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Institute of Physics eBooks

Quantum states

Paul Skrzypczyk, Daniel Cavalcanti

Published Mar 1, 2023• 0 citations
Publisher

Abstract

Chapter 3 addresses one of most basic problems in quantum physics, how to determine the properties of quantum states produced by an uncharacterised source. There are numerous variants of this problem, the most well-known being quantum state tomography, that is, estimating exactly which state the source is emitting. In this chapter we review this and other variants of state estimation, including the so-called quantum marginal problem. This will allow us to study various aspects of semidefinite programs, including how to obtain certificates of feasibility and the prevalent notion of a ‘witness’.

DOI
10.1088/978-0-7503-3343-6ch3

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Chapter 3 addresses one of most basic problems in quantum physics, how to determine the properties of quantum states produced by an uncharacterised source. There are numerous variants of this problem, the most well-known being quantum state tomography, that is, estimating exactly which state the source is emitting. In this chapter we review this and other variants of state estimation, including the so-called quantum marginal problem. This will allow us to study various aspects of semidefinite programs, including how to obtain certificates of feasibility and the prevalent notion of a ‘witness’.
In chapter 4, we shift focus to quantum measurements. We will first discuss the problem of estimating which measurement a given measuring device is performing. We then study quantum state discrimination, the problem of determining—by performing a measurement—which state, out of a set of possibilities, a given source is producing. We show how to gain new insight into this problem by using the key concept of semidefinite programming duality, and how the tool of complementary slackness can be used to derive optimality conditions.
In recent work [Phys. Rev. X 16, 021050], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefinite-programming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. These theory-dependent methods complement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.
Abstract In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/ process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantumstate and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.
A fundamental problem in quantum physics is to establish whether a multiparticle quantum state can be uniquely determined from its local marginals. In theory, this problem has been addressed in the exact case where the marginals are perfectly known. In practice, however, experiments only have access to finite statistics and therefore can only determine the marginals of a quantum state up to an error. In this Letter, we prove that unique determinability universally survives such local imperfections: specifically, for every uniquely determined state, we show that deviations of local marginals propagate to global states strictly bounded by a power law with exponent α∈(0,1]α\in(0,1]α∈(0,1]. This result induces a classification of multipartite quantum states by their power-law exponents, with linear scaling α=1α=1α=1 as the most favorable regime. We derive a necessary and sufficient criterion for linear robustness and translate it into an executable semidefinite programming certification. Applying our theory, we prove that stabilizer states are at least square-root robust and provide a complete robustness classification for the Dicke family. Finally, we exploit these results to construct a scalable two-local genuine multipartite entanglement witness, demonstrating the viability of this framework for broad practical applications.
The aim of this chapter is to introduce the reader to various aspects of quantum entanglement. A detailed summary of the jargon of important mathematical notions, and concepts used in entanglement theory is provided. Various methods of entanglement generation are discussed followed by an introduction to the methods of detection and quantification of quantum entanglement in terms of various witnesses and measures. A brief account of some well-known applications of quantum entanglement is presented. Finally, a set of problems with solutions is provided, illustrating various important concepts discussed in the chapter.