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arXiv:2010.09519 · Physical Review A

Optimal quantum channels

Rocco Duvenhage

Published Sep 7, 2021• 16 citations
arXivHTMLPDF

Abstract

A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal transport. To attach a clear conceptual meaning to the cost, channels are viewed in terms of what we call elementary transitions, which are analogous to point-to-point transitions between classical systems. The role of entanglement in optimization of cost is emphasized. We also show how our approach can be applied to theoretically search for channels performing a prescribed set of tasks on the states of a system, while otherwise disturbing the state as little as possible.

DOI
10.1103/physreva.104.032604

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A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal transport. To attach a clear conceptual meaning to the cost, channels are viewed in terms of what we call elementary transitions, which are analogous to point-to-point transitions between classical systems. The role of entanglement in optimization of cost is emphasized. We also show how our approach can be applied to theoretically search for channels performing a prescribed set of tasks on the states of a system, while otherwise disturbing the state as little as possible.
The capability of faithfully transmit quantum states and entanglement through quantum channels is one of the key requirements for the development of quantum devices. Different solutions have been proposed to accomplish such a challenging task, which, however, require either an ad hoc engineering of the internal interactions of the physical system acting as the channel or specific initialization procedures. Here we show that optimal dynamics for efficient quantum-state and entanglement transfer can be attained in generic quantum systems with homogeneous interactions by tuning the coupling between the system and the two attached qubits. We devise a general procedure to determine the optimal coupling, and we explicitly implement it in the case of a channel consisting of a spin-12\frac{1}{2}21​XY\mathit{XY}XY chain. The quality of quantum-state and entanglement transfer is found to be very good and, remarkably, almost independent of the channel length.
Spatial search and walk processes. Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database. One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk. Let S , with jS j D n, be a finite set of states. Assume that a subset M S of states are marked. We are given a procedure C that, on input x 2 S and an associated data structure d.x/, checks whether the state x is marked. The goal is either to find a marked state when promised that M ¤ ; (search version) or to determine whether M is nonempty (decision version). The algorithm progresses in stages. In the setup stage, we access some state of S (usually a random state). In the walk stage we move from state to state, performing a spatial walk as described below. The moves are called updates. In addition, in the walk stage we perform checks to see if the current state is marked at steps selected by the algorithm. In the classical setting, the transition probabilities of the spatial walk are described by a stochastic matrix P D .px;y/x;y2S . This makes the walk a Markov chain. In every move the algorithm must perform a random transition according to P . The possible x ! y moves, i.e., those with px;y ¤ 0, form the edges of a (directed) graph G, and we say that the Markov chain P has locality structure G. We define the search problem in the classical setting, which carries over to the quantum case with little modification:
We consider a realistic setting of quantum tasks that generate shared bipartite quantum states. Suppose \alice and \bob are located at different places and need to produce a target shared quantum state ρρρ. In order to save quantum communication, they can choose to share a proper smaller quantum state σσσ first, and then turn σσσ to ρρρ by performing only local quantum operations and classical communications (LOCC). We hope σσσ is the optimal such that the quantum communication needed is as little as possible, which is called the generation cost of ρρρ. In this paper, for an arbitrary bipartite ρρρ, we characterize its generation cost completely by proving that it is exactly equivalent to the logarithm of the Schmidt number of ρρρ. Similar quantum schemes where classical communication is not allowed have actually been considered. By comparing the two settings, we are able to look into the role that classical communication plays in these fundamental tasks, where we exhibit some instances in which classical communication is not helpful completely.
Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel, such as classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural cases in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality on which we elaborate in depth.