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Related Concept Videos

Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
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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...
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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic...
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Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...
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Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
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Quantum entropy and central limit theorem.

Kaifeng Bu1, Weichen Gu2, Arthur Jaffe1,3

  • 1Department of Physics, Harvard University, Cambridge, MA 02138.

Proceedings of the National Academy of Sciences of the United States of America
|June 12, 2023
PubMed
Summary
This summary is machine-generated.

We present a new framework for discrete-variable quantum systems using qudits, introducing a mean state (MS) and a novel convolution. This framework reveals a maximal entropy principle and a quantum convolution thermodynamics law.

Keywords:
central limit theoremconvolutionentropy

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Area of Science:

  • Quantum Information Theory
  • Quantum Computing
  • Discrete-Variable Quantum Systems

Background:

  • Quantum systems are often described using continuous variables, but discrete-variable (DV) systems offer unique advantages.
  • Studying DV quantum systems, particularly those based on qudits, requires specialized theoretical frameworks.
  • Existing methods may not fully capture the unique properties and behaviors of qudit-based systems.

Purpose of the Study:

  • To introduce a novel theoretical framework for analyzing discrete-variable quantum systems utilizing qudits.
  • To define and explore key concepts such as the mean state (MS), minimal stabilizer-projection state (MSPS), and a new convolution operation.
  • To establish fundamental principles and laws governing these DV quantum systems and their operations.

Main Methods:

  • Development of a new mathematical framework based on qudits.
  • Introduction of the concepts of mean state (MS) and minimal stabilizer-projection state (MSPS).
  • Definition and application of a novel quantum convolution operation for DV systems.
  • Analysis of quantum entropies and Fisher information using the new convolution.
  • Establishment of a central limit theorem for iterated convolutions of quantum states.

Main Results:

  • The mean state (MS) is identified as the closest MSPS to a given state, demonstrating a 'maximal entropy principle in DV systems'.
  • A 'second law of thermodynamics for quantum convolutions' is derived, providing inequalities for quantum entropies and Fisher information.
  • The convolution of two stabilizer states is proven to be a stabilizer state.
  • A central limit theorem shows that iterated convolutions of a zero-mean quantum state converge to its MS, with convergence rate determined by the 'magic gap'.

Conclusions:

  • The introduced framework provides powerful tools for understanding DV quantum systems and qudit-based information processing.
  • The discovered principles, including the maximal entropy principle and the second law for quantum convolutions, offer new insights into quantum thermodynamics and information.
  • The framework and its associated concepts, like the magic gap, are applicable to practical quantum devices such as DV beam splitters and amplifiers.