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Related Concept Videos

Entropy02:39

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...
Entropy01:18

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...
The Entropy as a State Function01:14

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...
Entropy and the Second Law of Thermodynamics01:20

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 Thermodynamics01:26

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...
Absolute Entropies and the Third Law of Thermodynamics01:23

Absolute Entropies and the Third Law of Thermodynamics

Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...

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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
09:23

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

Wigner separability entropy and complexity of quantum dynamics.

Giuliano Benenti1, Gabriel G Carlo, Tomaž Prosen

  • 1Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia and Center for Nonlinear and Complex Systems, Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy. giuliano.benenti@uninsubria.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary
This summary is machine-generated.

We introduce Wigner separability entropy to quantify quantum state complexity. This measure connects quantum entanglement to classical complexity, proving its broad applicability across both realms.

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Area of Science:

  • Quantum Information Theory
  • Statistical Mechanics
  • Quantum Chaos

Background:

  • Quantifying the complexity of quantum states is crucial for understanding quantum systems.
  • Entanglement entropy is a key measure, but its direct classical analogue is not always clear.
  • The Wigner function provides a phase-space representation of quantum states.

Purpose of the Study:

  • To introduce a novel measure of quantum state complexity: the Wigner separability entropy.
  • To establish the relationship between Wigner separability entropy and other established entanglement measures.
  • To explore the quantum-classical correspondence of this complexity measure.

Main Methods:

  • Definition of Wigner separability entropy based on Schmidt decomposition of the Wigner function.
  • Mathematical proof establishing its equivalence to operator space entanglement entropy.
  • Numerical simulations of chaotic maps to demonstrate quantum-classical correspondence.

Main Results:

  • Wigner separability entropy quantifies the effective number of terms in the Wigner function's Schmidt decomposition.
  • It is proven to be equal to operator space entanglement entropy and twice the entropy of entanglement for pure states.
  • Numerical results show a clear correspondence between Wigner separability entropy and classical separability entropy.

Conclusions:

  • Wigner separability entropy serves as a robust quantifier of complexity for quantum states.
  • The study bridges quantum and classical complexity measures through phase-space representations.
  • Separability entropy emerges as a universal complexity quantifier applicable in both classical and quantum physics.