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Lobachevskii Journal of Mathematics

Different Types of Capacity for Channels Connected with Elementary Quantum Random Walks

S. Grishin

Published Jun 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1995080225607829

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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.
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​) .
Entanglement and entanglement assisted are useful resources to enhance the mutual information of the Pauli channels, when the noise on consecutive uses of the channel has some partial correlations. In this paper, we study quantum-communication channels in ddd-dimensional systems and derive the mutual information of the quantum channels for maximally entangled states and product states coding with correlated noise. Then, we compare fidelity between these states. Our results show that there exists a certain fidelity memory threshold, which depends on the dimension of the Hilbert space (d)(d)(d) and the properties of noisy channels. We calculate the classical capacity of a particular correlated noisy channel and show that in order to achieve Holevo limit, we must use ddd particles with ddd degrees of freedom. Our results show that entanglement is a useful means to enhance the mutual information. We choose a special nonmaximally entangled state and show that in the quasiclassical depolarizing and quantum depolarizing channels, maximum classical capacity in the higher memory channels is given by the maximally entangled state. Hence, our results show that for high error channels in every degree of memory, maximally entangled states have better mutual information.
We study the capacity of ddd-dimensional quantum channels with memory modeled by correlated noise. We show that, in agreement with previous results on Pauli qubit channels, there are situations where maximally entangled input states achieve higher values of mutual information than product states. Moreover, a strong dependence of this effect on the nature of the noise correlations as well as on the parity of the space dimension is found. We conjecture that when entanglement gives an advantage in terms of mutual information, maximally entangled states saturate the channel capacity.
Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and is well understood for channels modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. This is because the formula for the quantum capacity involves maximizing the coherent information over an unbounded number of channel uses. In fact, entanglement across channel uses can even increase the coherent information from zero to non-zero. Here we study the number of channel uses necessary to detect positive coherent information. In all previous known examples, two channel uses already sufficed. It might be that only a finite number of channel uses is always sufficient. We show that this is not the case: for any number of uses, there are channels for which the coherent information is zero, but which nonetheless have capacity. The transmission of quantum information through channels is a fundamental step for future quantum communication technologies. Cubitt et al.now show that there exist channels whose potential for transmitting quantum information requires an unbounded number of usages to be detected.
We consider explicitly two examples of d-dimensional quantum channels with correlated noise and show that, in agreement with previous results on Pauli qubit channels, there are situations where maximally entangled input states achieve higher values of the output mutual information than product states. We obtain a strong dependence of this effect on the nature of the noise correlations as well as on the parity of the space dimension, and conjecture that when entanglement gives an advantage in terms of mutual information, maximally entangled states achieve the channel capacity.
We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multiletter characterizations of two different two-dimensional capacity regions. The first region comprises the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.