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IEEE Transactions on Information Theory

Quantum Asymptotic Spectra of Graphs and Non-Commutative Graphs, and Quantum Shannon Capacities

Yinan Li, Jeroen Zuiddam

Published Oct 22, 2020• 6 citations
Publisher

Abstract

We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen's spectral theorem (J. Reine Angew. Math., 1988) in order to obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We then exhibit spectral points in the new quantum asymptotic spectra and discuss their relations with the asymptotic spectrum of graphs. In particular, we prove that the (fractional) real and complex Haemers bounds upper bound the quantum Shannon capacity, which is defined as the regularization of the quantum independence number (Mančinska and Roberson, J. Combin. Theory Ser. B, 2016), and that the fractional real and complex Haemers bounds are elements in the quantum asymptotic spectrum of graphs. This is in contrast to the Haemers bounds defined over certain finite fields, which can be strictly smaller than the quantum Shannon capacity. Moreover, since the Haemers bound can be strictly smaller than the Lovász theta function (Haemers, IEEE Trans. Inf. Theory, 1979), we find that the quantum Shannon capacity and the Lovász theta function do not coincide. As a consequence, two well-known conjectures in quantum information theory, namely: 1) the entanglement-assisted zero-error capacity of a classical channel is equal to the Lovász theta function and 2) maximally entangled states and projective measurements are sufficient to achieve the entanglement-assisted zero-error capacity, cannot both be true.

DOI
10.1109/tit.2020.3032686

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We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen's spectral theorem (J. Reine Angew. Math., 1988) in order to obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We then exhibit spectral points in the new quantum asymptotic spectra and discuss their relations with the asymptotic spectrum of graphs. In particular, we prove that the (fractional) real and complex Haemers bounds upper bound the quantum Shannon capacity, which is defined as the regularization of the quantum independence number (Mančinska and Roberson, J. Combin. Theory Ser. B, 2016), and that the fractional real and complex Haemers bounds are elements in the quantum asymptotic spectrum of graphs. This is in contrast to the Haemers bounds defined over certain finite fields, which can be strictly smaller than the quantum Shannon capacity. Moreover, since the Haemers bound can be strictly smaller than the Lovász theta function (Haemers, IEEE Trans. Inf. Theory, 1979), we find that the quantum Shannon capacity and the Lovász theta function do not coincide. As a consequence, two well-known conjectures in quantum information theory, namely: 1) the entanglement-assisted zero-error capacity of a classical channel is equal to the Lovász theta function and 2) maximally entangled states and projective measurements are sufficient to achieve the entanglement-assisted zero-error capacity, cannot both be true.
We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen's spectral theorem (J. Reine Angew. Math., 1988) in order to obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We then exhibit spectral points in the new quantum asymptotic spectra and discuss their relations with the asymptotic spectrum of graphs. In particular, we prove that the (fractional) real and complex Haemers bounds upper bound the quantum Shannon capacity, which is defined as the regularization of the quantum independence number (Mančinska and Roberson, J. Combin. Theory Ser. B, 2016), and that the fractional real and complex Haemers bounds are elements in the quantum asymptotic spectrum of graphs. This is in contrast to the Haemers bounds defined over certain finite fields, which can be strictly smaller than the quantum Shannon capacity. Moreover, since the Haemers bound can be strictly smaller than the Lovász theta function (Haemers, IEEE Trans. Inf. Theory, 1979), we find that the quantum Shannon capacity and the Lovász theta function do not coincide. As a consequence, two well-known conjectures in quantum information theory cannot both be true.
We continue the study of the quantum channel version of Shannon's zero-error capacity problem. We generalize the celebrated Haemers bound to noncommutative graphs (obtained from quantum channels). We prove basic properties of this bound, such as additivity under the direct sum and submultiplicativity under the tensor product. The Haemers bound upper bounds the Shannon capacity of noncommutative graphs, and we show that it can outperform other known upper bounds, including noncommutative analogues of the Lovász theta function (Duan-Severini-Winter, IEEE Trans. Inform. Theory, 2013 and Boreland-Todorov-Winter, arXiv preprint, 2019).
We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (overC\mathbb {C}C) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes’ embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.
Li and Zuiddam's entanglement-assisted asymptotic spectrum of graphs consists of the normalized graph parameters that are additive under disjoint unions, multiplicative under strong products and monotone under entanglement-assisted cohomomorphisms; the entanglement-assisted Shannon capacity is their pointwise minimum. To our knowledge, the Lovász number ϑ\varthetaϑ was its only known element. We find a continuum of others. Three quantities of Duan and Winter, log⁡2ϑ\log_2\varthetalog2​ϑ, the minimal Holevo capacity Cmin⁡C_{\min}Cmin​ and the logarithm of the no-signalling simulation cost ΣΣΣ, are the smallest information radii of classical-quantum channels compatible with a graph, measured with Petz-Rényi divergences of orders 000, 111 and 222. At order zero this is due to Schrijver (as credited by Dalai) and to Duan and Winter; at order one it is the min-max formula for the Holevo capacity. Interpolating in the order gives a family ϑα\vartheta_αϑα​, 0≤α≤20\leα\le20≤α≤2, and we prove that every member lies in the spectrum. The main step, multiplicativity under strong products for channels that need not factor, follows from an exact nesting identity for compressions. Sandwiched Rényi radii of order β∈[1/2,∞]β\in[1/2,\infty]β∈[1/2,∞] give the same family, at α=2−1/βα=2-1/βα=2−1/β. On the five-cycle the family takes every value in [5,5/2][\sqrt5,5/2][5​,5/2]. A Hilbert-Schmidt inequality for the pentagon gives Cmin⁡(C5)=log⁡2(5/2)C_{\min}(C_5)=\log_2(5/2)Cmin​(C5​)=log2​(5/2), proving a conjecture of Duan and Winter. We also settle the two open links of their inequality chain: the graph-level no-signalling simulation rate always equals its single-letter value log⁡2Σ(G)\log_2Σ(G)log2​Σ(G), and a 32-vertex graph from the Mermin-Peres magic square has Cmin⁡<log⁡2ΣC_{\min}<\log_2ΣCmin​<log2​Σ, certified in exact rational arithmetic. Finally, ΣΣΣ never exceeds the projective rank of the complement; a rank inequality for the same square makes this strict on a 1440-vertex graph.
The zero-error capacity of a classical channel is a parameter of its confusability graph, and is equal to the minimum of the values of graph parameters that are additive under the disjoint union, multiplicative under the strong product, monotone under homomorphisms between the complements, and normalized. We show that any such function either has uncountably many extensions to noncommutative graphs with similar properties, or no such extensions at all. More precisely, we find that every extension has an exponent that characterizes its values on the confusability graphs of identity quantum channels, and the set of admissible exponents is either an unbounded subinterval of [1,∞)[1,\infty)[1,∞) or empty. In particular, the set of admissible exponents for the Lovász number, the projective rank, and the fractional Haemers bound over the complex numbers are maximal, while the fractional clique cover number does not have any extensions.
Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.