Abstract
Suppose a quantum experiment includes one or more random processes. Then the results of repeated measurements may appear consistent with irreversible decoherence even if the system’s evolution prior to measurement is reversible and unitary. Two thought experiments are constructed as examples.
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Notes
The quantum Loschmidt paradox should not be confused with the cosmological time-reversal paradox, which may or may not be related. See e.g. [2] for a discussion of both.
To avoid the difficulty of flipping thousands of coins without attracting the suspicion of her advisor, suppose she automates this process with a quantum RNG instead of a coin.
The notation \(\hat{H}\) is chosen to suggest “heads,” not “Hamiltonian.”
The physical source of imprecision in ϕ is left to readers’ imaginations; perhaps it is seismic vibrations, flexibility of the beamsplitters’ mounting brackets, or some other nuisance.
Bob must also assume that detection events for different trials are independent.
If ϕ is the sum of very many independent random variables with finite mean and variance, then this assumption is justified by the central limit theorem.
Diagonalization does not determine the overall phase of each eigenvector, but these phases are arbitrary and do not represent any physically observable quantity.
Unorthodox theories (e.g. Bohmian mechanics or the stochastic-spacetime interpretation) may consider this information accessible in principle but missing from quantum theory.
A discrete distribution is degenerate iff its support consists of exactly one value.
Jaynes’ original papers on subjective statistical mechanics address this issue [18].
The usual method of MaxEnt quantum thermodynamics is: given a Hilbert space and a set of expectation values {〈F i 〉}, define an equilibrium mixture \(\bar{\rho}_{T}\) as the density operator which maximizes S vN −∑λ i 〈F i 〉. Here {〈F i 〉} is called a macrostate and {λ i } are Lagrange multipliers. The von Neumann entropy of \(\bar{\rho}_{T}\) is then identified with S T for that macrostate [15].
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Acknowledgements
This paper was inspired by discussions of Wheeler’s delayed-choice experiment with Prof. Robert Gilmore and graduate student Allyson O’Brien at Drexel University. Additional discussion and examples of decoherence arising from an averaging process can be found under the heading “fake decoherence” in [19].
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Kennerly, S. Illusory Decoherence. Found Phys 42, 1200–1209 (2012). https://doi.org/10.1007/s10701-012-9664-6
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DOI: https://doi.org/10.1007/s10701-012-9664-6