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arXiv:1904.11769

The Role of Entanglement in Quantum Communication, and Analysis of the Detection Loophole

Thomas Cope

Published Apr 26, 2019• 2 citations
arXivHTMLPDF

Abstract

We begin this thesis by expanding the technique of teleportation simulation, which adds noise to the entangled resource state to mimic channel effects. By introducing classical noise in the communication step, we show it is possible to simulate more than just Pauli channels using teleportation. This new class is characterised, and studied in detail for a particular resource state, leading to a family of simulable channels named "Pauli-Damping channels" whose properties are analysed. Also introduced are a new family of quantum states, "phase Werner" states, whose entanglement properties relate to the interesting conjecture of bound entangled states with a negative partial transpose. Holevo-Werner channels, to which these states are connected, are shown to be teleportation covariant. We exploit this to present several interesting results, including the optimal estimation of the channel-defining parameter. The minimal binary-discrimination error for Holevo-Werner channels is bounded for the first time with the analytical form of the quantum Chernoff bound. We also consider the secret key capacity of these channels, showing how different entanglement measures provide a better upper bound for different regions of these channels. Finally, a method for generating new Bell inequalities is presented, exploiting nonphysical probability distributions to obtain new inequalities. Tens of thousands of new inequivalent inequalities are generated, and their usefulness in closing the detection loophole for imperfect detectors is examined, with comparison to the current optimal construction. Two candidate Bell inequalities which may equal or beat the best construction are presented.

DOI
10.48550/arxiv.1904.11769

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We begin this thesis by expanding the technique of teleportation simulation, which adds noise to the entangled resource state to mimic channel effects. By introducing classical noise in the communication step, we show it is possible to simulate more than just Pauli channels using teleportation. This new class is characterised, and studied in detail for a particular resource state, leading to a family of simulable channels named "Pauli-Damping channels" whose properties are analysed. Also introduced are a new family of quantum states, "phase Werner" states, whose entanglement properties relate to the interesting conjecture of bound entangled states with a negative partial transpose. Holevo-Werner channels, to which these states are connected, are shown to be teleportation covariant. We exploit this to present several interesting results, including the optimal estimation of the channel-defining parameter. The minimal binary-discrimination error for Holevo-Werner channels is bounded for the first time with the analytical form of the quantum Chernoff bound. We also consider the secret key capacity of these channels, showing how different entanglement measures provide a better upper bound for different regions of these channels. Finally, a method for generating new Bell inequalities is presented, exploiting nonphysical probability distributions to obtain new inequalities. Tens of thousands of new inequivalent inequalities are generated, and their usefulness in closing the detection loophole for imperfect detectors is examined, with comparison to the current optimal construction. Two candidate Bell inequalities which may equal or beat the best construction are presented.
Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the resource for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation of a less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels.
We consider the simulation of a quantum channel by two parties who share a resource state and may apply local operations assisted by classical communication (LOCC). One specific type of such LOCC is standard teleportation, which is however limited to the simulation of Pauli channels. Here we show how we can easily enlarge this class by means of a minimal perturbation of the teleportation protocol, where we introduce noise in the classical communication channel between the remote parties. By adopting this noisy protocol, we provide a necessary condition for simulating a non-Pauli channel. In particular, we characterize the set of channels that are generated assuming the Choi matrix of an amplitude damping channel as a resource state. Within this set, we identify a class of Pauli-damping channels for which we bound the two-way quantum and private capacities.
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 Werner states have a host of interesting properties, which often serve to illuminate the unusual properties of quantum information. Starting from these states, one may define a family of quantum channels, known as the Holevo–Werner channels, which themselves afford several unusual properties. In this paper we use the teleportation covariance of these channels to upper bound their two-way assisted quantum and secret-key capacities. This bound may be expressed in terms of relative entropy distances, such as the relative entropy of entanglement, and also in terms of the squashed entanglement. Most interestingly, we show that the relative entropy bounds are strictly subadditive for a sub-class of the Holevo–Werner channels, so that their regularization provides a tighter performance. These information-theoretic results are first found for point-to-point communication and then extended to repeater chains and quantum networks, under different types of routing strategies.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.
Understanding the characteristics of a quantum systems when affected by noise is one of the biggest challenges for quantum technologies. The general Pauli error channel is an important lossless channel for quantum communication. In this work we consider the effects of a Pauli channel on a pure four-qubit state and simulate the Pauli channel experimentally by studying the action on polarization encoded entangled photons. When the noise channel acting on the photons is correlated, a set spanned by four orthogonal bound entangled states can be generated. We study this interesting case experimentally and demonstrate that products of Bell states can be brought into a bound entangled regime. We find states in the set of bound entangled states which experimentally violate the CHSH inequality while still possessing a positive partial transpose.