# Questions tagged [quasiprobability-distributions]

For questions about quasiprobability distributions in quantum mechanics

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### Is there a non-negative normalized Wigner function that doesn't correspond to a physical state?

This is related to Is the Wigner function non-negative only for convex mixtures of Gaussian states? and Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator ...
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### Is the Wigner function non-negative only for convex mixtures of Gaussian states?

Hudson's theorem, the result usually cited in this context, tells us that for a pure state, the Wigner is non-negative iff the state is Gaussian, but doesn't in general say anything about mixed states....
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### How is Hudson's theorem for the Wigner function proved?

Hudson's theorem tells us that a pure state has non-negative Wigner function iff it's Gaussian. This was originally proven in [Hudson 1974], and then generalised to multidimensional systems in [Soto ...
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### What is a "reciprocal ordering" when discussing quasiprobability distributions?

The Wikipedia page about the optical equivalence theorem mentions that, if we denote with $f_{\Omega}(\hat a,\hat a^\dagger)$ an "operator that is expressible as a power series in the creation ...
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### Quasi-distribution for the composite (fermions + bosons) system

Let us have a system which consists of $N$ electrons with the spin (i. e. fermionic subspace has a dimension of $4^N$) and $K$ bosonic modes (let us consider $K=1$ for simplicity). Let us say, we ...
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### Quantum Behavior and Negativity of Wigner Functions

Let us consider a scenario where we have a dataset $\mathbf{X}$, which is a collection of vectors $\mathbf{x}_i \in \mathbb{R}^n$. We encode each component $x_j \in \mathbb{R}$ of $\mathbf{x}$ in a ...
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### Phase distribution of coherent states

I am studying the phase distribution for coherent states, as is defined in quantum optics. (See, for example, Introductory Quantum Optics by Gerry and Knight, pages 46–48). In this situation, we seek ...
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### Distributions "more singular than a Dirac delta" must have negativity

I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
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### Is it possible to relate the expectation values of an operator w.r.t. two different density matrices if the matrices are related up to a displacement?

I'm trying to derive a general relationship between the expectation values of two different operators with respect to some unspecified state $E_{j}(\lambda)=\mathrm{Tr}[\rho \hat{O}_{j}(\lambda)]$ (...
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### Do we actually need negative probabilities in quantum mechanics?

I was reading this thread and I'm a bit confused. The answer says negative probabilities can account for destructive wave interference and the events cancelling out. But if events just cancel out, ...
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### Is there a way to find the superposition of two Wigner functions without finding their density operators?

Say I have two Wigner functions $W_1(x,p)$ and $W_2(x,p)$ representing pure states on optical phase space, and I want to know what the Wigner function of their superposition $W_{1+2}(x,p)$ looks like (...
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### Intuitively, why does Quantum Mechanics involve a sum over all possibilities?

I understand that one can just mathematically derive the path integral from the Schrodinger equation. I'm looking for an intuitive explanation in contrast with classical mechanics. Consider a ...
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### Is there a relationship between the phase space path integral and phase space quantum mechanics?

I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space? I also ...
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### An equation for Wigner's quasi-probability distribution?

I learned that Moyal's evolution equation is the equation for the time-evolution of Wigner's quasi-probability distribution. However, I couldn't say I perfectly understand the meaning of this ...
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### Non-uniqueness of Glauber-Sudarshan $P$-function

For a state $\rho$ acting on single bosonic mode with coherent states $|\alpha\rangle$, one can always define a $P$-function to furnish a diagonal representation of the state in the coherent-state ...
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### Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)

I am moving into a new field and after thorough literature research need help appreciating what is out there. In the continuos variable formulation of optical state space. (Quantum mechanical/Optical) ...
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### What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?

(a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$: The diagonal elements \$\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \...
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