Complex-valued Wigner entropy of a quantum state
- URL: http://arxiv.org/abs/2310.19296v1
- Date: Mon, 30 Oct 2023 06:30:03 GMT
- Title: Complex-valued Wigner entropy of a quantum state
- Authors: Nicolas J. Cerf, Anaelle Hertz, Zacharie Van Herstraeten
- Abstract summary: We argue the merits of defining a complex-valued entropy associated with any Wigner function.
It is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: It is common knowledge that the Wigner function of a quantum state may admit
negative values, so that it cannot be viewed as a genuine probability density.
Here, we examine the difficulty in finding an entropy-like functional in phase
space that extends to negative Wigner functions and then advocate the merits of
defining a complex-valued entropy associated with any Wigner function. This
quantity, which we call the complex Wigner entropy, is defined via the analytic
continuation of Shannon's differential entropy of the Wigner function in the
complex plane. We show that the complex Wigner entropy enjoys interesting
properties, especially its real and imaginary parts are both invariant under
Gaussian unitaries (displacements, rotations, and squeezing in phase space).
Its real part is physically relevant when considering the evolution of the
Wigner function under a Gaussian convolution, while its imaginary part is
simply proportional to the negative volume of the Wigner function. Finally, we
define the complex-valued Fisher information of any Wigner function, which is
linked (via an extended de Bruijn's identity) to the time derivative of the
complex Wigner entropy when the state undergoes Gaussian additive noise.
Overall, it is anticipated that the complex plane yields a proper framework for
analyzing the entropic properties of quasiprobability distributions in phase
space.
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