Complexity and Capacity Bounds for Quantum Channels Rupert H

Home , Quantum channel

Complexity and capacity bounds for quantum channels Rupert H. Levene∗, Vern I. Paulsen∗, and Ivan G. Todorov‡ ∗School of Mathematics and Statistics, University College Dublin, Belﬁeld, Dublin 4, Ireland †Institute for Quantum Computing and Dept. of Pure Math., University of Waterloo, Waterloo, Ontario, Canada ‡Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom

Abstract—We generalise some well-known graph parameters Thus, we are lead to define the quantum complexity γ( ) of S to operator systems by considering their underlying quantum an operator subsystem of Mn as the least positive integer channels. In particular, we introduce the quantum complexity k for which there existsS a quantum channel Φ: M M as the dimension of the smallest co-domain Hilbert space a n → k such that = Φ. quantum channel requires to realise a given operator system as S S its non-commutative confusability graph. We describe quantum The goal of this paper is to study this and other, closely re- complexity as a generalised minimum semidefinite rank and, in lated, measures of complexity and to derive their relationships the case of a graph operator system, as a quantum intersection with various measures of capacity for classical and quantum number. The quantum complexity and a closely related quantum channels. We will show, in particular, that the measures of version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and complexity we introduce give upper bounds on the zero-error we construct examples for which these bounds beat the best capacity of a quantum channel. previously known general upper bound for the capacity of One of the most useful general bounds on the Shannon quantum channels, given by the quantum Lovasz´ theta number. capacity of a classical channel comes from ϑ, the Lovasz´ theta function [Lo79]. While, for classical channels, the complexity based bound is outperformed by the Lovasz´ number (see [Lo79, Theorem 11]), we will show that there exist quantum I.INTRODUCTION channels for which the quantum complexity bound on capacity we suggest is better than the bounds arising from the non- In 1956, Shannon [Sha56] initiated zero-error information commutative analogue ϑ of the Lovasz´ number introduced theory, introducing the concept of the confusability graph G in [DSW13]. In fact, we will show that there exist quantum of a finite input-output information channel : X YN. channels Φk for k Nefor which the ratio of the quantum The vertex set of G is the input alphabetN X, and→ two ∈ N Lovasz´ theta number ϑ(Φk) to the quantum complexity γ(Φk) vertices form an edge in G if they can result in the same introduced herein is arbitrarily large, while the upper bound symbol from the output alphabetN Y after transmission via γ(Φk) for the quantume Shannon zero-error capacity Θ(Φk) is . Shannon showed that the zero-error behaviour of , and N N accurate to within a factor of two (see Corollary V.3). various measures of its capacity, depend only on the graph We will see that the classical complexity of a graph G G . In particular, if two channels have the same confusability N is a familiar parameter which coincides with its intersection graphs, then they have the same one-shot zero-error capacity number (provided G lacks isolated vertices). For operator and the same Shannon capacity. It is not hard to see that systems, the measure of quantum complexity we propose has every graph with vertex set X is the confusability graph not been previously studied. We will characterise it in several of some—and in fact many—information channels. Thus, a different ways. Since every graph G gives rise to a canonical arXiv:1710.06456v1 [quant-ph] 17 Oct 2017 natural measure of the complexity of a graph G on X is the operator system G, it can in addition be endowed with a minimal cardinality of Y over all realisations of G as the quantum complexity,S which can be strictly smaller than its confusability graph of a channel : X Y . N → classical counterpart, and can be equivalently characterised as Similarly, Duan, Severini and Winter [DSW13] showed that a quantum intersection number of G. every quantum channel Φ: Mn Mk, where Mm denotes the set of complex m m matrices,→ has an associated non- The paper is organised as follows. In SectionII, we begin by × commutative confusability graph Φ, which they defined as a recalling the graph theoretic parameters needed in the sequel S certain operator subsystem of Mn. In the case Φ is a classical and show that our measure of the classical complexity of a channel, Φ coincides with the graph operator system of graph coincides with its intersection number. In Section III, S GΦ (see [DSW13, equation (3)]). As in Shannon’s case, they we turn to the quantum complexity of a graph, and show proved that many natural measures of the quantum capacity that it coincides with its minimum semi-definite rank (modulo of such a channel depend only on the operator system . It is any isolated vertices). In SectionIV, we achieve a parallel S again the case that every operator subsystem of Mn arises as development for operator systems, considering simultaneously the non-commutative confusability graph of potentially many a closely related notion of subcomplexity that coincides with quantum channels. the quantum chromatic number introduced in [Sta16]. We 2 show that our operator system parameters are genuine ex- and the quantum subcomplexity tensions of the graph theoretic ones (Theorem IV.10) and k n β(G) = min k N: G(x) G for some x (C 0 ) explore similarities and differences between their behaviour { ∈ ⊆ ∈ \{ } } on commutative and non-commutative graphs. In SectionV, = min γ(H): H is a subgraph of G ; { } we establish the bounds on capacities in terms of complexities (c) the intersection number int(G) of G, given by (Theorem V.1) and show by example that these bounds can improve dramatically on the Lovasz´ ϑ bound. Finally, in int(G) = min k N: G(x) = G { ∈ AppendixA we establish the partial ordering among various k n for some x (R+ 0 ) . bounds on the quantum Shannon zero-error capacity, from this ∈ \{ } } paper and elsewhere. Remark II.1. The graph parameters defined above are well In the sequel, we employ standard notation from linear known, and will be generalised to non-commutative graphs in SectionIV. algebra: we denote by Mk,n the space of all k by n matrices with complex entries, and set Mn = Mn,n. We let X be (i) The independence number α(G) is standard in graph theory the operator norm of a matrix X M , so that kX k2 is [GR01]. ∈ k,n k k the largest eigenvalue of X∗X. We equip Mn with the inner (ii) Writing Gc for the complement of G, we have β(G) = product given by X,Y = tr(Y ∗X), where tr(Z) is the trace ξ(Gc) where ξ is the orthogonal rank (see, for exam- h i of a matrix Z Mn. We write In (or simply I) for the identity ple, [SS12]). The parameter γ(G) is the minimum vector ∈ matrix in Mn. The positive cone of Mn (that is, the set of rank of G in the terminology of [JMN08], and is equal to all positive semi-definite n by n matrices) will be denoted msr(G) + iso(G) where msr(G) is the classical minimum + + + k | | by M ; if Mn, we let = M . We write R+ semidefinite rank of G [FH13], [HPRS15] and iso(G) is the n S ⊆ S S ∩ n for the cone of all vectors in Rk with non-negative entries, set of isolated vertices of G. and let k be the standard basis of k. If n, (ei)i=1 C v, w C (iii) Let int (G) be the set-theoretic intersection number of G; we denote by the rank one operator on n given∈ by st vw∗ C thus, int (G) is the smallest positive integer m for which , n. The cardinality of a set will st (vw∗)(z) = z, w v z C S there exist non-empty sets R [m], i = 1, . . . , n, such that be denoted byh i. ∈ i S i j if and only if R R ⊆= . (Note that usually in the | | 'G i ∩ j 6 ∅ literature one relaxes the assumption that the sets Ri be non- II.GRAPH PARAMETERS empty [MM99]; however, it is more convenient for us to work In this section, we recall some graph theoretic parameters with the definition above.) We claim that int(G) = intst(G). Indeed, first suppose that iso(G) = , and let m = intst(G). and point out their connection with Shannon’s confusability ∅ graphs and channel capacities. We start by establishing nota- Choose Ri [m] for i [n], so that i G j Ri R = . Note⊆ that, since∈iso(G) = , we' have R⇐⇒= for∩ tion and terminology. Unless otherwise stated, all graphs in j 6 ∅ ∅ i 6 ∅ every i. Defining xi = er for i [n], we have that xi this paper will be simple graphs: undirected graphs without r Ri k ∈ ∈ ∈ R+, i [n], and G(x) = G; hence int(G) k. Conversely, loops and at most one edge between any pair of vertices. Let ∈ P ≤ n and let G be a graph with vertex set [n] := 1, . . . , n . suppose that x = (x1, . . . , xn) is a tuple of non-zero vectors N k ∈ { } in R+ such that G(x) = G. Let Ri = l [k]: xi, el = 0 . For i, j [n] we write i j or i G j to denote non-strict { ∈ h i 6 } adjacency:∈ either i = j, or'G contains' the edge ij. We denote The non-negativity of the entries of xi, i = 1, . . . , n, implies by Gc the complement of the graph G; by definition, Gc has that i G j if and only if Ri Rj = ; thus, intst(G) int(G) and so'int (G) = int(G). ∩ 6 ∅ ≤ vertex set [n] and, for distinct i, j [n], we have i c j if st ∈ 'G and only if i Gc j. For graphs H,G with vertex set [n], we It is straightforward from the definitions that write H G6', and say that H is a subgraph of G, if every ⊆ α(G) β(G) γ(G) int(G) (1) edge of H is an edge of G. An independent set in G is a ≤ ≤ ≤ subset of its vertices between which there are no edges of G. for every graph G. We note that these inequalities will be gen- Let k N, and for an n-tuple x = (x1, . . . , xn), where each eralised to arbitrary operator systems in Mn in Theorem IV.4 ∈k xi C is a non-zero vector, we define G(x) to be the non- below. orthogonality∈ graph of x, with vertex set [n] and adjacency relation given by We now review some of Shannon’s ideas [Sha56]. Suppose that we have a finite set X, which we view as an alphabet that i G(x) j xj, xi = 0. we wish to send through a noisy channel in order to obtain ' ⇐⇒ h i 6 symbols from another alphabet, say Y . WeN let p(y x) denote Let G be a graph with vertex set [n]. Consider the following the probability that, if we started with the symbol| x X, graph parameters: then after this process, the symbol y Y is received.∈ We (a) the independence number α(G) of G, given by require that every x X is transformed∈ into some y Y , that ∈ ∈ is, y Y p(y x) = 1, for all x X. The column-stochastic α(G) = max S : S is independent in G ; matrix ∈ = (p| (y x)), indexed by∈Y X, is often referred to as {| | } P N | × (b) the quantum complexity the noise operator of the channel. We will write : X Y to indicate the matrix (p(y x)), and refer to suchN matrices→ as k n | γ(G) = min k N: G(x) = G for some x (C 0 ) (classical) channels. The confusability graph of is the graph { ∈ ∈ \{ } } N 3

G with vertex set X for which, given two distinct x, x0 X, Fix n N. Given k N, we write (k) for the set N ∈ ∈ ∈ P the pair xx0 is an edge if and only if there exists y Y such of n-tuples P = (P1,...,P ) where each P is a non- ∈ n i that p(y x)p(y x0) > 0. Equivalently, x x0 if and only zero projection in M . Let (k) denote the subset of (k) | | 'GN k Pc P if there exists y Y such that the symbols x and x0 can be consisting of the elements P = (P1,...,P ) with commuting ∈ n transformed into the same y via and hence confused. entries: PiPj = PjPi for all i, j. To any P (k) we The one-shot zero-error capacityN of , denoted α( ), is associate the non-orthogonality graph G(P ) with vertex∈ P set [n] N N defined to be the cardinality of the largest subset X1 of X such and edges defined by the relation that, whenever an element of X1 is sent via , no matter N i j P P = 0. which element of Y is received, the receiver can determine 'G(P ) ⇐⇒ i j 6 with certainly the input element from X1. It is straightforward We define the quantum intersection number qint(G) of a graph that α( ) = α(G ). G with vertex set [n] by letting N N Definition II.2. Let G be a graph with vertex set [n]. The qint(G) = min k N: G(P ) = G for some P (k) . complexity plex(G) of G is the minimal cardinality of a set { ∈ ∈ P } Y such that G = G for some channel :[n] Y . The next proposition explains the choice of terminology. N N → If : X Y is a channel, we set plex( ) = plex(G ) Proposition III.1. Let G be a graph with vertex set [n]. Then and callN it this→ parameter the complexity of N. N N int(G) = min k N: G(P ) = G for some P c(k) . Proposition II.3. Let G be a graph with vertex set [n]. Then { ∈ ∈ P }(2) plex(G) = int(G). In other words, if : X Y is a channel then plex( ) = int(G ). N → Proof. Let l be the minimum on the right hand side of (2), and N N suppose that G(x) = G for some n-tuple x of non-zero vectors Proof. Let :[n] Y be a channel so that G = G . For k N → N in R+. Letting Pi be the orthogonal projection onto the linear 1 i n set Ri = y Y : p(y i) = 0 . Then Ri is non- ≤ ≤ { ∈ | 6 } span of er : xi, er = 0 yields a tuple P = (P1,...,Pn) empty for each i and i G j Ri Rj = . This shows { h i 6 } ∈ c(k) with G(P ) = G; thus, l int(G). that int(G) is a lower bound' for⇐⇒ the complexity∩ 6 ∅ of G. P ≤ Conversely, suppose that P = (P1,...,Pn) c(l) is such Conversely, suppose that R1,...,Rn are non-empty subsets ∈ P that G(P ) = G. Simultaneously diagonalising the Pi’s with of [k] such that i G j if and only if Ri Rj = Set respect to a basis b : r [l] and defining R = r ' ∩ 6 ∅ { r ∈ } i { ∈ [l]: Pibr = 0 , i [n], we see that i G j if and only if 1/ Ri , y Ri 6 } ∈ ' p(y i) = | | ∈ ; Ri Rj = , and it follows that int(G) l. | (0, y / Ri ∩ 6 ∅ ≤ ∈ n n Let t = (t1, . . . , tn) N , and write t = ti. then = (p(y i)) is a channel from [n] to [k] with G = G. ∈ | | i=1 N | N Extending ideas from [HPRS15], let This shows that int(G) is an upper bound for the complexity P of G. (k, t) = (P1,...,P ) (k): rank P = t , i [n] P { n ∈ P i i ∈ } Remark II.4. (i) We will discuss later (Remark IV.11) the and define natural way to view a classical channel as a quantum + channel; we will see that the quantum complexityN of , mt (G) = min k N: G(P ) = G for some P (k, t) . N { ∈ ∈ P } studied in SectionIV, coincides with γ(G ). Consider each element A M as a block matrix A = N t c ∈ | | (ii) It is well-known that β(G) χ(G ) for any graph G (here [Ai,j]i,j [n] where Ai,j Mti,tj . We define ∈ ∈ χ(H) denotes the chromatic number≤ of a graph H [GR01]), so c + + it is natural to ask if χ(G ) fits into chain of inequalities (1). t (G) = A M t : Ai,j = 0 i j, F ∈ | | 6 ⇔ ' In fact, it does not: one can check using a computer program n c and rank(A ) = t for each i [n] that χ(G ) γ(G) for all graphs on 7 or fewer vertices, but i,i i ∈ this inequality≤ fails in general, for example if is a Kochen- x and o Specker set and G = G(x) (see [HPSWM11, Section 1.2]). + + (iii) For each π α, β, γ, int , we have that π(G) = 1 if t (G) = A t (G): Ai,i = Iti for each i [n] . ∈ { } H ∈ F ∈ and only if G is a complete graph. Indeed, if G is a complete + + + + We write (G) = 1 (G) and (G) = 1 (G), where graph, then π(G) int(G) = 1, so π(G) = 1; and if G is 1 F . NoteF that in [HPRS15H ], + H and + ≤ c = (1, 1,..., 1) mt (G) t (G) not a complete graph, then G contains at least one edge, so were defined in the special case where t = (r, r, . . .H , r) for π(G) α(G) > 1. some r N. ≥ ∈ III.THEQUANTUMINTERSECTIONNUMBER Proposition III.2. Let G be a graph with vertex set [n] and let t Nn. Then In this section we show that the graph parameter γ, dis- ∈ cussed in SectionII, has a reformulation in terms of projective + + mt (G) = min rank A : A t (G) colourings of the graph G, which leads to a parameter that we ∈ F + = min rank B : B (G) . call the quantum intersection number of G. This will allow a ∈ Ht key step in the proof of Theorem IV.10, where we show that Proof. The proof is an adaptation of [HPRS15, Theorem 3.10]. γ has a natural operator system generalisation. Suppose first that A +(G) and rank A = k. Then there ∈ Ft 4

exists a matrix X Mk, t such that A = X∗X. Write X = Note that rank A = rank Wε k. Let us write Wε = ∈ | | ≤ [X1 X2 Xn], where Xi Mk,ti , i = 1, . . . , n. We have [W1 W2 Wn] and A = [Ai,j]i,j [n], with block sizes given ··· ∈ ··· ∈ by s = (t1 1, t2, . . . , tn). Note that if 2 i, j n, then rank Xi = rank(Xi∗Xi) = rank(Aii) = ti, i [n]. − ≤ ≤ ∈ Wi = Xi and Wj = Xj, hence Ai,j = Wi∗Wj = Xi∗Xj = Writing P = (P1,...,P ) where P M is the orthogonal Bi,j = 0 i j. Let i [n]. We have n i ∈ k 6 ⇐⇒ ' ∈ projection onto the range of Xi, we have P (k, t). A = X∗(Y + εZ ) = X∗Y + εX∗Z . (3) Additionally, ∈ P i,1 i 1 1 i 1 i 1

If i 1, then Xi∗Y1 is a submatrix of Xi∗X1 = 0, and Xi∗Z1 P P = 0 X∗X = 0 A = 0 i j, 6' i j 6 ⇐⇒ i j 6 ⇐⇒ i,j 6 ⇐⇒ ' is the matrix with every column equal to the first column of so G(P ) = G. Hence Xi∗X1 = 0. Hence if i 1, then Ai,1 = 0 = A1,i. Now 6' suppose that i 1. Since Xi∗Y1 and Xi∗Z1 are submatrices m+(G) min rank A : A +(G) . ' t ≤ ∈ Ft of X∗Y1 and X∗Z1, respectively, by (3) and our choice of ε, + + if Xi∗Y1 has any non-zero entry, then the corresponding entry Since t (G) t (G), the inequality H ⊆ F of Ai,1 is also non-zero. On the other hand, if Xi∗Y1 = 0, + + min rank A : A (G) min rank B : B (G) then since i 1 yields X∗X1 = 0, we must have X∗Z1 = 0, ∈ Ft ≤ ∈ Ht ' i 6 i 6 hence A 1 = 0; since A = A∗, we also have A1 = 0. This holds trivially. i, 6 ,i 6 shows that for any ε > 0, the matrix A = Aε satisfies Now suppose that P = (P1,...,P ) (k, t) with n ∈ P G(P ) = G, and for each i [n] let Xi Mk,ti be a Ai,j = 0 i j, for any i, j [n]. matrix whose columns form an∈ orthonormal∈ basis for the 6 ⇐⇒ ' ∈ Since Wε Y as ε 0, we see that Aε converges to the range of Pi. Define X = [X1 X2 Xn] Mk, t and → → + ··· ∈ | | matrix B with the first row and column removed; in particular, let B = X∗X M t , so that rank B = rank X k. ∈ | | ≤ the top left (t1 1) (t1 1) block of Aε converges to It1 1. Note that the ti tj block Bi,j coincides with Xi∗Xj. Since − × − 1 1 − × Hence by choosing ε (0, 2 ab− ) sufficiently small, we may i j PiPj = 0 Xi∗Xj = 0, we have ∈ + ' ⇐⇒ 6 ⇐⇒ 6 ensure that A1,1 has rank t1 1, hence A s (G). By Bij = 0 i j. Moreover, the condition on the columns +− ∈ F + 6 ⇐⇒ ' + Proposition III.2, we have ms (G) rank A k = mt (G). of Xi implies that Bi,i = Iti for each i. So B t (G), ≤ ≤ hence ∈ H + + min rank B : B t (G) mt (G). ∈ H ≤ IV. QUANTUMCHANNELSANDOPERATORSYSTEM Theorem III.3. For any graph G, we have qint(G) = γ(G). PARAMETERS

Proof. Directly from the definitions, we have We recall that an operator subsystem of Mn is a subspace + + Mn such that I and X = X∗ . In this qint(G) = min m (G) m1(G) = γ(G). S ⊆ ∈ S ∈ S ⇒ ∈ S n t t N ≤ paper we will sometimes refer to such a self-adjoint unital ∈ subspace M simply as an operator system; we refer the Let t n. We claim that if t 2 and s = (t n N 1 1 reader toS [Pau02 ⊆ ] for the general theory of operator systems 1, t , t ∈, . . . , t ), then m+(G) m+(G≥). By symmetry and in-− 2 3 n s t and completely bounded maps. A linear map Φ: M M is duction, this yields γ(G) = m+≤(G) m+(G) for any t n, n k 1 t N called a quantum channel if it is completely positive and→ trace- hence qint(G) = γ(G). ≤ ∈ + preserving. By theorems of Choi and Kraus, Φ is a quantum To establish the claim, suppose that t1 2, let k = mt (G), ≥ + channel if and only if there exists m N and matrices and use Proposition III.2 to choose B t (G) with m ∈ ∈ H A1,...,Am Mk,n, satisfying i=1 Ai∗Ai = In, so that rank B = k. We may write B = X∗X where X Mk, t . ∈ ∈ | | m Write X = [X1 X2 Xn] where Xi Mk,ti , let P ··· ∈ Φ(X) = A XA∗,X M . Y Mk, t 1 be X with the first column deleted and let i i ∈ n ∈ | |− i=1 Z1 Mk,t1 1 be the matrix with every column equal to the X ∈ − first column of X. Let Z = [Z1 0 ... 0] Mk, t 1. This realisation of Φ is called a Choi-Kraus representation ∈ | |− Let Y1 Mk,t1 1 consist of the first (t1 1) columns of Y . and the matrices Ai are called its Kraus operators. The Choi- ∈ − − Note that X∗Y1 contains X1∗Y1 as a submatrix, which is equal Kraus representation is far from unique, but it was shown in to I with the first column removed. In particular, X∗Y1 = 0. [DSW13] that the subspace of M spanned by the set A∗A : t1 6 n { i j Similarly, the first column of X∗Z1 is the first column of I , 1 i, j m is independent of it. Consequently, [DSW13] 1 t1 ≤ ≤ } so X∗Z1 = 0. Define set 6 Φ := span Ai∗Aj : 1 i, j m , a = min w : w = 0, w is an entry of X∗Y1 S { ≤ ≤ } {| | 6 } m b = max w : w is an entry of X∗Z1 where Φ(X) = i=1 AiXAi∗ is any Choi-Kraus representa- {| | } tion of Φ. This space, easily seen to be an operator system, is 1 1 and let ε (0, ab− ). Define called the non-commutativeP confusability graph of . ∈ 2 Φ Regarding operator subsystems of Mn as non-commutative Wε = Y + εZ Mk, t 1 ∈ | |− confusability graphs, we wish to define operator system ana- and logues of the graph parameters considered in SectionII. Just + A = Aε = Wε∗Wε M t 1. as every graph is the confusability graph of some classical ∈ | |− 5

channel, [DSW13] showed that the map Φ Φ from We will refer to γ(Φ) as the quantum complexity of Φ and 7→ S quantum channels with domain Mn to operator subsystems β(Φ) as the quantum subcomplexity of Φ. Given a channel Φ, of Mn, is surjective. We will need the following estimate on we set π(Φ) = π( Φ) for π β, γ, int . the dimension of the target Hilbert space. S ∈ { } Remark IV.3. (i) A set of quantum states can be perfectly Proposition IV.1. Let n N. If Mn is an operator distinguished by a measurement system if and only if they are ∈ S ⊆ system, then there exists k N and a quantum channel orthogonal. Consequently, [DSW13] defined the one-shot zero- ∈ Φ: Mn Mk such that = Φ. In fact, if m N is such error capacity α(Φ) of a quantum channel Φ: Mn Mk m → S S ∈ that dim 1, then we can take k = mn. to be the maximum cardinality of a set v , ..., v → n 2 ≥ S − 1 p C orthogonal unit vectors, such that { } ⊆ Proof. Let d = dim and let In,S1,...,Sd 1 be a basis of . Suppose that mS dim 1. Then we− can form a S 2 ≥ S − tr(Φ(vivi∗)Φ(vjvj∗)) = 0, i = j. hermitian m m block matrix H = [Hij]i,j [m] Mm(Mn) 6 × ∈ ∈ so that It was shown in [DSW13] (see also [Pau16]) that α(Φ) = Hii = In for i [m] α( Φ). ∈ S and (ii) Let be an operator system. The quantum chromatic S S1,...,Sd 1 = Hij : 1 i < j m . number χq( ⊥) of the orthogonal complement ⊥ of was { − } { ≤ ≤ } S S S 1 introduced by D. Stahlke in [Sta16]. It is straightforward that For sufficiently small ε > 0, the matrix X = (Imn + εH) m β( ) = χq( ⊥). is positive semi-definite, hence X = C∗C for some C S S ∈ (iii) For an operator system Mn and π β, γ, int , we Mm(Mn), and the block entries of X span . The mn n S ⊆ ∈ { } block columns of C are then Kraus operatorsS for a quantum× have π( ) = 1 = Mn. Indeed, the trace tr: Mn C is a non-cancellingS ⇐⇒ quantum S channel since it has the entry-→ channel Φ: Mn Mmn for which Φ is spanned by the → S wise non-negative Kraus operators e∗, . . . , e∗ (where e∗ is the entries of X, so Φ = . 1 n i S S functional corresponding to the vector e ), so 1 π(M ) i ≤ n ≤ We now define parameters of operator systems which, as int(Mn) = 1 and hence π(Mn) = 1. Conversely, π( ) = 1 we will shortly see, generalise the graph parameters above. implies that β( ) = 1; the trace is the only scalar-valuedS S Let Mn be an operator system. As usual, we write quantum channel on M , so M = tr M , that is, S ⊆ n n S ⊆ S ⊆ n = Mn. ⊥ = A Mn : tr(A∗S) = 0 for all S . S S { ∈ ∈ S} Note that, in contrast with Remark II.4 (iii), it is not true (a) Let Mn be an operator system. Recall [DSW13] that M is the only operator system with α( ) = 1; see S ⊆ n that an -independent set of size m is an m-tuple x = Proposition IV.12. S S (x , . . . ,S x ) with each x a non-zero vector in n, so 1 m i C (iv) We claim that that xpxq∗ ⊥ whenever p, q [m] with p = q. The independence∈ S number α( ) is then∈ defined by6 letting S α(CIn) = β(CIn) = γ(CIn) = int(CIn) = n. α( ) = max m N: an -independent set of size m . S { ∈ ∃ S } Indeed, one sees immediately that α(CIn) n by consid- ≥ (b) We define the quantum complexity γ( ) by letting ering the CIn-independent set (e1, . . . , en), and since the S identity channel M M is non-cancelling, we have that n → n γ( ) = min k N: Φ = int(CIn) n, so an appeal to Theorem IV.4 below establishes S { ∈ S S ≤ for some quantum channel Φ: M M the claim. n → k} and the quantum subcomplexity β( ) by letting (v) Let Mn. Using (iv), we have S S ⊆ β( ) = min k N: Φ β( ) = min γ( ): γ(CIn) = n. S { ∈ S ⊆ S S { T T ⊆ S} ≤ for some quantum channel Φ: M M n → k} On the other hand, γ( ) may exceed n, even for n = 2 (see = min γ( ): is an operator subsystem . Proposition IV.12). S { T T ⊆ S } (c) A quantum channel Φ which has a set of Kraus operators (vi) There exist operator systems M with int( ) = , S ⊆ n S ∞ each of which is of the form AD for some entrywise non- so the infimum in the definition of int( ) cannot be replaced S negative matrix A and an invertible diagonal matrix D with a minimum. For example, it is not difficult to see that this will be said to be a non-cancelling. We define is the case for the two-dimensional operator system M4 0 X S1 ⊆ 1 spanned by the identity and H = [ X 0 ] where X = 1 1 . − int( ) = inf k N: Φ = for some S { ∈ S S Theorem IV.4. Let M be an operator system. Then non-cancelling quantum channel Φ: M M . S ⊆ n n → k} α( ) β( ) γ( ) int( ). Corollary IV.2. Let Mn be an operator system. Then S ≤ S ≤ S ≤ S γ( ) 2n2. S ⊆ S ≤ Proof. Suppose that Φ: Mn Mk is a quantum channel 2 → Proof. Since dim n , we can take m = 2n in Proposition with Φ ; let Ai Mk,n, i = 1, . . . , d, be its Kraus IV.1. S ≤ operators.S ⊆ Let S (x )m ∈be an -independent set of size m. p p=1 S 6

For each p [m], let Ep be the projection in Mk onto the Hence by (ii), π( 1) π( 1 2) and the lower bound span of A x∈ : i = 1, . . . , d . Since follows. S ≤ S ⊗ S { i p } (iv) Let A : p [m ] be a family of Kraus operators d d { p,i ∈ i } 2 in Mk ,n with i = span A∗ Aq,i : p, q [mi] , i = 1, 2. Aixp = Ai∗Ai xp, xp = 1, i p,i k k Ap,S1 0 ∈ i=1 * i=1 ! + Set Bp,1 = and Bp,2 = A , viewed as elements of X X 0 p,2 M . Then 1 B : i = 1, 2, p [m ] is a family of we have that Ep = 0 for each p. On the other hand, k1+k2,n √2 p,i i 6 m ∈ since Aj∗Ai for all i, j = 1, . . . , d and (xp)p=1 is - Kraus operators withn o independent,∈ we S have that S span Bp,i∗ Bq,j p,q,i,j = span Bp,i∗ Bq,i p,q,i = , A x ,A x = A∗A x , x = 0, p = q, i, j = 1, . . . , d. { } { } S h i p j qi h j i p qi 6 so γ( ) k1 + k2. Hence, γ( ) γ( 1) + γ( 2). Thus, E1,...,Em are pairwise orthogonal projections in Mk; S ≤ S ≤ S S (v) If , then . Suppose that it follows that m k and hence α( ) β( ). = 1 2 n = n1 +n2 γ( ) = ≤ S ≤ S k, so thatS thereS ⊕S exists a family C = [A B ]: p [mS] of The inequalities β( ) γ( ) int( ) hold trivially. { p p p ∈ } S ≤ S ≤ S k n-Kraus operators, where Ap Mk,n1 and Bp Mk,n2 , ∗ ∗ In the next proposition, we collect some properties of the × m ∈ ApAq ∈ApBq with = span Cp∗Cq p,q=1. Since Cp∗Cq = ∗ ∗ operator system parameters introduced above. S { } Bp Aq Bp Bq ∈ 1 2, we have A∗B = 0, hence the ranges of A and B S ⊕ S p q h p i q Proposition IV.5. Let M and M , i = 1, 2 be are orthogonal for every p, q. The projections P1 and P2 onto S ⊆ n Si ⊆ ni operator systems. the linear span of the ranges of A1,...,Am and B1,...,Bm

(i) If π α, β, γ and U Mn is unitary, then are therefore orthogonal, so if k1 = rank P1 and k2 = k ∈ { } ∈ k k1 k2 − π(U ∗SU) = π( ); rank P1, then there is a unitary U : C C C for which S A0 → ⊕ 0 (ii) If and is a projection of rank , UA = p for some k1 n matrices A0 , and UB = 0 π α, β, γ P Mn r p 0 p p Bp then,∈ viewing { } as an∈ operator subsystem of , we × P P Mr for somehk2 i n matrices Bp0 . Now Cp∗Cq = (UCp)∗(UChq) =i S 0 ∗ 0 × have π(P P ) π( ); Ap Aq 0 0 ∗ 0 , and it follows that for . S ≤ S 0 B B γ( i) ki i = 1, 2 (iii) If π β, γ , n = n1n2 and = 1 2, then p q S ≤ ∈ { } S S ⊗ S hHence, γ( 1)+iγ( 2) k1 +k2 = k = γ( ). Combined with S S ≤ S max π( 1), π( 2) π( ) π( 1)π( 2); (iv), this shows that γ( 1 2) = γ( 1) + γ( 2). { S S } ≤ S ≤ S S S ⊕ S S S (iv) If π β, γ, int , n = n1 = n2 and = span( 1 2), ∈ { } S S ∪ S Remark IV.6. (i) Let π α, β, γ and d N. Then then ∈ { } ∈ π(Md( )) = π( ). Indeed, by Proposition IV.5 (ii), we have π( ) π( 1) + π( 2). S S S ≤ S S π( ) π(Md( )), and the reverse inequality for π β, γ S ≤ S ∈ { } (v) If π β, γ , then π( 1 2) = π( 1) + π( 2). follows from Proposition IV.5 (iii) and Remark IV.3 (iii). To ∈ { } S ⊕ S S S see the corresponding result for π = α, suppose that ξ m Proof. The proofs for π = α are easy and are left to the p p=1 is an independent set for M . Then for X,Y M {, A} reader. We give the proofs for π = γ; the other proofs follow d d and p = q, we have S⊗ ∈ ∈ S identical patterns. 6 (i) If A m are Kraus operators in M for which p p=1 k,n (A I)(I X)ξp, (I Y )ξq = 0 (4) { m} m span A∗A = , then A U are Kraus operators h ⊗ ⊗ ⊗ i { p q}p,q=1 S { p }p=1 in Mk,n such that Let Qp be the projection onto span (I X)ξp : X Md ; { ⊗ ∈ n} m m then Qp = Ep Id for some non-zero projection Ep on C , span (ApU)∗AqU p,q=1 = U ∗ span Ap∗Aq p,q=1U ⊗ { } { } and (4) implies that Eq Ep = 0 provided p = q. If vp is a S { } 6 = U ∗ U, unit vector with Epvp = vp, p [m], we therefore have that S m ∈ vp is an independent set for . It follows that α( ) so γ(U ∗ U) γ( ); the reverse inequality follows by p=1 S ≤ S {α(M} ( )) and hence we have equality.S S ≥ symmetry. d S m (ii) If Ap p=1 are Kraus operators in Mk,n for which (ii) The parameter γ is neither order-preserving nor order- { m} span Ap∗Aq p,q=1 = , then after identifying the range of P reversing for inclusion. For example, CI2 M2 where { r } S m ⊆ S ⊆ S with C , we see that ApP =1 are Kraus operators in Mk,r is the operator system of Proposition IV.12, and these operator { }p with systems have γ-values 2, 3, 1, respectively. m m span (A P )∗A P = P span A∗A P = P P, We will now show that like its graph-theoretic counterpart, { p q }p=1 { p q}p,q=1 S namely, the minimum semidefinite rank, γ( ) is the solution so γ(P P ) γ( ). to a rank minimisation problem. S (iii)S Suppose≤ S that A : p [m ] M is a { p,i ∈ i } ⊆ ki,ni family of Kraus operators for i = 1, 2, so that i = Proposition IV.7. For any operator system M , we have S S ⊆ n span A∗ A : p, q [m ] , i = 1, 2. Set B := A 1 { p,i q,i ∈ i } p,r p, ⊗ Ar,2; then Bp,r : p [m1], r [m2] is a family of + { ∈ ∈ } γ( ) = min rank B : B = [Bi,j] Mm( ) with Kraus operators in Mk1k2,n with = span B∗ Bq,s : p, q S m ∈ S S { p,r ∈ ∈N [m1], r, s [m2] . It follows that γ( ) γ( 1)γ( 2). m ∈ } S ≤ S S If we set P1 = I Q where Q M is a rank one span Bi,j : i, j [m] = and Bi,i = In n1 ⊗ ∈ n2 { ∈ } S projection, then we have that P1( 1 2)P1 = 1 C 1. i=1 S ⊗ S S ⊗ ≡ S X 7 and quantum channel. The assertion about non-cancelling channels follows trivially. + β( ) = min rank B : B = [Bi,j] Mm( ) S m N ∈ S k ∈ Proposition IV.9. Let n, k N, xi be a non-zero vector in C , m ∈ i = 1, . . . , n, and x = (x1, . . . , xn). Then G(x) = ∆x . and Bi,i = In . S S 1 i=1 Proof. Let = and = ∆ . Set xˆ = x − x X S SG(x) T S x i k ik i Moreover, the minima on the right hand sides are achieved and A = [ˆx1 xˆn], and note that is spanned by the ··· T for m not exceeding 2n3. operators DA∗AD0 for D = diag(d1, . . . , dn) and D0 = diag(d , . . . , d ) in ∆. For i, j [n], we have + 10 n0 Proof. Suppose that m N and that B = [Bi,j] Mm( ) ∈ satisfies the relations span∈ B m = and ∈m B S = i,j i,j=1 i=1 i,i DA∗AD0 = D A∗A D0 I . Then B = A A for{ some} A =S [A ...A ] n ∗ 1 m D,D0 ∆, D ∆, D0 ∆, P ∈ ∈ X0∈ dXi=1 X0∈ M1,m(Mk,n) where k = rank B. Since Bi,j = Ai∗Aj, we di=dj =1 dj =1 see that A1,...,Am are Kraus operators for a quantum n 1 { } = 4 − Ei,iA∗AEj,j channel Φ with Φ = ; thus, γ( ) k. n 1 S S S ≤ = 4 − xˆj, xî Ei,j. Conversely, let k = γ( ), m N and A1,...,Am Mk,n S ∈ ∈ h i be Kraus operators for a quantum channel Φ with Φ = . + S S If Ei,j , then i G(x) j, so xˆj, xî = 0, hence Ei,j . Set B := [Ai∗Aj] Mm( ) ; we have span Ai∗Aj : ∈ S ' h i 6 ∈ T ∈ m S { Thus . On the other hand, ⊥ is spanned by the matrix i, j [m] = , i=1 Ai∗Ai = In and rank B = S ⊆ T S ∈ } S units Ei,j with i G(x) j. For such i, j and any D,D0 ∆, rank[A1 ...Am] k. Hence the minimum rank in the first 6' ∈ ≤ P we have (DA∗AD0)ji = dj xî, xˆj di0 = 0, so Ei,j ⊥. expression is no greater than γ( ). h i ∈ T S Hence ⊥ ⊥ as required. To see that some m 2n3 attains this minimum, set k = S ⊆ T ≤ γ( ). Then there exists a quantum channel Φ: Mn Mk S 2 → Theorem IV.10. For any graph G with vertex set [n] and with Φ = and, by Corollary IV.2, k 2n . By [Ch75, S S ≤ π α, β, γ, int , we have π( G) = π(G). Remark 6], the channel Φ can be realised using at most nk ∈ { } S 2n3 Kraus operators. Since m is precisely the number of Kraus≤ Proof. The case π = α is known [DSW13]. operators in the preceding argument, we see that the minimum We next consider the case π = γ. If k = γ(G), then there in the expression for γ( ) is attained for some m 2n3. exists x = (x , . . . , x ), where x k 0 , i = 1, . . . , n, so S ≤ 1 n i C The expression for β( ) follows from the fact that β( ) = that G(x) = G, and hence =∈ .\{ By} Proposition IV.9, S S SG(x) SG min γ( ): . Since the bound on m for γ, namely G(x) is the operator system of a quantum channel Mn Mk, 3 { T T ⊆ S} S → 2n , is independent of the operator system Mn, this fact hence γ( G) k = γ(G). 3S ⊆ S ≤ shows that here we may also take m 2n . Now let k = γ( ), so that there are Kraus operators ≤ G A ,...,A M Sfor a quantum channel Φ: M M Our next task is to show that the operator system parameters 1 m k,n n k with = span∈ A A : p, q [m] = . Since the column→ just defined generalise the graph parameters of SectionII. Φ q∗ p G operatorS with entries{ A ,...,A∈ }is anS isometry, for each Recall that if G is a graph with vertex set [n], we let 1 m i [n] we have m A e 2 = e = 1. In particular, ∈ p=1 k p ik k ik = span E : i j Apei = 0 for at least one p [m]. Thus, the projection SG { i,j ' } 6 P ∈ Pi Mk onto the span of Apei : p [m] is non-zero. be the associated operator subsystem of Mn. ∈ { ∈ } Consider the tuple P = (P1,...,P ) (k). Since Φ = n ∈ P S Lemma IV.8. Let n, k , and let ∆ be the group of diagonal G, for i, j [n] we have N S ∈ n n matrices whose∈ diagonal entries are each either 1 or × i j P P = 0 1. Let x = (x1, . . . , xn) be an n-tuple of non-zero vectors 6'G(P ) ⇐⇒ i j − k in C , and let A = [ˆx1 xˆn] be the k n matrix whose Apei,Aqej = tr(Eij∗ Aq∗Ap) = 0, ··· 1× ⇐⇒ h i i-th column is the unit vector xî = xi − xi. Then the map for all p, q [m] k k ∈ n ∆ : M M , ∆ (X) = 2− ADXDA∗, Eij Φ⊥ = G⊥ i G j. x n → k x ⇐⇒ ∈ S S ⇐⇒ 6' D ∆ X∈ Hence G(P ) = G. Using Theorem III.3, we obtain γ(G) = k is a quantum channel. Moreover, if xi R+ 0 , i = qint(G) k = γ( ). ∈ \{ } ≤ SG 1, . . . , n, then ∆x is non-cancelling. The case π = β is similar to the case π = γ; alternatively,

Proof. For D ∆, let di 1, 1 be the i-th diagonal entry see [Sta16, Theorem 12]. of D. We have∈ ∈ { − } Let π = int. Set k = int(G); then there exists a tuple x = (x , . . . , x ) ( k 0 )n with G(x) = G. By n n 1 n R+ 2− (DA∗AD)ij = 2− xˆj, xˆi didj = δij, Proposition IV.9, ∈ \{ } . By Lemma IV.8, is h i ∆x = G(x) = G ∆x D ∆ D ∆ S S S X∈ X∈ a non-cancelling quantum channel; therefore, int( G) k. since if i = j then the sum reduces to 0 by symmetry, whereas In particular, int( ) < . Now if k = int(S )≤, let 6 SG ∞ SG if i = j then every term in the sum is 1. Since each xˆi is a Φ: Mn Mk be a non-cancelling quantum channel such n → unit vector, we obtain 2− D ∆ DA∗AD = In, so ∆x is a that Φ = G. Fix Kraus operators A1D1,...,AmDm for Φ ∈ S S P 8

where each Ap Mk,n is entrywise non-negative and each If γ( ) 2, then there are Aj = [vj wj] M2 ∈ S ≤ 2 ∈ Dp is an invertible diagonal n n matrix. For i [n], define for some vj, wj C which are the Kraus operators of a × ∈ ∈ quantum channel with operator system . We have vj, vi = Ri = r [k]: Apei, er = 0 for some p [m] . S h i { ∈ h i 6 ∈ } (A∗Aj)11 = (A∗Aj)22 = wj, wi for each i, j. It follows i i h i Since the column operator with entries A1D1,...,AmDm is that there is a 2 2 unitary U with Uvj = wj, j = 1, 2. m 2 2 × an isometry, we have A D e = 1, and so R is Recall that A∗Ai = I2. This is equivalent to p=1 k p p ik i i=1 i non-empty for all i [n]. Let x = (x1, . . . , xn) where xi = ∈ P 2 2 e . For i, j [nP], we have 2 2 r Ri r v = w = 1 ∈ ∈ k ik k ik i=1 i=1 Pi G j Ej,j GEi,i = 0 X X ' ⇐⇒ S 6 { } and p, q [m] such that A D e ,A D e = 0 2 2 ⇐⇒ ∃ ∈ h p p i q q ji 6 p, q [m] such that A e ,A e = 0. wi, vi = Uvi, vi = 0. p i q j h i h i ⇐⇒ ∃ ∈ h i 6 i=1 i=1 Now X X In particular, 0 lies in the numerical range of U. However, the A e ,A e = A e , e e ,A e h p i q ji h p i rih r q ji numerical range of a normal matrix is the convex hull of its r [k] X∈ spectrum, and hence σ(U) = α, α for some α T. Thus { 2 − } ∈ and every term in the latter sum is non-negative. It follows αU is hermitian, and so U = α U ∗. Now that vj ,vi Uvj ,vi A∗A = h ∗ i h i i j U vj ,vi Uvj ,Uvi i G j p, q [m], r [k] with Apei, er = 0 h i h i ' ⇐⇒ ∃ ∈ ∈ h i 6 0 α2 = vj, vi I + U ∗vj, vi 1 0 , and Aqej, er = 0 h i h i h i 6 hence , a contradiction. Ri Rj = xj, xi = 0 3 = dim = dim span Ai∗Aj 2 ⇐⇒ ∩ 6 ∅ ⇐⇒ h i 6 To see that γS( ) < γ( {)2 = 9}, consider≤ the isometries i G(x) j. S ⊗ S S ⇐⇒ ' Vi M8,4 given by ∈ Hence G = G(x), so int(G) k = int( G). ≤ S V1 = [e1 e2 e3 e4],V2 = [e2 e5 e4 e6], Remark IV.11. Let (p(y x)) be a k n non-negative column- V3 = [e3 e4 e7 e8],V4 = [e6 e7 e5 e1]. stochastic matrix defining| a classical× channel :[n] [k] N → 1 with confusability graph G = G . The canonical quantum Let W = 2 [V1 V2 V3 V4] M8,16; then N ∈ channel q : Mn Mk associated with is defined by 0 0 0 0 0 0 0 0 0 0 0 1 N → N 1 0 0 0 0 0 0 0 0 0 0 0 setting q(Ex,x) = y [k] p(y x)Ey,y and q(Ex,x0 ) = 0 if I4 0 0 0 0 1 0 0 0 0 0 0 0 N ∈ | N 0 0 1 0 0 1 0 0 0 0 0 0 x = x0. We have that q = G [DSW13] (see also [Pau16]).   6 PSN S 0 1 0 0 0 0 0 0 0 0 0 0 So we see that γ(G) = γ( G) is the quantum complexity of 0 0 0 0 0 0 0 0 0 0 1 0 S  0 0 0 1 I4 0 1 0 0 0 0 0 0  the classical channel when viewed as a quantum channel. 1  0 0 0 0 0 0 0 0 1 0 0 0  N W ∗W =   4  0 0 1 0 0 0 0 0 0 0 0 0  Let G H denote the strong product of the graphs G and  0 0 0 1 0 0 1 0 0 0 0 0   0 0 0 0 0 0 0 0 I4 0 1 0 0  H [Sa60], in which (x, y) (x0, y0) if and only if x G  0 0 0 0 0 0 0 0 0 0 0 0  'GH '   x0 and y H y0. Note that G H = G H . If G, H are  0 0 0 0 0 0 0 1 0 0 0 0  ' S S ⊗ S  0 0 0 0 0 0 0 0 0 0 1 0  graphs and n is the number of vertices of G, then  0 0 0 0 0 1 0 0 0 0 0 0 I4   1 0 0 0 0 0 0 0 0 0 0 0  (i) α(G) = 1 if and only if G = K , i.e., if and only if   n Writing B for the (i, j)-th 4 4 block of W W , we = M ; i,j ∗ G n observe that × (ii) Sγ(G) n; and ≤ 4 (iii) γ(GH) γ(G)γ(H), but it is unknown whether strict ≤ span B : i, j [4] = and B = I4. inequality can occur. { i,j ∈ } S ⊗ S i,i i=1 The following proposition shows that the parameters for X By Proposition IV.7, general operator systems M behave quite differently, S ⊆ n with respect to the latter properties, than their graph theoretic γ( ) rank W ∗W = rank W = 8. counterparts. S ⊗ S ≤

λ a V. APPLICATIONS TO CAPACITY Proposition IV.12. Let = b λ : a, b, λ C . Then α( ) = 1, β( ) = 2, γ(S) = int( ) = 3 and γ∈( ) < Let : X Y be a classical information channel with N → γ(S)2. S S S S ⊗ S confusability graph G. Its parallel use r times can be expressed S as a channel r : Xr Y r, for which 1 1 0 × Proof. The Kraus operators A1 = 0 0 and A2 = N → √2 0 1 r 1 0 0 r r 0 1 yield a non-cancelling quantumh channeli with op- p((ys)s=1 (xs)s=1) = p(ys xs), √2 1 0 | | s=1 eratorh systemi , so γ( ) int( ) 3. Since ⊥ is Y 1S 0 S ≤ S ≤ S for xs X, ys Y and s = 1, . . . , r. Note that spanned by 0 1 , it contains no rank one operators, and ∈ ∈ hence α( ) = 1−. By Remark IV.3 (v), β( ) 2 while, by G ×r = G G . Remark IV.3S (iii), β( ) = 1; thus, β( ) =S 2. ≤ N N ··· N S 6 S r times | {z } 9

The Shannon capacity of the channel : X Y (or These bounds are often difficult to compute. The quantity N → equivalently of the graph G) is the quantity r r limr ϑ( Φ⊗ ) requires evaluation of a limit, each term →∞ S of which may be intractable, and the possibly larger bound r r r r q Θ( ) = Θ(G) = lim α( × ) = lim α(G ). N r N r N ϑ( Φ) = supn ϑ( Φ Mn) requires the evaluation of a →∞ →∞ q S ∈N S ⊗ (Some authors prefer to usep the logarithm of the quantities supremum, although this parameter has the advantage of pos- defined above.) sessinge a reformulation as a semidefinite program [DSW13]. Similarly, if Φ: Mn Mk is a quantum channel, letting Theorem V.1. For any quantum channel Φ, we have r r r → Φ⊗ : Mn⊗ Mk⊗ be its r-th power, we find → α(Φ) Θ (Φ) β ( Φ) . ≤ ≤ S Φ⊗r = Φ Φ . S S ⊗ · · · ⊗ S Proof. Let Φ be a quantum channel and = Φ. The inequal- r times ity α Θ is well known, and follows immediatelyS S from the ≤ The analogue of the Shannon| capacity{z of} a quantum channel supermultiplicative property of α. Since β is submultiplicative introduced in [DSW13] is the parameter for tensor products (Proposition IV.5 (iii)) and α is dominated

r r by β (Theorem IV.4), we have Θ(Φ) = lim α(Φ⊗ ). r →∞ r r Θ (Φ) = Θ( ) = lim α ( ⊗ ) β ( ) . p r Lovasz´ [Lo79] introduced his famous ϑ-parameter of a S →∞ S ≤ S graph and proved that α( ) ϑ(G) and that ϑ is multi- In the remainder of the section,p we will exhibit operator N ≤ plicative for strong graph product; hence, systems for which β( ) ϑ( ). For k N, let S S ∈ k Θ( ) = Θ(G ) ϑ(G), k = (ai,j)i,j=1 Mk : a1,1 = a2,2 = = ak,k . N N ≤ S { ∈ ··· } for any classical channel, thus giving a bound on the Shannon It is easy to show directly that capacity of classical channels. He also proved [Lo79, Theo- ϑ( k) = k. rem 11] that S For any m N, applying the canonical shuffle which identifies ϑ(G) β(G), ∈ ≤ Mk Mm with Mm Mk, we have so that his ϑ-bound is a better bound on the capacity of ⊗ ⊗ k classical channels than any of the bounds that we derived k Mm = (Ai,j)i,j=1 Mk(Mm): from complexity considerations. However, as we will shortly S ⊗ ∈ A1,1 = A2,2 = = Am,m . show, for quantum channels, β yields a bound on capacity that ··· can outperform . We note that a different bound on , Thus, for any operator system Mm, we have ϑ Θ(G) S ⊆ based on ranks of Hermitian matrices in the operator system k k = (Ai,j) Mk( ): c , was introduced by Haemers in [Hae81]. It is an inter- i,j=1 SG S ⊗ S ∈ S esting open question to formulate general non-commutative A1 1 = A2 2 = = A . , , ··· m,m analogues of Haemers’ parameter. Theorem V.2. We have Lovasz´ gave many characterisations of his parameter, but 2 3 the most useful for our purposes is the expression β( 2 ) k < k ϑ( 2 ). Sk ⊗ Sk ≤ ≤ Sk ⊗ Sk + Proof. 2 ϑ(G) = max I + K : I + K Mn ,K G⊥ . Let ω be a primitive k-th root of unity. Let S Mk k k ∈ ∈ S 2 ∈ be given by Sei = ei+1, i = 1, . . . , k , where addition is The latter formula motivated [DSW13] to define, for any 2 modulo k , while D Mk2 be the diagonal matrix with operator subsystem of M , 2 ∈ k2 1 n diagonal (1, ω, ω , . . . , ω − ). Note that, if D0 M is S ∈ k + the diagonal matrix with diagonal (1, ω, ω2, . . . , ωk 1), then ϑ( ) = max I + K : I + K M ,K ⊥ ; − n j j j S k k ∈ ∈ S D = D0 D0. We have that D S = ω SD for any note that ϑ(G) = ϑ( ). It was shown in [DSW13] that, for ⊕ · · · ⊕ SG k times any quantum channel Φ, one has j Z, and hence ∈ | {z j }i ij i j α(Φ) = α( Φ) ϑ( Φ). D S = ω S D , i, j Z. (5) S ≤ S ∈ However, ϑ is only supermultiplicative for tensor products Any element A of Mk( k k2 ) has the form A = k 1 S ⊗ S (C ) − , where C 2 for all r, s = 0, . . . , k 1. of general operator systems. This motivated [DSW13] to r,s r,s=0 r,s ∈ Sk ⊗ Sk − introduce a “complete” version, denoted ϑ, which is multi- In view of the remarks before the statement of the theorem, we k 1 plicative for tensor products of operator systems and satisfies may write Cr,s = (Akr+i,ks+j)i,j−=0, where Akr+i,ks+j k2 for all r, s, i, j = 0, . . . , k 1, and ∈ S ϑ( ) ϑ( ). This allowed them to bounde the quantum − capacityS ≤ of aS quantum channel, since Akr+1,ks+1 = Akr+2,ks+2 = = A(r+1)k,(s+1)k, e ··· r r r r Θ(Φ) = lim α ⊗ lim ϑ ⊗ for all r, s = 1, . . . , k. r SΦ ≤ r SΦ →∞ q →∞ q Let r r lim ϑ ⊗ = ϑ ( Φ) . + r Φ kr i r ≤ →∞ S S ukr+i = S D , r, i = 0, . . . , k 1, q − e e 10 and Moreover, these statements hold if throughout we replace ϑ B = (u0, u1, u2, . . . , uk2 1) Mk2,k4 . by ϑ, the quantum Lovasz´ theta number. − ∈ k 1 Set B = (u u ) − ; then the matrix Proof. Since ϑ ϑ, it suffices to prove these statments for ϑ. r,s kr∗ +i ks+j i,j=0 e ≤ k 1 (i) For the first ratio, consider = β := k k2 and B∗B = (Br,s) −=0 = (u∗ + uks+j)r,s,i,j S T S ⊗ S r,s kr i apply Theorem V.2e. For the second ratio, let γ be the span 2 T is positive and has rank at most k . We will show the of the matrices Br,s k k2 appearing in the proof of ∈ S ⊗ S following: Theorem V.2. Then γ k k2 is an operator system and T ⊆ S ⊗ S the set Br,s is one of the terms that appear in the minimum (i) ukr∗ +iuks+j k2 , for all r, s, i, j = 0, 1, . . . , k 1; { } ∈ S − that defines γ( γ ). Hence, T (ii) ukr∗ +1uks+1 = ukr∗ +2uks+2 = = ukr∗ +k 1uks+k 1, ··· − − 2 3 for all r, s, i = 0, . . . , k 1, and γ( γ ) rank ((Br,s)) k < k ϑ( k k2 ) ϑ( γ ). k − T ≤ ≤ ≤ S ⊗ S ≤ T (iii) Br,r = kI, r=1 (ii) For π β, γ , consider π := π CIk2 Mk3+k2 . 2 ∈ { } R T ⊕ ⊆ whichP will imply that β( k k2 ) k . For k > 3, by Proposition IV.5 (v) and Remark IV.3 (iv), we To show (i), note that S ⊗ S ≤ have

r kr i ks+j s r ks kr+j i s 2 2 2 ukr∗ +iuks+j = D− S− − S D = D− S − − D . π( π) = π( π) + π(CIk2 ) k + k = 2k = 2α(CIk2 ) R T ≤ If ks kr + j i = 0, then u∗ u + has zero diagonal 2α( π) 2Θ( π). − − 6 kr+i ks j ≤ R ≤ R and thus belongs to 2 . Suppose that ks kr + j i = 0. k Since ϑ is order-reversing for inclusion of operator systems Then k (i j) and henceS i = j. If, in addition,− r =− s then | − 6 and π ( k k2 ) CIk2 ( k C) k2 (the second u u has zero diagonal and therefore belongs to 2 ; kr∗ +i ks+j k inclusionR ⊆ holdsS ⊗ up S to a⊕ unitary⊆ shuffleS ⊕ equivalence)⊗ S and it is if, on the other hand, r = s, then u u = I and henceS kr∗ +i ks+j easy to see that ϑ( ) ϑ( ) for any operator systems again belongs to 2 . S ⊕ T ≥ S Sk and , we have To show (ii), note that for i = 0, . . . , k 1, we have S T − 3 r kr i ks+i s r k(s r) s ϑ( π) ϑ( k C)ϑ( k2 ) k . ukr∗ +iuks+i = D− S− − S D = D− S − D . R ≥ S ⊗ S ≥ In order to show (iii), suppose that i, j 0, . . . , k 1 APPENDIX with i = j, and, using (5), note that ∈ { − } 6 In this appendix, we briefly summarise the order rela- k 1 k 1 − − tionships between various bounds on the quantum Shannon u u = D rS kr iSkr+jDr kr∗ +i kr+j − − − zero-error capacity. These may be succinctly described by r=0 r=0 X kX1 the directed graph in Figure1. The parameters αq, χq and − r j i r χ are defined in [DSW13], and [Sta16, Definition 11]; the = D− S − D reader should swap and ⊥ when translating between r=0 S S Xk 1 our non-commutative graphs and Stahlke’s “trace-free non- − r(j i) j i = ω− − S − . commutative graphs”. r=0 ! Let M be an operator system. As observed in Remark X S ⊆ n 1 (j i) IV.3 (ii), we have χq( ) = β( ). The first two inequalities in Since ω is a primitive root of unity, so is ω− . Thus, ω− − S S (j i) the chain is a k-th root of unity with ω− − = 1. It follows that k 1 r(j i) k 1 6 − ω− − = 0. Thus, − u∗ ukr+j = 0 whenever r=0 r=0 kr+i αq( ) ϑ( ) χq( ⊥) χ( ⊥) i = j. On the other hand, S ≤ S ≤ S ≤ S q P6 P appear in [Sta16], following Corollary 20, and the third k 1 k 1 e − − inequality is a simple consequence of his Proposition 9. The ukr∗ +iukr+i = I = kI, r=0 r=0 inequality αq( ) α( ) is immediate from the definitions X X S ≤ S and (iii) is proved. (and appears in [DSW13, Proposition 2]), and we have seen By [DSW13, Lemma 4], in Theorems IV.4 and V.1 that α Θ β γ int. The inequality Θ ϑ follows immediately≤ ≤ from≤ [DSW13≤ , 3 ≤ ϑ( k k2 ) ϑ( k)ϑ( k2 ) = k , Proposition 2 and Corollary 10], noting that in the notation of S ⊗ S ≥ S S √ and the proof is complete. that paper, log2 Θ = Ce0 C0E; and ϑ ϑ is trivial. It only remains to prove≤ the incomparability≤ assertions of Corollary V.3. (i) The ratios Figure1. These follow from the inequalitiese e already estab- ϑ( )/β( ) and ϑ( )/γ( ) lished and the examples below. S S S S Let G = C5 be the 5-cycle, and let = . Lovasz´ has • G can be arbitrarily large, as varies over all non- shown [Lo79] that ϑ(G) = √5 while,S forS graph operator commutative graphs. S systems, as pointed out in [DSW13], we have ϑ( G) = (ii) For π β, γ , the ratio ϑ( )/π( ) can be arbitrarily ϑ(G). It is not difficult to see that S large, as∈ { varies} over all non-commutativeS S graphs with e 1 π( ) SΘ( ) π( ). α( ) = α(G) = 2 < β(G) = β( ). 2 S ≤ S ≤ S SG SG 11

ACKNOWLEDGEMENTS int( ) S The authors are grateful to the Fields Institute and the χ( ⊥) Institut Henri Poincare´ for financial support to attend the S Workshop on Operator Systems in Quantum Information and γ( ) S the Workshop on Operator Algebras and Quantum Information Theory, respectively, greatly facilitating our work on this project. The first named author also wishes to thank Helena Smigocˇ and Polona Oblak for stimulating discussion of the χq( ⊥) = β( ) ϑ( ) S S S minimum semidefinite rank. e REFERENCES Θ( ) ϑ( ) S S [Ch75]M.D.C HOI, Completely positive linear maps on complex √ matrices, Lin. Alg. Appl. 10 (1975), 285–290. e [DSW13]R.D UAN,S.SEVERINI,A.WINTER, Zero-error communi- cation via quantum channels, non-commutative graphs and α( ) a quantum Lovasz´ θ function, IEEE Trans. Inf. Theory 59 S (2013), 1164–1174. [FH13]S.M.F ALLAT,L.HOGBEN, Minimum Rank, Maximum Nul- lity, and Zero Forcing Number of Graphs, Handbook of Linear Algebra, Chapman and Hall, 2013. αq( ) [GR01]C.G ODSILAND G.ROYLE, Algebraic graph theory, Springer- S Verlag, 2001. [Hae81] W. HAEMERS, An upper bound for the Shannon capacity of Fig. 1. A directed graph showing the partial order among various param- a graph, Colloq. Math. Soc. Janos´ Bolyai, 25, North-Holland, eters bounding the quantum Shannon zero-error capacity Θ(S) of a non- Amsterdam-New York, 1981 commutative graph S. The ordering π1(S) ≤ π2(S) for every operator [HPSWM11]G.H AYNES,C.PARK,A.SCHAEFFER,J.WEBSTER,L.H. system S ⊆ Mn is indicated by placing π1(S) below π2(S), joined with a MITCHELL, Orthogonal vector coloring, Elec. J. Combin. 17 path directed towards π1(S); the absence of a directed path between a pair (2010), no. 1, Research Paper 55, 18 pp. of vertices indicates that the corresponding parameters are incomparable. [HPRS15]L.H OGBEN, K. F. PALMOWSKI,D.E.ROBERSON,S.SEV- ERINI, Orthogonal representations, projective rank, and frac- tional minimum positive semidefinite rank: connections and So, in this example, new directions, Elec. J. Linear Algebra 32 (2017), 98–115. [JMN08] Y. JIANG,L.H.MITCHELL,S.K.NARAYAN, Unitary matrix digraphs and minimum semidefinite rank, Lin. Alg. Appl. 428 ϑ( ) < α( ) and ϑ( ) < β( ). S S S S (2008), 1685–1695. [Lo79]L.L OVASZ´ , On the Shannon capacity of a graph, IEEE Trans. Consider qG = Cc, the complement of the 6-cycle, and • e 6 e Inf. Theory 25 (1979), no. 1, 1–7. = G. It is easy to see directly that γ( ) = γ(G) > 2, [MM99] T. MCKEE, F. MCMORRIS, Topics in intersection graph S S S theory, SIAM Monographs on Discrete Mathematics and Ap- and χ( ⊥) = χ(C6) = 2, so in this case, S plications, 1999. [Pau02] V.I.PAULSEN, Completely bounded maps and operator alge- χ( ⊥) < γ( ). S S bras, Cambridge University Press, 2002. Let be the operator system of Proposition IV.12 (i.e., in [Pau16] V.I.PAULSEN, Entanglement and non-locality, http://www. • S math.uwaterloo.ca/∼vpaulsen/EntanglementAndNonlocality the notation of SectionV, = 2). Note that α( ) = 1. LectureNotes 7.pdf, 2016. S S S We claim that if is any operator system with α( ) = 1, [Sa60]G.S ABIDUSSI, Graph multiplication, Math. Z. 72 (1960), then α( ) =T 1. Indeed, may be identifiedT with 446–457. [SS12]G.S CARPA AND S.SEVERINI, Kochen-Specker sets and the S ⊗T S ⊗T TA all 2 2 block matrices of the form [ ] for T, A, B rank-1 quantum chromatic number, IEEE Trans. Inf. Theory × BT ∈ , and if x, y are non-zero vectors with xy∗ ( 58 (2012), no. 4, 2524–2529. T x1 y1 ∈ S ⊗ [Sha56]C.E.S HANNON, The zero error capacity of a noisy channel, )⊥, then writing x = [ ] and y = [ y ], we obtain T x2 2 IRE Trans. Inf. Theory 2 (1956), no. 3, 8–19. xy∗ = (xiyj∗)i,j=1,2 ( )⊥. By considering the off- [Sta16]D.S TAHLKE, Quantum zero-error source-channel coding and diagonal entries and∈ theS condition ⊗T α( ) = 1, it readily non-commutative graph theory, IEEE Trans. Inf. Theory 62 T (2016), no. 1, 554–577. follows that x1 = 0 or y2 = 0, and x2 = 0 or y1 = 0. If x1 = 0, then y1 = 0; hence, xy∗ = 0 x2y∗, so ⊕ 2 x2y2∗ ⊥, so x = y = 0, a contradiction. The other case proceeds∈ T to a similar contradiction, so α( ) = 1. Hence, in particular, Θ( ) = 1. On the otherS ⊗ Thand, ϑ( ) = 2 by [DSW13, p.S 1172]; thus, in this case we haveS e Θ( ) < ϑ( ). S S q Finally, let = CI2 to obtain an example for which • S e int( ) < ϑ( ), S S since the left hand side is 2 by Remark IV.3 (iv), and, as e observed in [DSW13], the right hand side is 4.