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

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. Oct 30 '20 at 14:18
• The assertion was written into the tag for "density-matrix". should someone correct it quantumcomputing.stackexchange.com/questions/tagged/… Oct 30 '20 at 14:44
• Yes, someone should correct it. Oct 30 '20 at 15:36
• 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
Nov 4 '20 at 8:35
• @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 ;) Nov 4 '20 at 10:39