Group-covariant extreme and quasi-extreme channels Laleh Memarzadeh1and Barry C. Sanders2y 1Department of Physics Sharif University of Technology Tehran 11365-9161 Iran

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Group-covariant extreme and quasi-extreme channels

Laleh Memarzadeh1, ∗and Barry C. Sanders2, †

1Department of Physics, Sharif University of Technology, Tehran 11365-9161, Iran

2Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada

(Dated: October 10, 2022)

Constructing all extreme instances of the set of completely positive trace-preserving (CPTP)

maps, i.e., quantum channels, is a challenging and valuable open problem in quantum information

theory. Here we introduce a systematic approach that despite the lack of knowledge about the

full parametrization of the set of CPTP maps on arbitrary Hilbert-spaced dimension, enables us to

construct exactly those extreme channels that are covariant with respect to a ﬁnite discrete group or

a compact connected Lie group. Innovative labeling of quantum channels by group representations

enables us to identify the subset of group-covariant channels whose elements are group-covariant

generalized-extreme channels. Furthermore, we exploit essentials of group representation theory to

introduce equivalence classes for the labels and also partition the set of group-covariant channels.

As a result we show that it is enough to construct one representative of each partition. We construct

Kraus operators for group-covariant generalized-extreme channels by solving systems of linear and

quadratic equations for all candidates satisfying the necessary condition for being group-covariant

generalized-extreme channels. Deciding whether these constructed instances are extreme or quasi-

extreme is accomplished by solving system of linear equations. Proper labeling and partitioning the

set of group-covariant channels leads to a novel systematic, algorithmic approach for constructing

the entire subset of group-covariant extreme channels. We formalize the problem of constructing and

classifying group-covariant generalized extreme channels, thereby yielding an algorithmic approach

to solving, which we express as pseudocode. To illustrate the application and value of our method, we

solve for explicit examples of group-covariant extreme channels. With unbounded computational re-

sources to execute our algorithm, our method always delivers a description of an extreme channel for

any ﬁnite-dimensional Hilbert-space and furthermore guarantees a description of a group-covariant

extreme channel for any dimension and for any ﬁnite-discrete or compact connected Lie group if

such an extreme channel exists.

∗memarzadeh@sharif.edu

†sandersb@ucalgary.ca; http://iqst.ca/people/peoplepage.php?id=4

arXiv:2210.03449v1 [quant-ph] 7 Oct 2022

I. INTRODUCTION

Quantum channels, which are completely positive trace-preserving linear maps belonging to the set of linear maps

on Banach spaces of operators, represent the most general allowed form of quantum dynamics [1–3] and they form

a convex subset. Characterizing quantum channels is important for representing general dynamics and for modeling

decoherence in quantum systems [4]. Based on such characterizations, eﬃcient simulation of quantum dynamics

becomes feasible. Another importance of characterizing the whole set of quantum channels is to describe general

quantum communication channels and to analyze rates of reliable classical and quantum information that can be

communicated through quantum channels [5, 6].

As full characterization of convex sets is feasible by just knowing the extreme points of a convex set, full channel

characterization can be accomplished by determining only the small subset comprising all extreme channels. The

conundrum is that extreme channels can be determined by knowing the convex set and vice versa. For quantum

channels acting on d-dimensional Hilbert-space, necessary and suﬃcient conditions for a channel to be extreme are

described in [7] using similar arguments to those presented in [8]. For qubit channels, full characterization of extreme

points is known [9–12]. On the other hand, for Hilbert-space dimension d > 2 (qubit channels), neither extreme

channels nor full characterizations of the set of quantum channels are known

Here we advance the understanding and characterization of extreme channels by treating channels restricted by

symmetries, which are speciﬁed by ﬁnite discrete or compact connected Lie groups. Compactness ensures that the

connected Lie group has ﬁnite-dimensional representations. We exploit this symmetry to construct exact forms of

extreme channels. Speciﬁcally, for any d, we develop an algorithmic approach to derive exactly a subset of extreme

points of the set of channels that have certain speciﬁed symmetries.

Only special types of channels have been fully characterized to date: a set of qubit channels including all extreme

points [9–12] and some extreme points for the set of unital channels [13, 14], which are quantum counterparts of

classical bistochastic processes. Studying unital channels is useful for investigating similarities between classical

and quantum processes, such as establishing a quantum version of Birkhoﬀ’s Theorem [13] and proving additivity

or superadditivity of quantities relevant to the communication capacity of channels [15–17]. Fully characterizing

quantum channels is quite challenging, which necessitates tackling restricted cases such as unitality.

Characterizing quantum channels has an important practical application to quantum simulation [18–20]. Quantum

simulation is an important application of quantum computing and typically is studied for Hamiltonian-generated

unitary evolution [19, 21–25], but, as general evolution is described by quantum channels, a fully developed theory

of quantum simulation could be based on simulating quantum channels. Whereas Hamiltonian simulation exploits

notions such as the Solovay-Kitaev theorem [26, 27] for gate decomposition and sparseness of Hamiltonians [22, 23], di-

rect quantum simulation of channels is challenging. However some progress has been made by exploiting knowledge of

extreme channels. Decomposing the single-qubit channel has been explored theoretically [28] and experimentally [29]

and exploits properties of extreme channels. Extreme channels are valuable as well for qudit-channel decomposi-

tion [30], including for dimension-altering channels [31], and for decomposing m-qubit to n-qubit channels [32]. This

latter result [32] emphasizes the importance of their constructive approach to channel decomposition, which guarantees

success of the channel-decomposition procedure. Those authors contrast their constructive approach to the qudit-

channel decomposition approach [30], which is provably not guaranteed to succeed based on an insuﬃcient number

of parameters. These theoretical [28, 31, 32] and experimental [29] advances point to the importance of determining

extreme channels to make quantum-channel simulation eﬃcient or at least tractable.

Solving for extreme channels is currently restricted to particular examples of unital channels, whereas our goal is to

establish a systematic, algorithmic approach to constructing extreme channels that are group-covariant [33], whether

unital or not. Beginning with the name of a ﬁnite-discrete group or a compact connected Lie group and d, we exploit

the possibility of being able to look up all inequivalent irreducible representations (irreps) of the group in order to be

able to construct all possible inequivalent d-dimensional representations of the group by direct sums of inequivalent

group irreps. For any two d-dimensional inequivalent representations of the group and any inequivalent group irrep

with dimension less than or equal to d, we solve a set of linear equations, obtained by applying the group-covariance

constraint, to construct Kraus representations of corresponding group-covariant generalized extreme channels. Our

method exploits the full power of representation theory; if we employed the obvious brute-force approach instead,

we would be solving a set of linear equations for all pairs of d-dimensional representations and for all d2-dimensional

representations of the given group, which is an uncountably inﬁnite number of candidates; instead, provided the

representation theory for the group is known, our approach yields only a ﬁnite number of candidates, making our

approach feasible algorithmically. Once we have identiﬁed these candidates, we then test if the obtained generalized

extreme channel satisﬁes the constraint for being extreme. This constraint is expressed as a system of linear equations

whose solution reveals whether the obtained generalized extreme channel is extreme or quasi-extreme.

Our systematic, algorithmic approach is described by a pseudocode that we deﬁne for this purpose. Typically,

pseudocode serves as a convenient way of representing the logical ﬂow of a program for implementation on a standard,

i.e., Turing-like, computer, but our pseudocode is quite diﬀerent: serving as a representation of the logical ﬂow for

our mathematical approach. Thus, we make it clear that, formally, our pseudocode applies to a real-number model

of computing; this model enables us to be rigorous with respect to the logic of our systematic, algorithmic approach

to solving group-covariant extreme and quasi-extreme channels.

We begin by presenting a full background to our work in §II, including state of the art and methods, and then we

proceed to describe our approach in §III. Our results are presented and fully explained in §IV followed by a discussion

of these results in §V. Finally, we conclude in §VI including an outlook on outstanding problems and potential future

work.

II. BACKGROUND

In this section we summarize the pertinent literature and provide basic concepts required for subsequent sections.

We begin by discussing quantum channels in §II A, including their Kraus and Choi representations. §II B is devoted

to group-covariant channels and the constraints on the Kraus operators of group-covariant quantum channels.

A. Quantum channels

In this subsection ﬁrst we review the deﬁnition of quantum channels. We then review Kraus representation and Choi

matrix representations for quantum channels. Following that, we recall the deﬁnition of speciﬁc subsets of quantum

channels, namely extreme channels, generalized-extreme channels, and quasi-extreme channels.

1. Channel representation

For Ha complex ﬁnite-dimensional Hilbert space and L(H) the space of linear operators acting on H, density

operators {ρ}are positive trace-class operators on H; i.e., they belong to the subset of L(H) denoted by

T(H) = {ρ∈ L(H)|ρ≥0,tr(ρ)=1}.(1)

For our purposes, the trace of the density operator is unity. A quantum channel is any completely positive trace-

preserving map Φ : T(H)→ T (H)

For Hrestricted to ﬁnite dimension d, i.e.,

d:= dim H,(2)

every quantum channel can be expressed as

Φ(•) =

k=1

Ak•A†

k,•∈T(H), K ≤d2.(3)

This expression is subject to the trace-preserving constraint

Ξ = 1(4)

for

Ξ :=

k=1

A†

kAk,(5)

which can be non-diagonal. Here the nonzero linear operators Ak∈ L(H) are called Kraus operators with each Kraus

operator expressible as a d×dcomplex matrix whose entries are

(Ak)ij := hei|Ak|eji ∈ C, i + 1, j + 1 ∈[d] := {1,2, . . . , d},(6)

for {|eii} an orthonormal basis of ﬁnite-dimensional H, where igoes from 0 to d−1. In summary, Φ can be represented

by the set {Ak}k∈[K]with each Akcomprising d2complex-valued matrix elements. Therefore, the channel is described

by up to Kd2complex-valued parameters.

Remark 1.For d= 0, the only vector in Hilbert space is zero, which has zero norm. Hence there is no allowed state

in this case. For d= 1, only one normalized state exists, which forms the normal basis for the Hilbert space. Kraus

operators of this channel are proportional to the projector onto the basis of the space satisfying the trace-preserving

condition.

The set of Kraus operators describing a map Φ is unique up to an isometry [27]. For a given map Φ, the minimum

number of Kraus operators, called the Choi rank, equals the rank of the Choi operator

CΦ:= 1

d(Φ ⊗1) (|ΨihΨ|),|Ψi:= 1

√d

d−1

i=0 |ei, eii(7)

and

(CΦ)mn,pq =hem, en|CΦ|ep, eqi=1

dhem|Φ(|eniheq|)|epi(8)

is the Choi matrix. Typically, the operator (7) is called the Choi matrix, but, due to our algorithmic approach,

we need to be extra careful in distinguishing operators from their matrix representations and we denote matrices of

size m×nwith entries drawn from the ﬁeld Fby Mm×n(F). Choi showed that a minimal set of Kraus operators can

be obtained from the eigenvectors of the Choi matrix with non-zero eigenvalus [8].

2. Extreme channels

The set of quantum channels SΦis convex and thus has extreme points.

Deﬁnition 1.Extreme points of the convex set SΦare called extreme channels. Extreme channels are channels that

cannot be written as a convex combination of any other two distinct channels in a non-trivial way.

We denote the set of extreme channels by SΦext ⊂SΦ. Despite the importance of extreme channels, characterization

of extreme channels is unknown except for the special case of d= 2 [9–11]. Results beyond d= 2 are restricted to

characterizing extreme points of the set of unital channels (Φ(1)≡1), which are not necessarily extreme points of

the set of all channels [13].

An important theorem on extreme channels speciﬁes necessary and suﬃcient conditions for a channel to be an

extreme one [8]:

Theorem 1.A channel represented by a set of Kraus operators {Ak}k∈[K]is extreme if and only if (iﬀ) the set of

operators

S:= {A†

kAl}k,l∈[K](9)

is linearly independent [7, 8].

Thus, the number Kof Kraus operators of an extreme channel has upper bound

K≤d. (10)

Therefore, the Choi rank of an extreme channel is bounded by d.

Clearly not all channels with Choi rank satisfying inequality (10) are extreme channels, but such channels are

interesting as well [11, 34]. The fact that other channels are interesting leads to deﬁning two further important

subsets of quantum channels. One subset is known as generalized-extreme channels and the other subset is known as

quasi-extreme channels, which we now deﬁne.

Deﬁnition 2.Channels with Choi rank not exceeding dare called generalized-extreme channels [11] and the set of

generalized-extreme channels is denoted by SΦgen

Deﬁnition 3.Generalized-extreme channels that are not extreme are called quasi-extreme channels [11], and the set

of quasi-extreme channels is denoted by SΦqe

Remark 2.Extreme and quasiextreme channels are mutually exclusive: SΦgen ∩SΦqe =∅.

B. Group-covariant channels

In this subsection, we elaborate on a speciﬁc class of channels, namely group-covariant channels. First we recall the

deﬁnition of equivalent channels. Based on this deﬁnition we explain that a group-covariant channel is a channel (3)

with the additional property that the channel’s action is invariant under pre- and post-unitary conjugations that

are described by group representations. Then we explain the constraints on the Kraus operators of group-covariant

channels and how two group-covariant channels under the same group, with respect to equivalent representations of

the group, are equivalent. Group-covariant channels have been studied in the context of channel capacity [35–39],

extreme points of unital channels [13], and channel characterization [40–42], and complementarity and additivity

properties of various covariant channels are discussed in [43].

Deﬁnition 4.A Channel Φ is unitarily equivalent to channel Φ0, denoted by Φ ∼Φ0, if there exist d-dimensional

unitary operators Uand Vsuch that

Φ∼Φ0:Φ=TU◦Φ0◦ TV(11)

where ◦denotes composition of maps and

TQ:L(H)→ L(H) : • 7→ Q•Q†, Q ∈ L(H).(12)

The equivalence relation ∼(11) partitions the set of all quantum channels for given Hilbert-space dimension dinto

equivalence classes of channels. Any quantum channel is equivalent to itself for U=V=1. However, for some

channels, each channel is equivalent to itself even if Uand Vare not identity operators. These channels, with

symmetric properties, are group-covariant channels deﬁned as below.

Deﬁnition 5.For a ﬁnite discrete group or a compact connected Lie group denoted G, with a pair of unitary repre-

sentations [33]

D(1), D(2) ∈ L(H),(13)

a channel Φ is group-covariant with respect to representations D(1), D(2) if

TD(2)(g)◦Φ◦ TD(1)(g)= Φ ∀g∈ G (14)

where TQis deﬁned in Eq. (12).

Suppose channel Φ, represented by Kraus operators {Ak}k∈[K](3), is group-covariant with respect to two d-

dimensional unitary representations of the group G, namely, D(1) and D(2). Then Eq. (14) implies that

D(2)†(g)AkD(1)(g) =

l=1

Ωkl(g)Al,∀k∈[K],∀g∈ G,(15)

subject to the trace-preserving condition (4), for Ω a unitary representation of the group Gon any K-dimensional

unitary space [35], [40]. The dimension of Ω is equal to the number of Kraus operators describing the channel [40].

Remark 3.The group-covariant channel Φ (14) with respect to D(1) and D(2) is not necessarily unique and each

distinct Φ should be suitably labelled. The role of Ω (15) is to label distinct group-covariant channels with respect

to D(1) and D(2); i.e. we can write ΦΩ.

Remark 4.If Ω is a reducible representation of the group, then the group-covariant channel with Kraus operators

satisfying Eq. (15) is a convex combination of other channels that are also group-covariant with respect to D(1)

and D(2) [40].

Remark 5.Let Φ (3) be group-covariant with respect to representations D(1) and D(2). Let D0(1) and D0(2) be two

representations of the same group. Suppose these two representations are respectively unitarily equivalent to D(1)

and D(2) so

UiD(i)(g)U†

i=D0(i)(g),∀g∈ G,(16)

for {Ui}i∈[2] d-dimensional unitary operators. Then channel Φ0, described by Kraus operators {A0

k=U2AkU†

1}k∈[K],

is group-covariant with respect to representations D0(i)s [40]

Φ0=TU2◦Φ◦ TU†

1.(17)

That is according to Deﬁnition 4, these channels are unitarily equivalent.

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Group-covariantextremeandquasi-extremechannelsLalehMemarzadeh1,andBarryC.Sanders2,y1DepartmentofPhysics,SharifUniversityofTechnology,Tehran11365-9161,Iran2InstituteforQuantumScienceandTechnology,UniversityofCalgary,Calgary,AlbertaT2N1N4,Canada(Dated:October10,2022)Constructingallextremeinstancesoft...

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Group-covariant extreme and quasi-extreme channels Laleh Memarzadeh1and Barry C. Sanders2y 1Department of Physics Sharif University of Technology Tehran 11365-9161 Iran

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