Jaynes' maximum entropy (MaxEnt) principle selects, from all probability distributions consistent with a set of constraints, the one that maximizes the Shannon entropy. The principle is well-defined for classical, real-valued probabilities, but extending it to complex-valued probability theories requires care. Youssef's complex probability framework, in which quantum mechanics is reformulated as a Bayesian theory with complex amplitudes and the observable probability is the squared modulus (Born rule), does not admit a natural MaxEnt extension: the L2 normalization of that framework is incompatible with the L1 normalization assumed by Shannon entropy, and the resulting complex-valued entropy functional cannot be maximized without additional structure absent from Youssef's axioms. The present paper proposes a different extension of Kolmogorov's axioms to the complex domain, replacing non-negativity with the condition that the real part of every probability be non-negative. This framework, which retains L1 normalization and recovers classical probability when the imaginary part vanishes, admits a natural MaxEnt principle: the real part of the complex Shannon entropy is a well-posed real functional, its Euler-Lagrange equations produce an explicit complex Gibbs distribution with complex Lagrange multipliers, and the classical Gibbs distribution is recovered as the special case of real multipliers. The die problem (Jaynes' Brandeis dice example) is worked out in detail for both the fair and the loaded cases, illustrating how imaginary constraints redistribute the observable probability mass in a way that has no counterpart in classical MaxEnt.
Jaynes' maximum entropy (MaxEnt) principle selects, from all probability distributions consistent with a set of constraints, the one that maximizes the Shannon entropy. The principle is well-defined for classical, real-valued probabilities, but extending it to complex-valued probability theories requires care. Youssef's complex probability framework, in which quantum mechanics is reformulated as a Bayesian theory with complex amplitudes and the observable probability is the squared modulus (Born rule), does not admit a natural MaxEnt extension: the L2 normalization of that framework is incompatible with the L1 normalization assumed by Shannon entropy, and the resulting complex-valued entropy functional cannot be maximized without additional structure absent from Youssef's axioms. The present paper proposes a different extension of Kolmogorov's axioms to the complex domain, replacing non-negativity with the condition that the real part of every probability be non-negative. This framework, which retains L1 normalization and recovers classical probability when the imaginary part vanishes, admits a natural MaxEnt principle: the real part of the complex Shannon entropy is a well-posed real functional, its Euler-Lagrange equations produce an explicit complex Gibbs distribution with complex Lagrange multipliers, and the classical Gibbs distribution is recovered as the special case of real multipliers. The die problem (Jaynes' Brandeis dice example) is worked out in detail for both the fair and the loaded cases, illustrating how imaginary constraints redistribute the observable probability mass in a way that has no counterpart in classical MaxEnt.
In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ, where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: i) allows the violation of the uncertainty principle [1], [2]; ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.
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.
Probability theory is fundamental for modeling uncertainty, with traditional probabilities being real and non-negative. Complex probability extends this concept by allowing complex-valued probabilities, opening new avenues for analysis in various fields. This paper explores the information-theoretic aspects of complex probability, focusing on its definition, properties, and applications. We extend Shannon entropy to complex probability and examine key properties, including maximum entropy, joint entropy, conditional entropy, equilibration, and cross entropy. These results offer a framework for understanding entropy in complex probability spaces and have potential applications in fields such as statistical mechanics and information theory.
The Andrey Nikolaevich Kolmogorov's classical system of probability axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set 𝒞 which is the sum of the real set ℛ with its corresponding real probability and the imaginary set ℳ with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Whatever the probability distribution of the input random variable in ℛ is, the corresponding probability in the whole set 𝒞 is always one, so the outcome of the random experiment in 𝒞 can be predicted totally and perfectly. The result indicates that chance and luck in ℛ is replaced now by total determinism in 𝒞 . This is the consequence of the fact that the probability in 𝒞 is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to Ludwig Boltzmann's classical concept of entropy in thermodynamics and in statistical mechanics.
The connection between the maximum entropy (MaxEnt) formalism and restricted Boltzmann machines (RBMs) is natural as both give rise to a Boltzmann-like distribution with constraints enforced by Lagrange multipliers, which correspond to RBM parameters. We integrate RBMs into quantum state tomography by using them as probabilistic models to approximate quantum states while satisfying MaxEnt constraints. Additionally, we employ polynomially efficient quantum sampling techniques to enhance RBM training, enabling scalable and high-fidelity quantum state reconstruction. This approach provides a computationally efficient framework for applying RBMs to MaxEnt-based quantum tomography. Furthermore, our method applies to the general and previously unaddressed case of reconstructing arbitrary mixed quantum states from incomplete and potentially noncommuting sets of expectations of observables while still ensuring maximal entropy.
There is a focus in classical statistical mechanics on maximizing an entropy function subject to constraints to obtain probability values. For a coin or die probabilities are given, yet one may still calculate a Shannon’s entropy. In the case of a Maxwell-Boltzmann (MB) gas, one may obtain probabilities using energy conservation and products of probabilities and then show this is equivalent to maximizing an entropy function subject to an energy constraint. We argue that the energy conservation approach yields an energy conservation equation and a product of probabilities representing a certain total energy equated to a product of different probabilities representing the same total energy. The two equations are linked to solve for probability in terms of energy and this is formally the same as a maximization problem subject to conservation of energy. A point we make is that this approach is based on a kind of static conservation, even in the presence of a potential V(x). In quantum mechanics, we argue one one has fluxes even though there is also conservation of energy. For light moving in the positive x direction which reflects and refracts (with each photon retaining its energy) there is both energy conservation and flux. The energy conservation equation in terms of flux/intensity terms is given in (1) as: AA/c1 = BB/c1 + CC/c2 where A,B,C refer to the incident, reflected and refracted rays. This single equation cannot solve for probabilities B,C in terms of A and there is not a separate product of probabilities equation as in the MB case. Rather quantum mechanics seems to be solved using square root fluxes as dynamical probabilities. In particular, the energy conservation equation is equivalent to two square root flux equations A+B=C and p1A -p1B = p2C where E=pc. Thus probabilities are determined through dynamics not static conservation or entropy maximization. In an earlier note we concluded that exp(ipx), a quantum free particle wavefunction, is also like a square root flux, and also acts as a dynamic probability. These ideas may be extended to a quantum bound state which we discuss in the note. Thus the approach to finding probabilities in the classical statistical mechanics and quantum realms seem to be completely different. One uses static conservation laws in the former and square root flux continuity/dynamical probability in the latter.