Is the singlet state just a special case of mixture?

[quantumphilosopher]

If the answer is no, why?

Is it possible to show that, given the state spaces we have for individual particles, the singlet state is not a mixture of product states? If yes, how?

 

[vanesch]

I guess you are talking about the 2-particle state:
|up>|down> - |down> |up>

and you wonder whether this is any different from a mixture of, say, 50% |up>|down> and 50% |down> |up>.

The short answer is that the difference is exactly what violates the Bell inequalities. The mixture wouldn't, the singlet state does. At least in an ideal experiment.

The long answer is that the density matrix is different. In the basis of the two-particle space:

{ |up> |up> ,
|up> |down>,
|down> |up>,
|down>|down> }

the singlet state has a non-diagonal matrix, while the mixture has a diagonal matrix. Note that the diagonal elements of both are the same (0, 1/2, 1/2, 0).

 

[denian]

The singlet state is not just a special case of mixture. A mixture is a state that can be described as a combination of multiple individual states, while the singlet state is a specific quantum state that cannot be decomposed into a mixture of product states.

To show that the singlet state is not a mixture of product states, we can use the concept of entanglement. Entanglement is a unique property of quantum systems where two or more particles are linked in a way that their states cannot be described independently. The singlet state is an entangled state of two particles, and therefore cannot be written as a simple combination of individual states.

Furthermore, the singlet state has specific properties that cannot be explained by a mixture of product states. For example, in a mixture of product states, measurements of one particle will not affect the state of the other particle. However, in the singlet state, measurements on one particle will instantaneously affect the state of the other particle, regardless of the distance between them. This phenomenon, known as quantum entanglement, is a clear indication that the singlet state is not a mixture of product states.

Therefore, it is not possible to show that the singlet state is a mixture of product states, as it possesses unique properties that cannot be explained by a mixture.