## What is the relation between density matrices and phase-space probability distributions?

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According to its tag description, a density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

How can I reconcile the concept and intuition behind a "probability distribution based on both position and momentum" found in statistical mechanics, with that of a matrix ensemble of 1's and 0's found in the quantum density matrix?

The assertion in the first paragraph is false. A good analogue to the phase-space probability distribution is the Wigner function, not the density matrix.

– Mateus Araújo – 2020-10-30T14:18:10.247

The assertion was written into the tag for "density-matrix". should someone correct it https://quantumcomputing.stackexchange.com/questions/tagged/density-matrix

– develarist – 2020-10-30T14:44:32.583

Yes, someone should correct it. – Mateus Araújo – 2020-10-30T15:36:48.223

I edited the tags. Note that there is also a discrete analogue of the Wigner function for finite-dimensional systems. Anyway, would you reformulate/state your question more precisely? – Markus Heinrich – 2020-10-30T16:01:03.477

i didn't mean edit the tag in the question, but edit the tag tag's description in the link i gave – develarist – 2020-10-30T16:02:25.833

Ah right, sorry, I completely misread your sentence. You can certainly debate this assertion. But I guess this should happen in Meta. – Markus Heinrich – 2020-10-30T16:15:25.990

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I would say that the "quantum-mechanical analogue" to a phase-space distribution would be a quasiprobability distribution, not a density matrix. A density matrix, like a state, gives you the amplitude in a single basis (say, the position or the momentum). You need to use something like a Wigner function for a description in terms of phase-space variables. Unless you mean something more specific with "quantum-mechanical analogue" here.

– glS – 2020-11-04T08:35:13.830

no one has fixed the density-matrix tag's description yet? – develarist – 2020-11-04T08:45:09.777

@glS well, as I said, one can certainly debate this. Nevertheless, the quasiprobability distribution has to origin from a quantum state, which is not the case for an arbitrary one (psd constraint!). Thus, the "quantum-mechanical analogue" are those distributions coming from quantum states which are 1-to-1 with quantum states. It's just a matter of representation. In the end, the density matrix contains the information about the state of the system, as does the phase space measure in the classical case. – Markus Heinrich – 2020-11-04T10:04:27.447

@MarkusHeinrich actually it seems to me like we are saying the same thing. I agree that they are both equally valid representations for a physical state. I don't see why the possibility of describing finite-dimensional systems with quasiprobability distributions affects this statement – glS – 2020-11-04T10:16:51.517

1@glS I just wanted to point out that while probability measures on phase space are always valid classical states, quasiproability measures are not necessarily valid quantum states. Thus saying that they are the quantum analogue is not entirely correct. But I agree that this point of view is often helpful. I'm working in the (discrete) phase space picture the whole day ;) – Markus Heinrich – 2020-11-04T10:39:12.413