Entropy 2015, 17(2), 772-789; doi:10.3390/e17020772
Quantropy
1
Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
2
Department of Mathematics, University of California, Riverside, CA 92521, USA
3
Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
*
Author to whom correspondence should be addressed.
Received: 17 December 2014 / Accepted: 30 January 2015 / Published: 9 February 2015
(This article belongs to the Section Quantum Information)
Abstract
There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to \(\exp(-E/kT)\), where \(E\) is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to \(\exp(-S/i\hbar)\), where \(S\) is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results. View Full-TextKeywords:
path integration; variational principles; quantum mechanics; entropy
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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MDPI and ACS Style
Baez, J.C.; Pollard, B.S. Quantropy. Entropy 2015, 17, 772-789.