On an analyticity property in (complex) phase space

Leon W. Cohen

Nishan C. Jayarama, Viktor Svensson

Circuit complexity has been used as a tool to study various properties in condensed matter systems, in particular as a way to probe the phase diagram. However, compared with measures based on entanglement, complexity has been found lacking. We show that when imposing penalty factors punishing nonlocality, it becomes a much stronger probe of the phase diagram, able to probe more subtle features. We do this by deriving analytical solutions for the complexity in the XY chain with transverse field.

P. D. Drummond

Stochastic processes that correspond to equations of arbitrary differential order are obtained.We show that stochastic paths in a complex extension of the original phase-space allow an implementation of such higher-order derivative terms.The resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments.However, the correspondence has unusual properties.Only the analytic moments are convergent, while non-analytic moments such as the complex moduli are non-convergent.These results unify previous approaches that transform higher-derivative equations into probabilistic stochastic equations.Larger ensembles are required as time-steps are reduced, giving these equations unusual convergence properties.This type of process is relevant to the question of how to obtain stochastic quantum simulations using the Wigner representation.

Sergio A Albeverio, Ambar N. Sengupta

Nicolas J. Cerf, Anaelle Hertz, Zacharie Van Herstraeten

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 thecomplex 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.

Huzihiro Araki