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arXiv:2203.05511 · IEEE International Symposium on Information Theory (ISIT)

Multi-Party Quantum Purity Distillation with Bounded Classical Communication

Touheed Anwar Atif, S. Sandeep Pradhan

Published Jun 26, 2022• 3 citations
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Abstract

We consider the task of distilling local purity from a noisy quantum state ρABC, wherein we provide a protocol for three parties, Alice, Bob and Charlie, to distill local purity (at a rate P) from many independent copies of a given quantum state ρABC. The three parties have access to their respective subsystems of ρABC, and are only allowed to use local unitary operations. In addition, Alice and Bob can communicate with Charlie using a one-way multiple-access dephasing channel of link rates R1and R2, respectively. The objective of the protocol is to minimize the usage of the dephasing channel (in terms of rates R1and R2) while maximizing the asymptotic purity that can be jointly distilled from ρABC. To achieve this, we employ ideas from distributed measurement compression protocols, and in turn, characterize a set of sufficient conditions on (P, R1,R2) in terms of quantum information theoretic quantities such that P amount of purity can be distilled using rates R1and R2.

DOI
10.1109/isit50566.2022.9834596

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We consider the task of distilling local purity from a noisy quantum state ρABC, wherein we provide a protocol for three parties, Alice, Bob and Charlie, to distill local purity (at a rate P) from many independent copies of a given quantum state ρABC. The three parties have access to their respective subsystems of ρABC, and are only allowed to use local unitary operations. In addition, Alice and Bob can communicate with Charlie using a one-way multiple-access dephasing channel of link rates R1and R2, respectively. The objective of the protocol is to minimize the usage of the dephasing channel (in terms of rates R1and R2) while maximizing the asymptotic purity that can be jointly distilled from ρABC. To achieve this, we employ ideas from distributed measurement compression protocols, and in turn, characterize a set of sufficient conditions on (P, R1,R2) in terms of quantum information theoretic quantities such that P amount of purity can be distilled using rates R1and R2.
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In this paper we revisit the problem of distributed purity distillation in the one-shot setting. The problem considers a bipartite quantum state ρABshared between two parties Alice and Bob, who are allowed local unitary operations and one-way classical communication. The goal is to extract pure qubit states using these given resources, in their respective laboratories. A caveat of this problem is that all known protocols which recover the optimal number of pure qubit states require some pure qubits at the beginning at the protocol, to be used as ancilla. In this paper, we present a one-shot purity distillation protocol which requires a supply of ancilla qubits at 0 rate in the asymptotic iid setting, and little to no ancilla in the one-shot setting. We show that our protocol outperforms the best known purity distillation algorithms in terms of requirement of additional ancilla, while maintaining the same rate of pure states distilled.
Local pure states represent a fundamental resource in quantum information theory. In this work we obtain one-shot achievable bounds on the rates for local purity distillation, in the single-party and in the two-party cases. In both situations, local noisy operations are freely available, while in the two-party case also one-way classical communication can be used. In addition, in both situations local pure ancillas can be borrowed, as long as they are discounted from the final net rate of distillation. The one-shot rates that we obtain, written in terms of mutual information-like quantities, are shown to recover in the limit the asymptotic i.i.d. rates of Devetak [PRA, 2005], up to first order analysis.
Pure states are an important resource in many quantum information processing protocols. However, even making a fixed pure state, say ∣0⟩|0\rangle∣0⟩, in the laboratory requires a considerable amount of effort. Often one ends up with a mixed state ρρρ whose classical description is nevertheless known. Hence it is important to develop protocols that extract a fixed pure state from a known mixed state. In this work, we study the problem of extracting a fixed pure state ∣0⟩A′∣0⟩B′|0\rangle^{A'} |0\rangle^{B'}∣0⟩A′∣0⟩B′ from a known pure state ρABρ^{AB}ρAB distributed between two parties AAA and BBB. Here, A′A'A′, B′B'B′ are subspaces of AAA, BBB and the total amount of purity extracted is log⁡∣A′∣+log⁡∣B′∣\log |A'| + \log |B'|log∣A′∣+log∣B′∣. The parties can borrow local pure ancilla, apply local unitary operations and send a message from AAA to BBB through a dephasing channel. If local pure ancilla is borrowed, it must be subtracted in order to properly account for the purity extracted. We obtain the most efficient achievable bounds on one shot distributed purity extraction, in terms of the rate of local ancilla borrowed by the protocol, while distilling pure qubits at the best known rate. Our protocols borrow little to no local pure ancilla. Our bounds improve upon the existing bounds for this problem in both one shot as well as asymptotic iid settings. In particular they subsume all the asymptotic iid results of Devetak and Krovi-Devetak. In addition, we derive upper bounds for the rate of distillation in the one shot setting, which nearly match our achievable bounds.
Quantum network states are multipartite states built from distributing pairwise entanglement among parties and underpin the paradigm of quantum networks for quantum information processing. In this work we introduce the problem of partial distillability in noisy quantum networks. This corresponds to the possibility of distilling by local operations and classical communication an arbitrary pure state for an arbitrary subset of parties starting from a single copy of a quantum network state of an increasing number of parties that share noisy (i.e. mixed) bipartite entangled states. Here, distillation means that the target state is obtained with fidelity as close to 1 as desired as the size of the network increases when the pairwise entangled links support a constant amount of noise. While we prove an obstruction to multipartite distillation protocols after teleportation with channels with constant noise, we show that partial distillability is indeed possible if certain well-established graph-theoretic parameters that measure the connectivity in the network grow fast enough with its size. We give necessary as well as sufficient conditions for partial distillability in terms of these parameters and we moreover provide explicit constructions of networks with partial distillability and a relatively slow connectivity growth.
Quantum algorithms require quantum computers performing logical operations on a sufficiently large number of entangled qubits. Unfortunately, state-of-the-art quantum computers can only operate on tens of qubits. A solution to this scalability challenge is to employ the distributed paradigm, where a network of small-scale quantum computers are used in a distributed manner. It is the aim of my work to study the fundamental limits of distributed quantum problems and to further enhance their performance by exploiting the structure inherent to these problems employing asymptotically good Algebraic codes. This thesis consists of two parts. The first part studies the task of faithfully simulating a distributed quantum measurement, wherein we provide a protocol for the three parties, Alice, Bob and {Charlie}, to simulate a repeated action of a distributed quantum measurement using a pair of non-product approximating measurements by Alice and Bob, followed by a stochastic mapping at {Charlie}. The objective of the protocol is to utilize minimum resources, in terms of classical bits needed by Alice and Bob to communicate their measurement outcomes to {Charlie}, and the common randomness shared among the three parties, while faithfully simulating independent repeated instances of the original measurement. We characterize a set of sufficient communication and common randomness rates required for asymptotic simulatability in terms of single-letter quantum information quantities. We further improve the results obtained in the above by exploiting the structure present in the Charlie's stochastic bivariate mapping using random structured POVMs based on asymptotically good algebraic codes. The algebraic structure of these codes is matched to that of the bivariate function that models the action of Charlie. This leads to the computation being performed on the fly, thus obviating the need to reconstruct individual measurement outcomes at Charlie. We provide examples to illustrate the information-theoretic gains attained by endowing POVMs with algebraic structure. As an application of the distributed measurement compression problem, we also demonstrate a multi-party purity distillation protocol. Concluding this part, we consider the lossy quantum source coding problem where the objective is to compress a given quantum source below its von Neumann entropy. Inspired by the duality connections between the rate-distortion and channel coding problems in the classical setting, we propose a new formulation for the lossy quantum source coding problem. We require that the reconstruction of the compressed quantum source fulfill a global error constraint and employ the notion of a ``posterior reference map'' to measure the reconstruction error. Using these, we characterize the asymptotic performance limit of this problem in terms of single-letter coherent information of the given posterior reference map. In the second part of this thesis, we study the advantage algebraic structured codes can provide, to the class of the classical-quantum network problems. In particular, we investigate two problems. Firstly, we consider the problem of communicating a general bivariate function of two classical sources observed at the encoders of a classical-quantum multiple access channel. We propose and analyze a coding scheme based on algebraic structured coset codes that enables the decoder to recover the desired function without recovering the sources themselves. We derive a new set of sufficient conditions that are weaker than the current known for identified examples. In addition, we analyze the performance of these algebraic codes toward studying the capacity of a special class of 3-user classical-quantum interference channel.