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arXiv:1905.11920 · Physical Review A

Quantum-capacity bounds in spin-network communication channels

Stefano Chessa, Marco Fanizza, Vittorio Giovannetti

Published Sep 9, 2019• 8 citations
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Abstract

Using the Lieb-Robinson inequality and the continuity property of the quantum capacities in terms of the diamond norm, we derive an upper bound on the values that these capacities can attain in spin-network communication i.i.d. models of arbitrary topology. Different from previous results we make no assumptions about the encoding mechanisms that the sender of the messages adopts in loading information on the network.

DOI
10.1103/physreva.100.032311

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Using the Lieb-Robinson inequality and the continuity property of the quantum capacities in terms of the diamond norm, we derive an upper bound on the values that these capacities can attain in spin-network communication i.i.d. models of arbitrary topology. Different from previous results we make no assumptions about the encoding mechanisms that the sender of the messages adopts in loading information on the network.
The Lieb-Robinson bound sets a theoretical upper limit on the speed at which information can propagate in nonrelativistic quantum spin networks. In its original version, it results in an exponentially exploding function of the evolution time, which is partially mitigated by an exponentially decreasing term that instead depends upon the distance covered by the signal (the ratio between the two exponents effectively defining an upper bound on the propagation speed). In the present paper, by properly accounting for the free parameters of the model, we show how to turn this construction into a stronger inequality where the upper limit only scales polynomially with respect to the evolution time. Our analysis applies to any chosen topology of the network, as long as the range of the associated interaction is explicitly finite. For the special case of linear spin networks we present also an alternative derivation based on a perturbative expansion approach which improves the previous inequality. In the same context we also establish a lower bound to the speed of the information spread which yields a nontrivial result at least in the limit of small propagation times.
We consider quantum and private communications assisted by repeaters, from the basic scenario of a single repeater chain to the general case of an arbitrarily-complex quantum network, where systems may be routed through single or multiple paths. In this context, we investigate the ultimate rates at which two end-parties may transmit quantum information, distribute entanglement, or generate secret keys. These end-to-end capacities are defined by optimizing over the most general adaptive protocols that are allowed by quantum mechanics. Combining techniques from quantum information and classical network theory, we derive single-letter upper bounds for the end-to-end capacities in repeater chains and quantum networks connected by arbitrary quantum channels, establishing exact formulas under basic decoherence models, including bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels. For the converse part, we adopt a teleportation-inspired simulation of a quantum network which leads to upper bounds in terms of the relative entropy of entanglement. For the lower bounds we combine point-to-point quantum protocols with classical network algorithms. Depending on the type of routing (single or multiple), optimal strategies corresponds to finding the widest path or the maximum flow in the quantum network. Our theory can also be extended to simultaneous quantum communication between multiple senders and receivers.
A new upper bound for the quantum capacity of the d-dimensional depolarizing channels is presented. Our derivation makes use of a flagged extension of the map where the receiver obtains a copy of a state σ_{0} whenever the messages are transmitted without errors, and a copy of a state σ_{1}, when instead the original state gets fully depolarized. By varying the overlap between the flag states, the resulting transformation nicely interpolates between the depolarizing map (when σ_{0}=σ_{1}), and the d-dimensional erasure channel (when σ_{0} and σ_{1} have orthogonal support). We find sufficient conditions for degradability of the flagged channel, which let us calculate its quantum capacity in a suitable parameter region. From this last result we get the upper bound for the depolarizing channel, which by a direct comparison appears to be tighter than previous available results for d>2, and for d=2 it is tighter in an intermediate regime of noise. In particular, in the limit of large d values, our findings present a previously unnoticed O(1) correction.
Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing(ε,ν)(\varepsilon ,\nu)(ε,ν)–degradable channels. These are$\varepsilon –closeindiamondnormand–close in diamond norm and–closeindiamondnormand\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.