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IEEE Transactions on Information Theory

Addendum to "Strong converse for identification via quantum channels"

Rudolf Ahlswede, Andreas Winter

Published Jan 1, 2003• 14 citations
Publisher

Abstract

The paper discusses Conjecture 21 formulated by Ahlswede and Winter (see ibid., vol.48, p.569-79, Mar. 2002) about finite families of self-adjoint operators A/sub i/ and B/sub i/ noting that for commuting operators equality holds. It adds that the other statements made in connection with this conjecture, though not logically disproved, should be regarded with caution.

DOI
10.1109/tit.2002.806161

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The paper discusses Conjecture 21 formulated by Ahlswede and Winter (see ibid., vol.48, p.569-79, Mar. 2002) about finite families of self-adjoint operators A/sub i/ and B/sub i/ noting that for commuting operators equality holds. It adds that the other statements made in connection with this conjecture, though not logically disproved, should be regarded with caution.
Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs G,HG,HG,H with χ(G×H)≤1538<1539=min⁡(χq(G),χq(H)). χ(G\times H) \leq 1538 < 1539 = \min(χ_q(G),χ_q(H)).χ(G×H)≤1538<1539=min(χq​(G),χq​(H)).The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital C∗C^*C∗-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and C∗C^*C∗-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.
We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum communication channels of Ahlswede's (1979, 1992) approach to classical channels. It involves a development of explicit large deviation estimates to the case of random variables taking values in self-adjoint operators on a Hilbert space. This theory is presented separately in an appendix, and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems.
A strong converse bound for the classical identification capacity of a quantum channel is an upper bound on the asymptotic identification rate of classical messages sent through the channel, such that, above this rate, the probability of an identification error necessarily converges to one. Converse bounds for identification are notoriously difficult to obtain for fully quantum channels. The only previously known converse bound, due to Atif, Pradhan and Winter [Int.~J.~Quantum Inf.~22(5):2440013, 2024], has the unsatisfactory feature of remaining strictly positive even for a completely noisy channel, for which identification is clearly impossible. We derive strong (and hence also weak) converse bounds, for the qubit depolarizing channel with noise parameter ppp, that vanish as p→1p\to 1p→1, thereby yielding the correct behavior in the completely noisy limit. Moreover, in the setting of simultaneous classical identification under the constraint of complete product measurements, our converse bound matches the corresponding achievability bound, and establishes that in this case the identification capacity equals the classical capacity of the channel.