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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MDPI and ACS Style
Baez, J.C.; Pollard, B.S. Quantropy. Entropy 2015, 17, 772-789.