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Lobachevskii Journal of Mathematics

Estimation of Capacity for Channels Connected with Elementary Quantum Random Walks

S. Grishin

Published Oct 1, 2024• 1 citation
Publisher

Abstract

Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length $$\log(\frac{e}{2})$$ .

DOI
10.1134/s1995080224605666

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Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity of such channels can be accounted via Holevo upper bound, which is evaluated here. Upper estimation is the difference between maximal and minimal output entropy, and lower estimation is given by an ensemble of states with minimal output entropy, such that the output entropy for medium state is as big as possible. For these computations, we use complementary channels because they are much easier to deal with. Both estimations coincide in case of dimension 2. Moreover, in this case the complementary channel is unital cubit, so that the Holevo upper bound equals to the capacity. In large dimensions, the asymptotical estimation is an interval of length log⁡(e2)\log(\frac{e}{2})log(2e​) .
Abstract Finite-dimensional restrictions of the channel describing elementary quantum random walk are discussed in this paper. Capacity with entangled state of such channels can be accounted via maximal mutual information, and quantum capacity can be computed via maximal coherent information. Both ones are evaluated here. There is some evidence that the states with maximal mutual and coherent information are diagonal. We managed to prove it in case of dimension 2. The channels turn out to be degradable, and this fact simplifies calculation of quantum capacity. We compared our estimations with classical capacity and concluded that entangled state benefits a lot, and quantum information is possible but slightly more difficult to transmit via such channels than classical one.
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3log(N)1/2ε-1) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.
We present a family of easily computable upper bounds for the Holevo quantity of ensemble of quantum states depending on a reference state as a free parameter. These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve existing estimates in the literature. We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemble. The specification of this upper bound for the multi-mode quantum oscillator is tight for large energy. The above results are used to obtain tight upper bounds for the Holevo capacity of finite-dimensional and infinite-dimensional energy-constrained quantum channels depending on metric characteristics of the channel output.
We establish unbounded two-use Holevo additivity gaps in finite dimensions. For each sufficiently large fixed integer KKK and all sufficiently large nnn, we construct channels TnT_nTn​ with output dimension KnK^nKn, input dimension exp⁡(ΘK(n2))\exp(Θ_K(n^2))exp(ΘK​(n2)), and χ(Tn⊗2)−2χ(Tn)≥n[log⁡2KK−2log⁡2(1+9/K)]−O(1/n). χ(T_n^{\otimes2})-2χ(T_n) \ge n\left[\frac{\log_2K}{K}-2\log_2(1+9/K)\right]-O(1/n). χ(Tn⊗2​)−2χ(Tn​)≥n[Klog2​K​−2log2​(1+9/K)]−O(1/n). The gap is linear in output qubits, with an explicit quadratic input-qubit cost and an explicit threshold on nnn. We also obtain channels whose single-use Holevo quantity tends to zero while their two-use Holevo information, and hence classical capacity, diverges. The channels arise from structured tensor products of the complementary mixed-unitary channels used in Collins's free-probabilistic proof. We establish minimum-output-entropy gaps by combining Collins--Youn's product-group Haagerup inequality with Bordenave--Collins's quantitative strong-convergence estimates. We also provide a Lean certificate for our proofs.
We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) NNN-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the NNN-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.