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Entropy

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...

Entropy

Entropy

The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

Entropy and the Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...

Entropy and the Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...

The Entropy as a State Function

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

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Related Experiment Video

Updated: Mar 13, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Additivity and Chain Rules for Quantum Entropies via Multi-index Schatten Norms.

Omar Fawzi¹, Jan Kochanowski², Cambyse Rouzé²

¹ENS Lyon, UCBL, LIP, Inria, 69342 Lyon Cedex 07, France.

Communications in Mathematical Physics

|March 12, 2026

Summary

This study proves additivity for optimized sandwiched Rényi entropy in quantum channels, crucial for quantum information and cryptography. This advances quantum key distribution security analysis, especially for time-adaptive protocols.

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Area of Science:

Quantum Information Theory
Quantum Cryptography
Mathematical Physics

Background:

Entropic measures are fundamental in quantum information, with additivity under tensor products being a key property.
The minimum output entropy of quantum channels is vital for information processing and security protocols.
A critical open question concerns the additivity of minimum output entropy under the tensor product of channels.

Purpose of the Study:

To establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels.
To generalize existing additivity results to multi-index Schatten norms.
To provide tools for analyzing quantum information processing tasks and cryptographic protocols.

Main Methods:

Generalization of Devetak et al.'s results to multi-index Schatten norms.
Establishment of additivity for optimized sandwiched Rényi entropy.
Development of chain rules for Rényi conditional entropies.

Main Results:

A general additivity statement for the optimized sandwiched Rényi entropy of quantum channels is established.
The additivity statement strengthens existing results for quantum key distribution security proofs.
New chain rules for Rényi conditional entropies are derived, analogous to the generalized entropy accumulation theorem.

Conclusions:

The established additivity provides a powerful tool for quantum information theory and cryptography.
The findings enhance the security analysis of quantum key distribution, particularly for time-adaptive protocols.
The derived chain rules offer new perspectives on entropy accumulation in quantum systems.