For questions about quasiprobability distributions in quantum mechanics
Questions tagged [quasiprobability-distributions]
29 questions
21
votes
4 answers
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, shouldn't that make the probability zero? Why…
Mikayla Eckel Cifrese
- 2,433
15
votes
1 answer
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) states are represented by quasi-probability…
ckrk
- 650
12
votes
2 answers
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 this wikipedia article on the Glauber…
twoform
- 307
6
votes
3 answers
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 $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \rangle$ give the populations.
The off-diagonal…
juanrga
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4
votes
2 answers
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 probability distribution on the classical phase space. We…
Ryder Rude
- 6,915
4
votes
1 answer
Is there a simple way to obtain the Quantum Fisher Information (QFI) from a Wigner function? (or any other quasiprobability distribution?)
In theory the Wigner function $W(q,p)$ contains all the same information as the density matrix $\rho$ associated with it, so it's definitely possible to write the QFI $\mathcal F_Q$ in terms of the Wigner function. But all the expressions I've found…
AndyFo
- 107
4
votes
3 answers
An example of a quantum system for which Wigner function transitions to negative values
I want to check my understanding of the Wigner transform and try to understand why and how exactly the probabilistic interpretation drops down as the function goes to zero and then to negative values
So, suppose we have a free quantum oscillator…
lurscher
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3
votes
1 answer
What are examples of (non-thermal) states with non-singular $P$ function?
We know that the $P$ function of (displaced) thermal states is smooth and positive. Indeed, as shown e.g. here, a thermal state $\rho_{\rm th}(x)\equiv (1-x)x^{a^\dagger a}$, $x\equiv e^{-\beta}$, has $P$ function
$$P_x(\nu) = \frac{1-x}{\pi x}…
glS
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3
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0 answers
How is Hudson's theorem for the Wigner function proved?
Hudson's theorem tells us that a pure state has non-negative Wigner function iff it's Gaussian. This was originally proven in [Hudson 1974], and then generalised to multidimensional systems in [Soto and Claverie 1983].
Accessing these old papers is…
glS
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3
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1 answer
What is the observable corresponding to the $P$ function?
The P-function of a state $\rho$ (focusing on the single-mode case) can be written as, using a notation analogous to the one in Gerry&Knight' book,
$$P_\rho(\alpha) = \int \frac{d^2\eta}{\pi^2} \chi_N(\eta) e^{\alpha\bar\eta-\bar\alpha\eta},
\quad…
glS
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3
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2 answers
What does it mean for $P$ functions to be "more singular than a delta"?
Consider the Glauber-Sudarshan $P$ representation of a state $\rho$, which is the function $\mathbb C\ni\alpha\mapsto P_\rho(\alpha)\in\mathbb R$ such that
$$\rho = \int d^2\alpha \, P_\rho(\alpha) |\alpha\rangle\!\langle\alpha|.$$
Something that is…
glS
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3
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2 answers
Why can the $Q$ function be equivalently defined as "Wigner convolved with a Gaussian" and "density matrix put into normal order"?
I read on Wikipedia two different descriptions of the "Husimi-Q representation."
One is that it is the Wigner function convolved with a Gaussian, which in particular results in a positive definite function.
The other is that it is "essentially"…
commutatertot
- 403
2
votes
1 answer
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 the phase probability distribution $P(\varphi)$…
2
votes
0 answers
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 understand that the phase space path integral is a tool…
Ryder Rude
- 6,915
2
votes
1 answer
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 basis
$$\rho=\int d^2\alpha |\alpha\rangle\langle…
Quantum Mechanic
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