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Quantum channel simulation and discrimination with applications to quantum communications

Jason L. Pereira

Published Oct 1, 2020• 0 citations
OpenAlex

Abstract

Quantum channel discrimination is a highly versatile task in quantum information. Almost any physical process can be modelled as a quantum channel, so discrimination between channels has broad applications across many fields of science. Since the processes modelled by quantum channels and the contexts in which we want to discriminate between them are so wide-ranging, it should come as no surprise that the possible protocols are equally varied. The most general discrimination protocols can use any sequence of operations allowed by physics. \n \nThis is what makes protocol stretching such a powerful mathematical tool. It allows the calculation of bounds on the performance of any discrimination protocol and can be applied to any situation in which the channels involved are jointly simulable by some quantum processor. \n \nCertain channels, such as the amplitude damping channels, cannot be simulated using standard teleportation. Others, like the lossy bosonic channels, can be (individually) simulated using teleportation, but two lossy channels with different losses cannot be jointly simulated. \n \nIn this thesis, we characterise port-based teleportation so that it can be used as a tool for channel simulation. Port-based teleportation is a variant of quantum teleportation that can simulate any channel in the asymptotic limit of infinite ports. For a finite number of ports, we can find resource states that simulate amplitude damping channels well. By combining our channel simulations with the technique of protocol stretching, we are able to tighten existing bounds on the discrimination of amplitude-damping channels. \n \nWe also address the relatively unstudied subfield of channel position finding. We use channel simulation to bound the performance of environment localisation, and we show the viability of idler-free channel position finding over pure loss channels. Finally, we calculate the secret key rate for a scenario of quantum hacking based on a side channel in the sender's device.

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Quantum channel discrimination is a highly versatile task in quantum information. Almost any physical process can be modelled as a quantum channel, so discrimination between channels has broad applications across many fields of science. Since the processes modelled by quantum channels and the contexts in which we want to discriminate between them are so wide-ranging, it should come as no surprise that the possible protocols are equally varied. The most general discrimination protocols can use any sequence of operations allowed by physics. \n \nThis is what makes protocol stretching such a powerful mathematical tool. It allows the calculation of bounds on the performance of any discrimination protocol and can be applied to any situation in which the channels involved are jointly simulable by some quantum processor. \n \nCertain channels, such as the amplitude damping channels, cannot be simulated using standard teleportation. Others, like the lossy bosonic channels, can be (individually) simulated using teleportation, but two lossy channels with different losses cannot be jointly simulated. \n \nIn this thesis, we characterise port-based teleportation so that it can be used as a tool for channel simulation. Port-based teleportation is a variant of quantum teleportation that can simulate any channel in the asymptotic limit of infinite ports. For a finite number of ports, we can find resource states that simulate amplitude damping channels well. By combining our channel simulations with the technique of protocol stretching, we are able to tighten existing bounds on the discrimination of amplitude-damping channels. \n \nWe also address the relatively unstudied subfield of channel position finding. We use channel simulation to bound the performance of environment localisation, and we show the viability of idler-free channel position finding over pure loss channels. Finally, we calculate the secret key rate for a scenario of quantum hacking based on a side channel in the sender's device.
In this thesis we investigate the fundamental limitations that the laws of the quantum nature impose on the performance of quantum communications, quantum metrology and quantum channel discrimination. In a quantum communication scenario, the typical tasks are represented by the simple transmission of quantum bits, the distribution of entangle- ment and the sharing of quantum secret keys. The ultimate rates for each of these protocols are given by the two-way quantum capacities of the quantum channel which are in turn defined by considering the most general adaptive strategies that can be implemented over the channel. To assess these quantum capacities, we combine the simulation of quantum channels, suitably generalized to systems of arbitrary dimension, with quantum telepor- tation and the relative entropy of entanglement. This procedure is called teleportation stretching. Relying on this, we are able to reduce any adaptive protocols into simpler block ones and to determine the tightest upper bound on the two-way quantum capacities. Re- markably, we also prove the existence of a particular class of quantum channel for which the lower and the upper bounds coincide. By employing a slight modification of the tele- portation scheme, allowing the two parties to share a multi-copy resource state, we apply our technique to simplify adaptive protocols for quantum metrology and quantum channel discrimination. In the first case we show that the modified teleportation stretching implies a quantum Cram ́er-Rao bound that follows asymptotically the Heisenberg scaling. In the second scenario we are able to derive the only known so far fundamental lower bound on the probability of error affecting the discrimination of two arbitrary finite-dimensional quantum channels.
Abstract What is the ultimate performance for discriminating two arbitrary quantum channels acting on a finite-dimensional Hilbert space? Here we address this basic question by deriving a general and fundamental lower bound. More precisely, we investigate the symmetric discrimination of two arbitrary qudit channels by means of the most general protocols based on adaptive (feedback-assisted) quantum operations. In this general scenario, we first show how port-based teleportation can be used to simplify these adaptive protocols into a much simpler non-adaptive form, designing a new type of teleportation stretching. Then, we prove that the minimum error probability affecting the channel discrimination cannot beat a bound determined by the Choi matrices of the channels, establishing a general, yet computable formula for quantum hypothesis testing. As a consequence of this bound, we derive ultimate limits and no-go theorems for adaptive quantum illumination and single-photon quantum optical resolution. Finally, we show how the methodology can also be applied to other tasks, such as quantum metrology, quantum communication and secret key generation.
Amplitude-damping (AD) channels are good models for many physical scenarios, and so the development of protocols to discriminate between them is an important task in quantum information science. It is therefore important to bound the performance of such protocols. Since adaptivity has been shown to improve the performance of discrimination protocols, bounds on the distinguishability of AD channels must take this into account. In this paper, we use both channel simulation and a bound based on the diamond norm to significantly tighten the upper bound on the trace norm between the possible outputs of binary channel discrimination protocols acting on AD channels (and hence the lower bound on the error probability of such protocols). The diamond norm between any two AD channels is found analytically, giving the optimal error probability for a one-shot discrimination protocol. We also present a tighter lower bound on the achievable trace norm between protocol outputs (and a corresponding upper bound on the achievable error probability). The upper and lower bounds are compared with existing bounds and then applied to quantum hacking and biological quantum sensing scenarios.
Abstract Port-based teleportation (PBT) is a teleportation protocol that employs a number of Bell pairs and a joint measurement to enact an approximate input-output identity channel. Replacing the Bell pairs with a different multi-qubit resource state changes the enacted channel and allows the PBT protocol to simulate qubit channels beyond the identity. The channel resulting from PBT using a general resource state is consequently of interest. In this work, we fully characterise the Choi matrix of the qubit channel simulated by the PBT protocol in terms of its resource state. We also characterise the PBT protocol itself, by finding a description of the map from the resource state to the Choi matrix of the channel that is simulated by using that resource state. Finally, we exploit our expressions to show improved simulations of the amplitude damping channel by means of PBT with a finite number of ports.
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.
The characterisation of Quantum Channel Discrimination (QCD) offers critical insight for future quantum technologies in quantum metrology, sensing and communications. The task of multi-channel discrimination creates a scenario in which the discrimination of multiple quantum channels can be equated to the idea of pattern recognition, highly relevant to the tasks of quantum reading, illumination and more. Whilst the optimal quantum strategy for many scenarios is an entangled idler-assisted protocol, the extension to a multi-hypothesis setting invites the exploration of discrimination strategies based on unassisted, multipartite probe states. In this work, we expand the space of possible quantum enhanced protocols by formulating general classes of unassisted multi-channel discrimination protocols which are not assisted by idler modes. Developing a general framework for idler-free protocols, we perform an explicit investigation in the bosonic setting, studying prominent Gaussian channel discrimination problems for real world applications. Our findings uncover the existence of strongly quantum advantageous, idler-free protocols for the discrimination of bosonic loss and environmental noise. This circumvents the necessity for idler assistance to achieve quantum advantage in some of the most relevant discrimination settings, significantly loosening practical requirements for prominent quantum sensing applications.