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arXiv:2408.16858

Invariants of the quantum graph of the partial trace

Wojciech Paupa, Piotr Mikołaj Sołtan

Published Aug 29, 2024• 0 citations
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Abstract

We compute the independence number, zero-error capacity, and the values of the Lovász function and the quantum Lovász function for the quantum graph associated to the partial trace quantum channel $\operatorname{Tr}_n\otimes\mathrm{id}_k\colon\operatorname{B}(\mathbb{C}^n\otimes\mathbb{C}^k)\to\operatorname{B}(\mathbb{C}^k)$.

DOI
10.48550/arxiv.2408.16858

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Recently, it was discovered that there is a connection between the growth factor of the partially commutative monoid, denoted as β(G), and the independence number of the graph.It was shown that β(G) ≥ α(G) and that log β(G) is as an upper bound for the classical zero-error capacity [1].In this paper, we demonstrate that β(G) is not only an upper bound for the Lovász number of G, but is also for the chromatic number of the complement G. Furthermore, we show that log β(G) is an upper bound for the quantum zero-error capacity.
The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lov\'asz theta function play an important role in zero-error communication. In this paper, we show that the Lov\'asz theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver.
A linear map LLL from Cn×n{\mathbb C}^{n \times n}Cn×n into Cn×n{\mathbb C}^{n \times n}Cn×n is called a quantum channel if it is completely positive and trace preserving. The set Ln{\cal L}_nLn​ of all such quantum channels is known to be a compact convex set. While the extreme points of Ln{\cal L}_nLn​ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix Z(L)Z(L)Z(L) associated with the quantum channel LLL, we compute the maximum dimension of a proper face of Ln{\cal L}_nLn​, and in addition the possible dimensions of faces generated by LLL when $rank \ Z(L)=2 $.
We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous ϑ function on general operator systems, as the norm-completion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovász' original ϑ in the classical case, give several applications, and propose to study the operator systems associated with channels as “noncommutative graphs,” using the language of Hilbert modules.