmetry constraints

directly (e.g. in thermodynamics). Along similar

lines, the techniques we have described may ﬁnd

application in the context of realising universal

quantum computation via the combination of re-

source states with simple gate-sets, for example

Cliﬀord operations and magic states.

6 Acknowledgements

We would like to thank Cristina Cirstoiu, Erick

Hinds Mingo, Zoe Holmes and Markus Frembs

for many useful discussions. TH is funded by

the EPSRC Centre for Doctoral Training in Con-

trolled Quantum Dynamics. DJ is supported by

the Royal Society.

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Accepted in Quantum 2019-01-23, click title to verify 18

A Proofs

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(E ⊗ F)  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 some probability 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

.

Proof. For g ∈ Iso(E) ∧ Iso(F), the global group action gives U

g

(E ⊗ F) = U

g

(E) ⊗ U

g

(F) = E ⊗ F,

therefore g ∈ Iso(E ⊗F). However E ⊗ F may have further symmetries from taking the tensor product

(for example Iso(|0i h0| ⊗ |0i h0|)  Iso(|0i h0|)), therefore Iso(E ⊗ F)  Iso(E) ∧ Iso(F).

The group action on E ◦ F is given by U

g

(E ◦ F) = U

g

◦ E ◦ F ◦ U

g

−1

= U

g

◦ E ◦ U

g

−1

◦ U

g

◦ F ◦ U

g

−1

=

U

g

(E)◦U

g

(F). Therefore if g ∈ Iso(E)∧Iso(F), then g ∈ Iso(E◦F), implying Iso(E◦F)  Iso(E)∧Iso(F).

If g ∈ Iso(E) ∧ Iso(F), then U

g

(pE + (1 − p)F) = p U

g

(E) + (1 − p) U

g

(F) = pE + (1 − p)F, implying

Iso(E) ∧ Iso(F) ≺ Iso(pE + (1 − p)F).

For h ∈ Iso(U

g

◦ E ◦ U

g

−1

), we have U

h

◦ U

g

◦ E ◦ U

g

−1

◦ U

h

−1

= U

g

◦ E ◦ U

g

−1

which simpliﬁes to

U

(g

−1

hg)

◦ E ◦ U

(g

−1

hg)

−1

. Therefore g

−1

hg ∈ Iso(E) or h ∈ g[Iso(E)]g

−1

. When gh

0

g

−1

∈ g[Iso(E)]g

−1

acts upon U

g

◦E ◦U

g

−1

, we have (U

g

◦U

h

0

◦U

g

−1

)◦(U

g

◦E ◦U

g

−1

)◦(U

g

◦U

h

0−1

◦U

g

−1

) = U

g

◦U

h

0

◦E ◦U

h

0−1

◦

U

g

−1

= U

g

◦E◦U

g

−1

, therefore gh

0

g

−1

∈ Iso(U

g

◦E◦U

g

−1

). Therefore Iso(U

g

◦E◦U

g

−1

) = g[Iso(E)]g

−1

.

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). Conversely, if E has isotropy group Iso(E) then there exists a

quantum system B and quantum state σ

B

that can be used to simulate E with Iso(σ

B

) = Iso(E).

Proof. Viewing the state σ

B

as a CPTP map, we can view the simulation as a composite channel

E = F ◦ σ

B

. Therefore Iso(E) = Iso(F ◦ σ

B

)  Iso(F) ∧ Iso(σ

B

) = Iso(σ

B

) since F is symmetric.

Therefore Iso(E)  Iso(σ

B

), so M(σ

B

)  M(E).

Alternatively, from the Covariant Stinespring theorem [45, 46], we can write E(ρ) = F(ρ ⊗ σ

B

) =

tr

BC

[V(ρ ⊗ σ

B

⊗ γ

C

)V

†

], where γ

C

= |ηihη| is some pure state, [V, U

A

(g) ⊗ U

B

(g) ⊗ U

C

(g)] = 0 and

Iso(γ

C

) = G. Supposing g ∈ Iso(σ

B

),

E(ρ) = tr

BC

[V (ρ ⊗ U

g

(σ

B

) ⊗ U

g

(γ

C

))V

†

] = tr

BC

[V (1 ⊗ U

g

⊗ U

g

)(ρ ⊗ σ

B

⊗ γ

C

)V

†

]. (29)

V is a symmetric unitary, so V (1⊗U

g

⊗U

g

) = V (U

g

⊗U

g

⊗U

g

)(U

†

g

⊗1⊗1) = (U

g

⊗U

g

⊗U

g

)V (U

†

g

⊗1⊗1),

giving

E(ρ) = tr

BC

[U

g

(V (U

g

−1

(ρ) ⊗ σ

B

⊗ γ

C

)V

†

)] = U

g

(tr

BC

[V (U

g

−1

(ρ) ⊗ σ

B

⊗ γ

C

)V

†

])

= U

g

◦ E ◦ U

g

−1

(ρ). (30)

Therefore Iso(σ

B

) ≺ Iso(E), and M(σ

B

)  M(E).

We now establish the second part of the theorem, and construct the protocol that realises E with

a quantum state that has the same isotropy subgroup as E. Firstly, we note that we can decompose

any E into irreducible process modes [25] as

E =

X

λ,k

α

λ

∗

,k

(x)Φ

λ,k

(31)

where λ labels irreps, including multiplicities and k labels the basis vector of the irrep. As shown in

[25] the coeﬃcients α

λ,k

(x) are non-normalised harmonic wavefunctions on the space M := G/Iso(E),

Accepted in Quantum 2019-01-23, click title to verify 19

which is diﬀeomorphic to the orbit of E under the group G, and where the point x corresponds to the

reference data needed to simulate E under the symmetry constraint.

For our reference system B, we take its Hilbert space to be L

2

(M, C), where σ

B

= |ψihψ| is a

reference state with wavefunction ψ(z) = δ(z − x), understood in the distributional sense. The

protocol to realise E via σ

B

involves a measurement of the position z of B, which provides the

necessary reference frame data to realise E on the system A. Explicitly, we have

E =

X

λ,k

α

λ

∗

,k

(x) Φ

λ,k

= tr

B

X

λ,k

|xi

B

hx| ⊗ α

λ

∗

,k

(x) Φ

λ,k

=

X

λ,k

Z

dz tr[M(z) σ

B

] α

λ

∗

,k

(z) Φ

λ,k

(32)

where {dzM(z)} is the position measurement on M, which is covariant since G acts transitively on

M. Finally, the action of G on σ

B

is given by the action of G on the quotient space G/Iso(E) where

at each point we have an isotropy group Iso(E), and thus Iso(σ

B

) = Iso(E) as required.

Lemma 2. Under a symmetric operation E, Iso(E(ρ))  Iso(ρ).

Proof. We can view the state ρ as a CPTP map 1 → ρ, therefore from Lemma 1, Iso(E(ρ)) =

Iso(E ◦ ρ)  Iso(E) ∧ Iso(ρ). Since G ∧ H = H for any subgroup H ≺ G, then Iso(E) ∧ Iso(ρ) = Iso(ρ),

therefore Iso(E(ρ))  Iso(ρ), and thus the group orbits also obey M(E(ρ)) ≺ M(ρ).

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

). (33)

Then d(C(H

1

), C(H

2

)) = 0 for all H

1

, H

2

≺ G.

Proof. For a state ρ

1

∈ C(H

1

), U

g

(ρ

1

+ (1 − ) σ) =  U

g

(ρ

1

) + (1 − ) U

g

(σ) =  U

g

(ρ

1

) + (1 − ) σ

when σ ∈ C(G). Hence Iso(ρ

1

+ (1 − )σ) = Iso(ρ

1

), and ρ

1

+ (1 − )σ ∈ C(H

1

). Likewise,

for ρ

2

∈ C(H

2

), the state ρ

2

+ (1 − )σ (with the same σ ∈ C(G)) is also in C(H

2

). As  → 0,

d(ρ

1

+ (1 − )σ, ρ

2

+ (1 − )σ) → 0, therefore d(C(H

1

), C(H

2

)) = 0 for all H

1

, H

2

≺ G.

Lemma 4. The map P

H

has the following properties:

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.

Proof. For states ρ ∈ C(W ) for W  H, the unitary U(h) for h ∈ H will leave the state unchanged.

Therefore P

H

(ρ) =

R

H

dh U

h

(ρ) =

R

H

dh ρ = ρ. Moreover, P

H

is a projector because P

H

◦ P

H

=

R

H

dh

R

H

dh

0

U

h

◦U

h

0

=

R

H

dh

R

H

dh

0

U

hh

0

=

R

H

dh

00

U

h

00

= P

H

. Furthermore, P

†

H

(ρ) =

R

H

dh U

h

−1

(ρ) =

R

H

dh U

h

(ρ) = P

H

(ρ), meaning P

H

is an orthogonal projector.

We follow the proof in [22]. By Klein’s Inequality, we have S(P

H

(ρ)||σ) ≥ 0, with equality iﬀ

σ = P

H

(ρ), therefore min

σ∈

ˆ

C(H)

S(P

H

(ρ)||σ) = 0. From the deﬁnition of relative entropy, S(P

H

(ρ)) =

min

σ∈

ˆ

C(H)

[−Tr(ρ log σ)] because log is analytic and σ ∈

ˆ

C(H). Using the idempotency of P

H

,

S(ρ||P

H

(ρ)) = −S(ρ) − Tr(ρ log P

H

(ρ)) = −S(ρ) + S(P

H

(ρ)) = −S(ρ) + min

σ∈

ˆ

C(H)

[−Tr(ρ log σ)] =

min

σ∈

ˆ

C(H)

S(ρ||σ).

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.

Accepted in Quantum 2019-01-23, click title to verify 20

Proof. The group action on P

H

is U

g

(P

H

) =

R

H

dh U

g

(U

h

) =

R

H

dh U

g

◦ U

h

◦ U

g

−1

=

R

H

dh U

ghg

−1

.

An element g ∈ G will therefore be a member of Iso(P

H

) iﬀ gHg

−1

= H. This is the deﬁnition of the

normalizer, so we have that Iso(P

H

) = N

G

(H). Averaging over a normal subgroup H  G is therefore

symmetric, because N

G

(H) = G by the deﬁnition of a normal subgroup.

B Bell States and Single Component T-States

The extremal points of the T-states are the Bell states, which can be expressed as the vectorisation

[47] of Pauli matrices:



φ

+

E

∝ |vec(1)i (34)



φ

−



∝ |vec(Z)i (35)



ψ

+

E

∝ |vec(X)i (36)



ψ

−



∝ |vec(Y)i . (37)

Elements of the isotropy subgroup of a Bell state therefore satisfy U

g

(|vec(σ

i

)i hvec(σ

i

)|) =

|vec(σ

i

)i hvec(σ

i

)|, hence

U(g) ⊗ U(g) |vec(σ

i

)i = e

iφ

|vec(σ

i

)i (38)

where φ ∈ R is some arbitrary phase. From the identity A ⊗ B |vec(M)i =



vec(AMB

T

)

E

[47], the

residual symmetries satisfy

U(g)σ

i

U

T

(g) = e

iφ

σ

i

. (39)

We now consider each of the Bell states in turn.

B.1 Iso(φ

+

)

∼

=

K

∞



φ

+



∝ |vec(1)i is invariant under group transformations satisfying U(g)U

T

(g) = e

iφ

1. SU(2) group

elements take the form U (g) = exp(iϕ[aX +bY +cZ]) with a

2

+b

2

+c

2

= 1 (a, b, c ∈ R) and 0 ≤ ϕ < 2π,

therefore

e

iϕ(aX+bY +cZ)

e

iϕ(aX−bY +cZ)

= e

iφ

1 =⇒ e

iϕ(aX+bY +cZ)

= e

iφ

e

iϕ(−aX+bY −cZ)

. (40)

Expansion of the matrix exponentials gives

cos ϕ1 + i sin ϕ(aX + bY + cZ) = e

iφ

[cos ϕ1 + i sin ϕ(−aX + bY − cZ)] (41)

which provides the conditions

cos ϕ(1 − e

iφ

) = 0 (42)

ia sin ϕ(1 + e

iφ

) = 0 (43)

ib sin ϕ(1 − e

iφ

) = 0 (44)

ic sin ϕ(1 + e

iφ

) = 0. (45)

The problem simpliﬁes because φ ∈ R is restricted in the values it may take. Suppose cos ϕ 6= 0, then

cos ϕ(1 − e

iφ

) = 0 implies φ = 0 (modulo 2π). Now suppose cos ϕ = 0, ensuring that sin ϕ 6= 0. Since

a

2

+ b

2

+ c

2

= 1, at least one of a, b or c is non-zero, therefore either 1 + e

iφ

= 0 or 1 − e

iφ

= 0 in order

for the conditions to hold. Together, these imply that φ = 0 or π (modulo 2π).

When φ = 0, the conditions reduce to

2ia sin ϕ = 0 (46)

2ic sin ϕ = 0. (47)

Accepted in Quantum 2019-01-23, click title to verify 21

One possible solution is a = c = 0, leaving ϕ unconstrained and therefore b = ±1 (without loss of

generality choose b = 1, since b = −1 gives ‘rotations’ of the opposite handedness). These solutions

take the form U(g) = e

iϕY

. The equations are also satisﬁed when sin ϕ = 0, so when ϕ = 0, π, which

give group transformations U (g) = ±1.

In the φ = π case, the conditions become

2 cos ϕ = 0 (48)

2ib sin ϕ = 0. (49)

If b 6= 0, then cos ϕ = sin ϕ = 0, which is not possible. Therefore the only solutions for these conditions

is b = 0 and cos ϕ = 0, which gives ϕ = π/2 or 3π/2. In this case a and c are only constrained by

a

2

+ c

2

= 1. The ϕ = π/2 solution corresponds to the SU(2) transformations

U(g) = i(aX + cZ) = (iZ)(c1 + iaY ) = (iZ)(cos θ1 + i sin θY ) = (iZ)e

iθY

, (50)

since a

2

+ c

2

= 1, so we parametrise in terms of cos θ and sin θ. Similarly U (g) = (−iZ)e

iθY

for the

ϕ = 3π/2 case. This shows that

Iso(φ

+

) = {(iZ)

α

e

iθY

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

. (51)

B.2 Iso(φ

−

) = K

∞

Analogous arguments hold for |φ

−

i ∝ |vec(Z)i. The condition for membership of Iso(φ

−

) is

e

iϕ(aX+bY +cZ)

Ze

iϕ(aX−bY +cZ)

= e

iφ

Z (52)

which simpliﬁes to

e

iϕ(aX+bY +cZ)

= e

iφ

e

iϕ(aX−bY −cZ)

. (53)

This gives the conditions

cos ϕ(1 − e

iφ

) = 0 (54)

ia sin ϕ(1 − e

iφ

) = 0 (55)

ib sin ϕ(1 + e

iφ

) = 0 (56)

ic sin ϕ(1 + e

iφ

) = 0, (57)

solved in the same way as the previous example. The isotropy subgroup is

Iso(φ

−

) = {(iY)

α

e

iθX

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

. (58)

B.3 Iso(ψ

+

) = K

∞



ψ

+



∝ |vec(X)i gives the condition

e

iϕ(aX+bY +cZ)

= e

iφ

e

iϕ(−aX−bY +cZ)

. (59)

This is satisﬁed when

cos ϕ(1 − e

iφ

) = 0 (60)

ia sin ϕ(1 + e

iφ

) = 0 (61)

ib sin ϕ(1 + e

iφ

) = 0 (62)

ic sin ϕ(1 − e

iφ

) = 0, (63)

which gives the isotropy subgroup as

Iso(ψ

+

) = {(iX)

α

e

iθZ

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

. (64)

Accepted in Quantum 2019-01-23, click title to verify 22

B.4 Iso(ψ

−

) = SU(2)

The singlet Bell state |ψ

−

i ∝ |vec(Y)i behaves diﬀerently to the other Bell states. The isotropy

subgroup condition is

e

iϕ(aX+bY +cZ)

= e

iφ

e

iϕ(aX+bY +cZ)

, (65)

which is satisﬁed for all elements of SU(2), therefore Iso(ψ

−

) = SU(2).

B.5 Iso(

1

4

(1 ⊗ 1 + τ

i

σ

i

⊗ σ

i

) = K

∞

For states, ρ =

1

4

(1 ⊗ 1 + τ

i

σ

i

⊗ σ

i

), the 1 ⊗ 1 term can be neglected because Iso(1 ⊗ 1 + ρ

0

) = Iso(ρ

0

).

For ρ =

1

4

(1 ⊗ 1 + τ

1

X ⊗ X), an SU (2) group element belongs to Iso(ρ) when

U

g

(X ⊗ X) = U

g

(X) ⊗ U

g

(X) = X ⊗ X, (66)

satisﬁed when

e

iϕ(aX+bY +cZ)

Xe

−iϕ(aX+bY +cZ)

= ±X, (67)

or equivalently

e

iϕ(aX+bY +cZ)

= ±e

iϕ(aX−bY −cZ)

. (68)

This is the same condition as for ψ

+

, therefore

Iso(X ⊗ X) = Iso(ψ

+

) = {(iY)

α

e

iθX

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

. (69)

Likewise, elements of Iso(Y ⊗ Y) satisfy

e

iϕ(aX+bY +cZ)

= ±e

iϕ(−aX+bY −cZ)

, (70)

the same as for φ

+

, therefore

Iso(Y ⊗ Y) = {(iZ)

α

e

iθY

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

, (71)

while the condition for membership of Iso(Z ⊗ Z) is

e

iϕ(aX+bY +cZ)

= ±e

iϕ(−aX+bY −cZ)

, (72)

showing that

Iso(Z ⊗ Z) = {(iX)

α

e

iθZ

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}

∼

=

K

∞

. (73)

C 2-Qubit Isotropy Subgroups

The isotropy subgroups of 2-qubit states under a tensor product SU(2) group action can be found

with projection operators P

H

. These do not indicate which subgroups of SU (2) will appear as isotropy

subgroups, however the following lemma restricts the possibilities.

Lemma 6. For a 2-qubit state ρ transforming under an SU(2) tensor product representation, any

cyclic subgroup of the isotropy subgroup Iso(ρ) must be either Z

2

, Z

4

or U(1).

Proof. A group element belongs to Iso(ρ) when U

g

(ρ) = ρ, or in vectorised form [47],

U(g) ⊗ U(g) ⊗ U

∗

(g) ⊗ U

∗

(g) |vec(ρ)i = |vec(ρ)i . (74)

For ρ to be invariant under the group transformation U

g

, it requires |vec(ρ)i to be in ker[U(g)⊗U(g) ⊗

U

∗

(g) ⊗ U

∗

(g) − 1].

Accepted in Quantum 2019-01-23, click title to verify 23

All groups contain cyclic subgroups from repeated application of a single generator, because g ∈ G

implies that g

n

∈ G. For any cyclic subgroup of Iso(ρ), there is some basis in which its representation

is diagonal, i.e. U (g) = e

iθ

|0i h0| + e

−iθ

|1i h1|. Therefore

U(g) ⊗ U(g) ⊗ U

∗

(g) ⊗ U

∗

(g) − 1

= (e

2iθ

− 1)(|1i h1| + |2i h2| + |7i h7| + |11i h11|) + (e

4iθ

− 1) |3i h3| + (e

−4iθ

− 1) |12i h12|

+ (e

−2iθ

− 1)(|4i h4| + |8i h8| + |13i h13| + |14i h14|), (75)

where we have assumed for simplicity that U(g) is diagonal in the computational basis (with for

example |2i = |0010i). Vectors in span(|0i , |5i , |6i , |9i , |10i , |15i) are invariant under this U(1)

subgroup. Vectors in span(|1i , |2i , |4i , |7i , |8i , |11i , |13i , |14i) are also in ker[U(g) ⊗ U(g) ⊗ U

∗

(g) ⊗

U

∗

(g) − 1] for θ = 0, π, and are therefore invariant under a Z

2

subgroup. Finally, the vectors of

span(|3i , |12i) are in ker[U(g) ⊗ U(g) ⊗ U

∗

(g) ⊗ U

∗

(g) − 1] when θ = 0, π/2, π, 3π/2, and hence

invariant under a Z

4

subgroup.

This holds for any cyclic subgroup of SU(2), therefore any cyclic subgroup of Iso(ρ) must be Z

2

, Z

2

or U(1).

This plays a similar role to the crystallographic restriction theorem in crystallography [1], and

reduces the number of subgroups to check. For example, any 2-qubit state invariant under a Z

6

must

be invariant under the U(1) subgroup with the same generator. This allows us to enumerate the

possible isotropy subgroups for 2-qubit states. It also suggests that an N-qubit state will have only

cyclic subgroups Z

2

, Z

4

, . . . Z

2N

, U(1) within Iso(ρ).

The channels P

H

provide a way to ﬁnd the sets C(H). Consider two subgroups H

1

and H

2

, with

H

1

≺ H

2

and there are no subgroups H that could be the isotropy subgroup of some state such that

H

1

≺ H ≺ H

2

. If a state ρ satisﬁes P

H

1

(ρ) = ρ but P

H

2

(ρ) 6= ρ, then Iso(ρ) = H

1

, and ρ ∈ C(H

1

).

We now use this technique to identify the isotropy subgroups of all 2-qubit states.

C.1 Z

2

≺ Iso(ρ)

All 2-qubit states are symmetric under Z

2

= {±1}, conﬁrmed by

P

Z

2

(ρ) =

1

2

[ U

1

(ρ) + U

−1

(ρ) ] = ρ (76)

for any 2-qubit state ρ.

C.2 Z

4

≺ Iso(ρ)

From Lemma 6, if a 2-qubit state has Z

3

≺ Iso(ρ), then it also has U(1) ≺ Iso(ρ). The next subgroup

to check is Z

4

= {±1, ±ir · σ}. For illustrative purposes, let us consider the particular Z

4

subgroup

{±1, ±iZ}. For a general 2-qubit state ρ,

P

Z

4

(ρ) =

1

4





1 ⊗ 1 + P

Z

4

(a · σ ⊗ 1) + P

Z

4

(1 ⊗ b · σ) +

3

X

i,j=1

T

ij

P

Z

4

(σ

i

⊗ σ

j

)





(77)

=

1

4





1 ⊗ 1 + a

z

Z ⊗ 1 + b

z

1 ⊗ Z + T

33

Z ⊗ Z +

2

X

i,j=1

T

ij

σ

i

⊗ σ

j





. (78)

Now consider the more general Z

4

= {±1, ±ir · σ} subgroup. The useful identities

(r · σ)(v · σ) = (r · v)1 + i(r × v) · σ (79)

(r · σ)(v · σ)(r · σ) = 2(r · v)(r · σ) − v · σ (80)

Accepted in Quantum 2019-01-23, click title to verify 24

allow us to calculate P

Z

4

(ρ). The local Bloch vectors give

P

Z

4

(a · σ ⊗ 1) =

1

2

[a · σ ⊗ 1 + (r · σ)(a · σ)(r · σ) ⊗ 1] (81)

=

1

2

[a · σ ⊗ 1 + (2(r · a)(r · σ) − a · σ) ⊗ 1] (82)

= (r · a)(r · σ) ⊗ 1 (83)

and similarly P

Z

4

(1 ⊗ b · σ) = (r · b)(1 ⊗ r · σ). The correlation terms may be written

3

X

i,j=1

T

ij

σ

i

⊗ σ

j

=

3

X

i,j=1

T

0

ij

c

i

· σ ⊗ c

j

· σ, (84)

where {c

i

} is an orthonormal basis for R

3

and we choose c

1

= r. It is clear that P

Z

4

(c

1

· σ ⊗ c

1

· σ) =

P

Z

4

(r · σ ⊗ r · σ) = r · σ ⊗ r · σ. The identity

(r · σ)(u · σ)(r · σ) ⊗ (r · σ)(v · σ)(r · σ) = 4(r · u)(r · v)r · σ ⊗ r · σ − 2(r · u)r · σ ⊗ v · σ

− 2(r · v)u · σ ⊗ r · σ + u · σ ⊗ v · σ. (85)

allows us to show that when j 6= 1,

P

Z

4

(c

1

· σ ⊗ c

j

· σ) = c

1

· σ ⊗ c

j

· σ + 2(r · c

1

)(r · c

j

)r · σ ⊗ r · σ − (r · c

1

)r · σ ⊗ c

j

· σ − (r · c

j

)c

1

· σ ⊗ r · σ

= r · σ ⊗ c

j

· σ − r · σ ⊗ c

j

· σ = 0 (86)

because r · c

j

= c

1

· c

j

= 0. The same holds for the c

j

· σ ⊗ c

1

· σ terms with j 6= 1, which also vanish

when P

Z

4

is applied. Finally, when i, j 6= 1,

P

Z

4

(c

i

· σ ⊗ c

j

· σ) = c

i

· σ ⊗ c

j

· σ + 2(r · c

i

)(r · c

j

)r · σ ⊗ r · σ − (r · c

i

)r · σ ⊗ c

j

· σ − (r · c

j

)c

i

· σ ⊗ r · σ

= c

i

· σ ⊗ c

j

· σ. (87)

Therefore

P

Z

4

(ρ) =

1

4





1 ⊗ 1 + (r · a)(r · σ) ⊗ 1 + (r · b)(1 ⊗ r · σ) + T

0

11

r · σ ⊗ r · σ +

3

X

i,j=2

T

0

ij

c

i

· σ ⊗ c

j

· σ





=

1

4

"

1 ⊗ 1 + (r · a)(r · σ) ⊗ 1 + (r · b)(1 ⊗ r · σ) + τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

#

,

(88)

where {r, c

i

} and {r, d

i

} are orthonormal bases of R

3

.

C.3 Iso(ρ) = Z

2

The image of P

Z

4

forms a proper subset of the image of P

Z

2

, therefore there exist 2-qubit states with

Iso(ρ) = Z

2

, and these arise in several ways:

• States of the form

ρ =

1

4

"

1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ +

3

X

i=1

τ

i

c

i

· σ ⊗ d

i

· σ

#

, (89)

where {c

i

} and {d

i

} are orthonormal bases of R

3

and c

i

6= d

i

for i = 1, 2, 3.

Accepted in Quantum 2019-01-23, click title to verify 25

• States of the form

ρ =

1

4

"

1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ + τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

#

, (90)

where {r, c

i

} and {r, d

i

} are orthonormal bases of R

3

but a and/or b are not parallel or anti-

parallel to r, i.e. there are no γ

1

, γ

2

∈ R such that both a = γ

1

r and b = γ

2

r.

• States of the form

ρ =

1

4





1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ +

3

X

i,j=1

T

ij

σ

i

⊗ σ

j





, (91)

where a 6= γb for any γ ∈ R, so the local Bloch vectors are neither parallel nor anti-parallel to

each other.

C.4 U(1) ≺ Iso(ρ)

The next subgroup to consider is U (1). We apply P

U(1)

to states P

Z

4

(ρ), because if Z

4

⊀ Iso(ρ) then

U(1) ⊀ Iso(ρ). For illustrative purposes, consider the U(1) subgroup {e

iθZ

: 0 ≤ θ < 2π}, which

contains Z

4

= {±1, ±iZ}. The states invariant under this U(1) subgroup are

P

U(1)

(ρ) = P

U(1)

◦ P

Z

4

(ρ) (92)

=

1

4





1 ⊗ 1 + a

z

Z ⊗ 1 + b

z

1 ⊗ Z + T

33

Z ⊗ Z +

2

X

i,j=1

T

ij

P

U(1)

(σ

i

⊗ σ

j

)





(93)

because the terms Z ⊗ 1, 1 ⊗ Z and Z ⊗ Z are certainly invariant under the U(1) subgroup generated

by Z. For the remaining correlation terms,

P

U(1)

(X ⊗ X) =

1

2π

Z

2π

0

dθ e

iθZ

Xe

−iθZ

⊗ e

iθZ

Xe

−iθZ

(94)

=

1

2π

Z

2π

0

dθ cos

2

2θ X ⊗ X + sin

2

2θ Y ⊗ Y − cos 2θ sin 2θ (X ⊗ Y + Y ⊗ X) (95)

=

1

2

(X ⊗ X + Y ⊗ Y ) = P

U(1)

(Y ⊗ Y ) (96)

and similarly P

U(1)

(X ⊗ Y ) = −P

U(1)

(Y ⊗ X) =

1

2

(X ⊗ Y − Y ⊗ X). Together these give

P

U(1)

(ρ) =

1

4



1 ⊗ 1 + a

z

Z ⊗ 1 + b

z

1 ⊗ Z + T

33

Z ⊗ Z +

1

2

(T

11

+ T

22

)(X ⊗ X + Y ⊗ Y )

+

1

2

(T

12

− T

21

)(X ⊗ Y − Y ⊗ X)



(97)

Now consider the more general U (1) = {e

ir·σ

: 0 ≤ θ < 2π}. This acts on a Bloch vector as

e

iθr·σ

(u · σ)e

−iθr·σ

= cos 2θ u · σ + sin 2θ (u × r) · σ + 2 sin

2

θ (r · u) r · σ, (98)

therefore

P

U(1)

(a · σ ⊗ 1) =

1

2π

Z

2π

0

dθ e

iθr·σ

(a · σ)e

−iθr·σ

⊗ 1 (99)

=

1

2π

Z

2π

0

dθ cos 2θ a · σ ⊗ 1 + sin 2θ (a × r) · σ ⊗ 1 + 2 sin

2

θ (r · a) r · σ ⊗ 1

= (r · a) r · σ ⊗ 1 (100)

Accepted in Quantum 2019-01-23, click title to verify 26

and similarly P

U(1)

(1 ⊗ b · σ) = (r · b) 1 ⊗ r · σ. The group action on the correlation terms is

e

iθr·σ

(u · σ)e

−iθr·σ

⊗ e

iθr·σ

(v · σ)e

−iθr·σ

= cos

2

2θ u · σ ⊗ v · σ + 4 sin

4

θ(r · u)(r · v)r · σ ⊗ r · σ + 2 cos 2θ sin 2θ [u · σ ⊗ (v × r) · σ

+ (u × r) · σ ⊗ v · σ] + 2 cos 2θ sin

2

θ[(r · v)u · σ ⊗ r · σ + (r · u)r · σ ⊗ v · σ]

+ 2 sin 2θ sin

2

θ [(r · v)(u × r) · σ ⊗ r · σ + (r · u)r · σ ⊗ (v × r) · σ]

+ sin

2

2θ(u × r) · σ ⊗ (v × r) · σ. (101)

This gives

P

U(1)

(c

i

· σ ⊗ c

j

· σ) =

1

2π

Z

2π

0

dθ e

iθr·σ

(c

i

· σ)e

−iθr·σ

⊗ e

iθr·σ

(c

j

· σ)e

−iθr·σ

(102)

=

1

2

[ c

i

· σ ⊗ c

j

· σ − (r · c

j

)c

i

· σ ⊗ r · σ − (r · c

i

)r · σ ⊗ c

j

· σ

+ (c

i

× r) · σ ⊗ (c

j

× r) · σ + 3(r · c

i

)(r · c

j

)r · σ ⊗ r · σ ] (103)

=

1

2

[ c

i

· σ ⊗ c

j

· σ − δ

1j

c

i

· σ ⊗ c

1

· σ − δ

1i

c

1

· σ ⊗ c

j

· σ

+ (c

i

× c

1

) · σ ⊗ (c

j

× c

1

) · σ + 3 δ

1i

δ

1j

c

1

· σ ⊗ c

1

· σ], (104)

therefore

P

U(1)

(ρ) =

1

4



1 ⊗ 1 + (r · a) r · σ ⊗ 1 + (r · b) 1 ⊗ r · σ + T

0

11

r · σ ⊗ r · σ

+

1

2

(T

0

22

+ T

0

33

) [c

2

· σ ⊗ c

2

· σ + c

3

· σ ⊗ c

3

· σ]

+

1

2

(T

0

23

− T

0

32

) [c

2

· σ ⊗ c

3

· σ − c

3

· σ ⊗ c

2

· σ]



(105)

=

1

4

"

1 ⊗ 1 + (r · a) r · σ ⊗ 1 + (r · b) 1 ⊗ r · σ + τ

1

r · σ ⊗ r · σ + τ

3

X

i=2

c

i

· σ ⊗ d

i

· σ

#

(106)

where {r, c

i

} and {r, d

i

} are orthonormal bases of R

3

.

C.5 K

2

≺ Iso(ρ)

Consider one such subgroup, K

2

= {±1, ±iX, ±iY, ±iZ}. For this subgroup,

P

K

2

(ρ) =

1

4





1 ⊗ 1 + P

K

2

(a · σ ⊗ 1) + P

K

2

(1 ⊗ b · σ) +

3

X

i,j=1

T

ij

P

K

2

(σ

i

⊗ σ

j

)





. (107)

Local Bloch vectors are eliminated, since P

K

2

(X ⊗ 1) = P

K

2

(Y ⊗ 1) = P

K

2

(Z ⊗ 1) = 0, and likewise

for the other local Bloch vector. For the correlation terms,

P

K

2

(X ⊗ X) =

1

4

[X ⊗ X + XXX ⊗ XXX + Y XY ⊗ Y XY + ZXZ ⊗ ZXZ] = X ⊗ X, (108)

and similarly P

K

2

(Y ⊗ Y ) = Y ⊗ Y and P

K

2

(Z ⊗ Z) = Z ⊗ Z. The remaining correlation terms vanish

since

P

K

2

(X ⊗ Y ) =

1

4

[X ⊗ Y + XXX ⊗ XY X + Y XY ⊗ Y Y Y + ZXZ ⊗ ZY Z] (109)

=

1

4

[X ⊗ Y − X ⊗ Y − X ⊗ Y + X ⊗ Y ] = 0 (110)

Accepted in Quantum 2019-01-23, click title to verify 27

and similarly for other σ

i

⊗ σ

j

terms with i 6= j. Therefore

P

K

2

(ρ) =

1

4

"

1 ⊗ 1 +

3

X

i=1

T

ii

σ

i

⊗ σ

i

#

, (111)

which are the T-states. This proves that all T-states ρ

T

have K

2

≺ Iso(ρ

T

).

Consider the more general K

2

subgroup, {±1, ±ir

1

· σ, ±ir

2

· σ, ±ir

3

· σ} where {r

1

, r

2

, r

3

} forms an

orthonormal basis of R

3

. We use that same basis for the correlation terms, such that 2-qubit states

can be expressed

ρ =

1

4





1 ⊗ 1 + a · σ ⊗ 1 + 1 ⊗ b · σ +

3

X

i,j=1

T

0

ij

r

i

· σ ⊗ r

j

· σ





. (112)

We ﬁrst consider the action of P

K

2

on the local Bloch vectors, a · σ =

P

3

i=1

(r

i

· a) r

i

· σ,

P

K

2

(a · σ ⊗ 1) =

1

4



a · σ ⊗ 1 +

3

X

i=1

(r

i

· σ)(a · σ)(r

i

· σ) ⊗ 1



(113)

=

1

4



a · σ ⊗ 1 +

3

X

i=1

(2(r

i

· a)r

i

· σ − a · σ) ⊗ 1



(114)

=

1

2

"

3

X

i=1

(r

i

· a) r

i

· σ ⊗ 1 − a · σ ⊗ 1

#

= 0, (115)

and similarly P

K

2

(1 ⊗ b · σ) = 0. The correlation terms give

P

K

2

(r

i

· σ ⊗ r

j

· σ) =

1

4

"

r

i

· σ ⊗ r

j

· σ +

3

X

k=1

(r

k

· σ)(r

i

· σ)(r

k

· σ) ⊗ (r

k

· σ)(r

j

· σ)(r

k

· σ)

#

(116)

=

1

4

"

r

i

· σ ⊗ r

j

· σ +

3

X

k=1



4(r

k

· r

i

)(r

k

· r

j

)r

k

· σ ⊗ r

k

· σ + r

i

· σ ⊗ r

j

· σ

− 2(r

k

· r

i

)r

k

· σ ⊗ r

j

· σ − 2(r

k

· r

j

)r

i

· σ ⊗ r

k

· σ



=

1

4

"

3

X

k=1

(4δ

ik

δ

jk

r

k

· σ ⊗ r

k

· σ − 2δ

ik

r

k

· σ ⊗ r

j

· σ − 2δ

jk

r

i

· σ ⊗ r

k

· σ) + 4r

i

· σ ⊗ r

j

· σ

#

= δ

ij

r

j

· σ ⊗ r

j

· σ (117)

Therefore

P

K

2

(ρ) =

1

4

"

1 ⊗ 1 +

3

X

i=1

τ

i

r

i

· σ ⊗ r

i

· σ

#

(118)

where {r

i

} is an orthonormal basis for R

3

.

C.6 Iso(ρ) = Z

4

We have now calculated P

H

(ρ) for the two subgroups directly above Z

4

on the subgroup lattice that

are possible isotropy subgroups. This identiﬁes the 2-qubit states in C(Z

4

). They are:

• States of the form

ρ =

1

4



1 ⊗ 1 + τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ



(119)

where τ

2

6= τ

3

and c

i

6= d

i

for i = 2, 3.

Accepted in Quantum 2019-01-23, click title to verify 28

• States of the form

ρ =

1

4

"

1 ⊗ 1 + ar · σ ⊗ 1 + b1 ⊗ r · σ + τ

1

r · σ ⊗ r · σ +

3

X

i=2

τ

i

c

i

· σ ⊗ d

i

· σ

#

(120)

where τ

2

6= τ

3

and c

i

6= d

i

for i = 2, 3, and a, b ∈ R.

• The states

ρ =

1

4

"

1 ⊗ 1 + ac

1

· σ ⊗ 1 + b1 ⊗ c

1

· σ +

3

X

i=1

τ

i

c

i

· σ ⊗ c

i

· σ

#

(121)

where a, b ∈ R. This gives a Z

4

isotropy subgroup when τ

1

= τ

2

6= τ

3

.

C.7 K

∞

≺ Iso(ρ)

The next subgroup to consider is K

∞

, which contains the K

2

subgroup. Therefore we need only apply

P

K

∞

to states P

K

2

(ρ). We can also say the same about states P

U(1)

(ρ), however the K

2

subgroup

projects onto a simpler subset of states.

As an illustrative example, consider the particular subgroup {(iX)

α

e

iθZ

: 0 ≤ θ < 2π, α = 0, 1, 2, 3}.

This contains the K

2

subgroup {±1, ±iX, ±iY, ±iZ}, so we need only apply P

K

∞

to the T-states. The

CPTP map P

K

∞

can be split up as P

K

∞

= P

U(1)

◦ P

Z

4

, therefore

P

K

∞

(ρ

T

) = P

U(1)

(ρ

T

) =

1

4



1 ⊗ 1 +

1

2

(τ

1

+ τ

2

) (X ⊗ X + Y ⊗ Y ) + τ

3

Z ⊗ Z



(122)

More generally, we consider

K

∞

= {(i r

j

· σ)

α

e

iθ r

k

·σ

: 0 ≤ θ < 2π, α = 0, 1, 2, 3} (123)

where {r

i

} is an orthonormal basis of R

3

and j 6= k. The projector P

K

∞

on a general 2-qubit state

gives

P

K

∞

(ρ) = P

U(1)

◦ P

K

2

(ρ) = P

U(1)

"

1

4

1 ⊗ 1 +

3

X

i=1

τ

i

r

i

· σ ⊗ r

i

· σ

!#

(124)

=

1

4





1 ⊗ 1 +

1

2





X

i6=k

τ

i









X

i6=k

r

i

· σ ⊗ r

i

· σ





+ τ

k

r

k

· σ ⊗ r

k

· σ





(125)

C.8 Iso(ρ) = K

2

These are the T-states and the states with maximally mixed marginals that can be reached from the

T-states through rigid SU(2) rotations:

ρ =

1

4



1 ⊗ 1 +

X

i

τ

i

c

i

· σ ⊗ c

i

· σ



, (126)

with τ

1

6= τ

2

6= τ

3

. There cannot be any local Bloch vectors.

C.9 Iso(ρ) = U(1)

These are states of the form

ρ =

1

4

"

1 ⊗ 1 + τ

1

r · σ ⊗ r · σ + τ

3

X

i=2

c

i

· σ ⊗ d

i

· σ

#

(127)

Accepted in Quantum 2019-01-23, click title to verify 29

where c

i

6= d

i

for i = 2, 3. Local Bloch vectors are permitted so long as they are either aligned or

anti-aligned with the shared Bloch vector in the correlation terms:

ρ =

1

4

"

1 ⊗ 1 + ar · σ ⊗ 1 + b1 ⊗ r · σ + τ

1

r · σ ⊗ r · σ + τ

3

X

i=2

c

i

· σ ⊗ d

i

· σ

#

(128)

where c

i

6= d

i

for i = 2, 3 and a, b ∈ R.

C.10 Iso(ρ) = SU(2)

Since G = SU(2), when we apply P

SU (2)

to a general 2-qubit state we get the form of a general 2-qubit

symmetric state. Since K

∞

≺ SU(2),

P

SU (2)

(ρ) = P

SU (2)

◦ P

K

∞

(ρ) =

1

4

1 ⊗ 1 +

3

X

i=1

τ

i

P

SU (2)

[c

i

· σ ⊗ c

i

· σ]

!

. (129)

A general SU (2) group transformation is U (g) = e

iϕr·σ

. We parameterise r =

(sin φ cos θ, sin φ sin θ, cos φ). Considering each of the terms individually,

P

SU (2)

(c

i

· σ ⊗ c

i

· σ)

=

1

2π

2

Z

2π

0

dθ

Z

π

0

dφ sin φ

Z

π

0

dϕ sin

2

ϕ



e

iϕr·σ

c

i

· σe

−iϕr·σ

⊗ e

iϕr·σ

c

i

· σe

−iϕr·σ



(130)

=

1

2π

2

Z

2π

0

dθ

Z

π

0

dφ sin φ

Z

π

0

dϕ sin

2

ϕ cos

2

2ϕ c

i

· σ ⊗ c

i

· σ

+ 2 sin

2

ϕ cos 2ϕ sin 2ϕ [c

i

· σ ⊗ (c

i

× r) · σ + (c

i

× r) · σ ⊗ c

i

· σ]

+ 2 cos 2ϕ sin

4

ϕ (r · c

i

) [c

i

· σ ⊗ r · σ + r · σ ⊗ c

i

· σ]

+ 2 sin 2ϕ sin

4

ϕ (r · c

i

) [(c

i

× r) · σ ⊗ r · σ + r · σ ⊗ (c

i

× r) · σ]

+ 4 sin

6

ϕ (r · c

i

)

2

r · σ ⊗ r · σ + sin

2

ϕ sin

2

2ϕ (c

i

× r) · σ ⊗ (c

i

× r) · σ (131)

=

1

8π

Z

2π

0

dθ

Z

π

0

dφ sin φ



c

i

· σ ⊗ c

i

· σ − 2(r · c

i

) [c

i

· σ ⊗ r · σ + r · σ ⊗ c

i

· σ]

+ (c

i

× r) · σ ⊗ (c

i

× r) · σ + 5(r · c

i

)

2

r · σ ⊗ r · σ



(132)

Performing the integrals over θ and φ gives

P

SU (2)

(ρ) =

1

4



1 ⊗ 1 + τ

3

X

i=1

σ

i

⊗ σ

i



. (133)

C.11 Iso(ρ) = K

∞

These are states in the image of P

K

∞

but not in that of P

SU (2)

. They have the form

ρ =

1

4

"

1 ⊗ 1 +

X

i

τ

i

c

i

· σ ⊗ c

i

· σ

#

(134)

where τ

1

= τ

2

6= τ

3

i.e. two of the τ

i

are the same, but not equal to the third.

Accepted in Quantum 2019-01-23, click title to verify 30

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