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 $[1,\infty)$ 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.
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,∞) 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.
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 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).
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 was its only known element. We find a continuum of others.
Three quantities of Duan and Winter, log2ϑ, the minimal Holevo capacity Cmin and the logarithm of the no-signalling simulation cost Σ, are the smallest information radii of classical-quantum channels compatible with a graph, measured with Petz-Rényi divergences of orders 0, 1 and 2. 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, 0≤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,∞] give the same family, at α=2−1/β.
On the five-cycle the family takes every value in [5,5/2]. A Hilbert-Schmidt inequality for the pentagon gives 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 log2Σ(G), and a 32-vertex graph from the Mermin-Peres magic square has Cmin<log2Σ, certified in exact rational arithmetic. Finally, Σ never exceeds the projective rank of the complement; a rank inequality for the same square makes this strict on a 1440-vertex graph.
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.
The one-shot zero-error classical capacity of a quantum channel is the maximum amount of classical information that can be transmitted with zero probability of error via a single channel use. This capacity is fundamentally characterized by the logarithm of the independence number of the noncommutative graph induced by the quantum channel. Consequently, the additivity of the one-shot zero-error classical capacity is equivalent to the multiplicativity of the independence number of the associated noncommutative graphs. As the independence number is not multiplicative in general, the specific conditions under which multiplicativity is preserved remain an open question in quantum information theory. Here, we establish some sufficient conditions for the multiplicativity of the independence number and provide explicit examples of quantum channels that satisfy these criteria. Furthermore, we investigate the block form of noncommutative graphs and derive the conditions under which the independence number remains multiplicative within this framework. These results offer new insights into the structural properties of noncommutative graphs and the fundamental limits of zero-error quantum communication.
We generalize some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realize a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalized minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lovász theta number.