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arXiv:1712.00145 · Physical Review A

Strong and uniform convergence in the teleportation simulation of bosonic Gaussian channels

Mark M. Wilde

Published Jun 5, 2018• 19 citations
arXivHTMLPDF

Abstract

In the literature on the continuous-variable bosonic teleportation protocol due to Braunstein and Kimble [Phys. Rev. Lett. 80, 869 (1998)], it is often loosely stated that this protocol converges to a perfect teleportation of an input state in the limit of ideal squeezing and ideal detection, but the exact form of this convergence is typically not clarified. In this paper, I explicitly clarify that the convergence is in the strong sense, and not the uniform sense, and furthermore that the convergence occurs for any input state to the protocol, including the infinite-energy Basel states defined and discussed here. I also prove, in contrast to the above result, that the teleportation simulations of pure-loss, thermal, pure-amplifier, amplifier, and additive-noise channels converge both strongly and uniformly to the original channels, in the limit of ideal squeezing and detection for the simulations. For these channels, I give explicit uniform bounds on the accuracy of their teleportation simulations. I then extend these uniform convergence results to particular multimode bosonic Gaussian channels. These convergence statements have important implications for mathematical proofs that make use of the teleportation simulation of bosonic Gaussian channels, some of which have to do with bounding their nonasymptotic secret-key-agreement capacities. As a by-product of the discussion given here, I confirm the correctness of the proof of such bounds from my joint work with Berta and Tomamichel from [Wilde, Tomamichel, and Berta, IEEE Trans. Inf. Theory 63, 1792 (2017)]. Furthermore, I show that it is not necessary to invoke the energy-constrained diamond distance in order to confirm the correctness of this proof.

DOI
10.1103/physreva.97.062305

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In the literature on the continuous-variable bosonic teleportation protocol due to Braunstein and Kimble [Phys. Rev. Lett. 80, 869 (1998)], it is often loosely stated that this protocol converges to a perfect teleportation of an input state in the limit of ideal squeezing and ideal detection, but the exact form of this convergence is typically not clarified. In this paper, I explicitly clarify that the convergence is in the strong sense, and not the uniform sense, and furthermore that the convergence occurs for any input state to the protocol, including the infinite-energy Basel states defined and discussed here. I also prove, in contrast to the above result, that the teleportation simulations of pure-loss, thermal, pure-amplifier, amplifier, and additive-noise channels converge both strongly and uniformly to the original channels, in the limit of ideal squeezing and detection for the simulations. For these channels, I give explicit uniform bounds on the accuracy of their teleportation simulations. I then extend these uniform convergence results to particular multimode bosonic Gaussian channels. These convergence statements have important implications for mathematical proofs that make use of the teleportation simulation of bosonic Gaussian channels, some of which have to do with bounding their nonasymptotic secret-key-agreement capacities. As a by-product of the discussion given here, I confirm the correctness of the proof of such bounds from my joint work with Berta and Tomamichel from [Wilde, Tomamichel, and Berta, IEEE Trans. Inf. Theory 63, 1792 (2017)]. Furthermore, I show that it is not necessary to invoke the energy-constrained diamond distance in order to confirm the correctness of this proof.
We consider the Braunstein–Kimble protocol for continuous variable teleportation and its application for the simulation of bosonic channels. We discuss the convergence properties of this protocol under various topologies (strong, uniform, and bounded-uniform) clarifying some typical misinterpretations in the literature. We then show that the teleportation simulation of an arbitrary single-mode Gaussian channel is uniformly convergent to the channel if and only if its noise matrix has full rank. The various forms of convergence are then discussed within adaptive protocols, where the simulation error must be propagated to the output of the protocol by means of a “peeling” argument, following techniques from PLOB [S. Pirandola et al., Nat. Comm. 8, 15043 (2017)]. Finally, as an application of the peeling argument and the various topologies of convergence, we provide complete rigorous proofs for recently claimed strong converse bounds for private communication over Gaussian channels.
We consider the Braunstein-Kimble protocol for continuous variable teleportation and its application for the simulation of bosonic channels. We discuss the convergence properties of this protocol under various topologies (strong, uniform, and bounded-uniform) clarifying some typical misinterpretations in the literature. We then show that the teleportation simulation of an arbitrary single-mode Gaussian channel is uniformly convergent to the channel if and only if its noise matrix has full rank. The various forms of convergence are then discussed within adaptive protocols, where the simulation error must be propagated to the output of the protocol by means of a "peeling" argument, following techniques from PLOB [arXiv:1510.08863]. Finally, as an application of the peeling argument and the various topologies of convergence, we provide complete rigorous proofs for recently-claimed strong converse bounds for private communication over Gaussian channels.
Abstract We review recent results on the simulation of quantum channels, the reduction of adaptive protocols (teleportation stretching), and the derivation of converse bounds for quantum and private communication, as established in PLOB (Pirandola et al 2017 Nat. Commun. 8 15043). We start by introducing a general weak converse bound for private communication based on the relative entropy of entanglement. We discuss how combining this bound with channel simulation and teleportation stretching, PLOB established the two-way quantum and private capacities of several fundamental channels, including the bosonic lossy channel. We then provide a rigorous proof of the strong converse property of these bounds by adopting a correct use of the Braunstein–Kimble teleportation protocol for the simulation of bosonic Gaussian channels. This analysis provides a full justification of claims presented in the follow-up paper WTB (Wilde et al 2017 IEEE Trans. Inf. Theory 63 1792–817) whose upper bounds for Gaussian channels would be otherwise infinitely large. Besides clarifying contributions in the area of channel simulation and protocol reduction, we also present some generalizations of the tools to other entanglement measures and novel results on the maximum excess noise which is tolerable in quantum key distribution.
We consider private communication over bosonic Gaussian channels via the most general adaptive protocols based on local operations and two-way classical communication. These protocols include all the possible strategies allowed by quantum mechanics where two remote parties have local quantum computers but do not share prior quantum entanglement. In this context, Pirandola-Laurenza-Ottaviani-Banchi (PLOB) [Nat. Commun. 8, 15043 (2017)] established weak converse upper bounds for the secret key capacity of these channels. These bounds were computed by combining teleportation stretching, able to simplify any adaptive protocol into a block form, and the channel’s relative entropy of entanglement, so that data-processing properties allow one to write simple single-letter quantities. Here we discuss an extension of these bounds to repeater-assisted quantum communications. Then, using an energy-constrained version of the diamond norm and the Braunstein-Kimble teleportation protocol, we can rigorously show the strong converse property of the bounds discovered by PLOB. Our analysis provides a full mathematical justification of recent claims appeared in the literature.
Upper bounds on the secret-key-agreement capacity of a quantum channel serve as a way to assess the performance of practical quantum-key-distribution protocols conducted over that channel. In particular, if a protocol employs a quantum repeater, achieving secret-key rates exceeding these upper bounds is evidence of having a working quantum repeater. In this paper, we extend a recent advance [Liuzzo-Scorpo et al., Phys. Rev. Lett. 119, 120503 (2017)] in the theory of the teleportation simulation of single-mode phase-insensitive Gaussian channels such that it now applies to the relative entropy of entanglement measure. As a consequence of this extension, we find tighter upper bounds on the nonasymptotic secret-key-agreement capacity of the lossy thermal bosonic channel than were previously known. The lossy thermal bosonic channel serves as a more realistic model of communication than the pure-loss bosonic channel, because it can model the effects of eavesdropper tampering and imperfect detectors. An implication of our result is that the previously known upper bounds on the secret-key-agreement capacity of the thermal channel are too pessimistic for the practical finite-size regime in which the channel is used a finite number of times, and so it should now be somewhat easier to witness a working quantum repeater when using secret-key-agreement capacity upper bounds as a benchmark.
Quantum teleportation of an unknown broadband electromagnetic field is investigated. The continuous-variable teleportation protocol by Braunstein and Kimble [Phys. Rev. Lett. 80, 869 (1998)] for teleporting the quantum state of a single mode of the electromagnetic field is generalized for the case of a multimode field with finite bandwith. We discuss criteria for continuous-variable teleportation with various sets of input states and apply them to the teleportation of broadband fields. We first consider as a set of input fields (from which an independent state preparer draws the inputs to be teleported) arbitrary pure Gaussian states with unknown coherent amplitude (squeezed or coherent states). This set of input states, further restricted to an alphabet of coherent states, was used in the experiment by Furusawa et al. [Science 282, 706 (1998)]. It requires unit-gain teleportation for optimizing the teleportation fidelity. In our broadband scheme, the excess noise added through unit-gain teleportation due to the finite degree of the squeezed-state entanglement is just twice the (entanglement) source's squeezing spectrum for its ``quiet quadrature.'' The teleportation of one half of an entangled state (two-mode squeezed vacuum state), i.e., ``entanglement swapping,'' and its verification are optimized under a certain nonunit gain condition. We will also give a broadband description of this continuous-variable entanglement swapping based on the single-mode scheme by van Loock and Braunstein [Phys. Rev. A 61, 10 302 (2000)].