Abstract

Currently, ‘time’ does not play any essential role in quantum information theory. In this sense, quantum information theory is underdeveloped similarly to how quantum physics was underdeveloped before Erwin Schrödinger introduced his famous equation for the evolution of a quantum wave function. In this review article, we cope with the problem of time for one of the central quantities in quantum information theory: entropy. Recently, a replica trick formalism, the so-called ‘multiple parallel world’ formalism, has been proposed that revolutionizes entropy evaluation for quantum systems. This formalism is one of the first attempts to introduce ‘time’ in quantum information theory. With the total entropy being conserved in a closed system, entropy can flow internally between subsystems; however, we show that this flow is not limited only to physical correlations as the literature suggest. The nonlinear dependence of entropy on the density matrix introduces new types of correlations with no analogue in physical quantities. Evolving a number of replicas simultaneously makes it possible for them to exchange particles between different replicas. We will summarize some of the recent news about entropy in some example quantum devices. Moreover, we take a quick look at a new correspondence that was recently proposed that provides an interesting link between quantum information theory and quantum physics. The mere existence of such a correspondence allows for exploring new physical phenomena as the result of controlling entanglement in a quantum device.

Highlights

  • Entropy is one of the central quantities in thermodynamics and, without its precise evaluation, one cannot predict what new phenomena are to be expected in the thermodynamics of a device.In quantum theory, entropy is defined as a nonlinear function of the density matrix, i.e., S = −Trρln ρ, in the units of the Boltzmann constant k B

  • Let us clarify that, in this paper, we study the flow of thermodynamic Renyi and von Neumann entropies between the heat baths and quantum system q

  • We discuss Shannon entropy SShannon, we have to distinguish between the Shannon entropy, which can be measured as a number of bits, and the rest of the paper in which we study von Neumann thermodynamic entropy measured in the unit Joule per Kelvin

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Summary

Introduction

Entropy is one of the central quantities in thermodynamics and, without its precise evaluation, one cannot predict what new phenomena are to be expected in the thermodynamics of a device. A physical quantity, such as energy or charge, can be measured in the lab in real time and can be defined in quantum theory to linearly depend on the density matrix. Let us consider the example of two heat baths A and B, both coupled through a quantum system q that contains discrete energies and allows for the superposition of states with long coherence time. In contrast to what has been so far presented in the literature [14], the second term in the entropy flow is not heat transfer—the average change of energy at the two times Q B ≡ h H (0)i B − h H (t)i B Instead, it is the difference of incoherent and coherent heat transfers [15], i.e., ( Q B,incoh (t) − Q B,coh (t)) −. We briefly report on the new correspondence that makes entropy flow directly measurably in the lab by monitoring physical quantities, i.e., the statistics of energy transfer

Classical Entropy
Renyi Entropy
Von Neumann and Renyi Entropy
Replica Trick
Time Evolution of Entropy
Extended Keldysh Diagrams
Calculating the Diagrams
Quantum Entropy Production
Example 1
Generalized KMS
Example 2
Example 3
Linking Information to Physics: A New Correspondence
Discussion
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