Why do entangled particles have to be in an indeterminate state?

[San K]

Why do (we assume that) entangled particles (say photons/electrons) have to be in an indeterminate state?

 

[Maui]

Why do (we assume that) entangled particles have to be in an indeterminate state?

Which 'particles' are in a determined state? There are no known 'particles' prior to measurement/interaction according to standard qm.

 

[Darwin123]

Why do (we assume that) entangled particles (say photons/electrons) have to be in an indeterminate state?

The probability function has to satisfy invariance with exchange of particles. The invariance of measurable properties with exchange of particle index is a major hypothesis in the theory. If the individual particles were in a state where their parameters could be determined precisely, then by definition there would be a measurement that could distinguish between particles.

The invariance of measurable parameters to particle exchange is an important physical symmetry. Sometime it is referred to as "exchange symmetry". It is an important physical postulate in quantum mechanics. One consequence of this condition is the existence of bosons and fermions.

Note that the wave function itself does not have to be invariant to particle exchange because one can never measure the entire wave function. The idea is that the expectation values (i.e., measurable parameters) have to be invariant to particle exchange. The difference between an expectation value being invariant to particle exchange and the wave function being invariant to particle exchange was confusing to me.

 

[San K]

The probability function has to satisfy invariance with exchange of particles. The invariance of measurable properties with exchange of particle index is a major hypothesis in the theory. If the individual particles were in a state where their parameters could be determined precisely, then by definition there would be a measurement that could distinguish between particles.

The invariance of measurable parameters to particle exchange is an important physical symmetry. Sometime it is referred to as "exchange symmetry". It is an important physical postulate in quantum mechanics. One consequence of this condition is the existence of bosons and fermions.

Note that the wave function itself does not have to be invariant to particle exchange because one can never measure the entire wave function. The idea is that the expectation values (i.e., measurable parameters) have to be invariant to particle exchange. The difference between an expectation value being invariant to particle exchange and the wave function being invariant to particle exchange was confusing to me.

Thanks Darwin. Well answered.

 

[Darwin123]

Thanks Darwin. Well answered.

Just one additional fact:
The definition of "entangled" is "being in a stationary state of the multibody Hamiltonian."

A stationary state is an eigenvector of a Hamiltonian.

Under the condition of exchange invariance symmetry, the particle exchange operator and the multibody Hamiltonian have to commute. Furthermore, all stationary states of the multibody Hamiltonian have to eigenvectors of the particle exchange operator.

The condition of particle exchange invariance is really the basis of second quantization. Quantum mechanics is taught in a historical rather than logical sequence. Therefore, one learns about first quantization in the introductory courses. This is quantum mechanics without the postulate of invariance to particle exchange.

Invariance to particle exchange becomes very important whenever the number of particles changes during an interaction. Without understanding that postulate, one can't understand quantum electrodynamics or any quantum field theory.

 

[DrChinese]

Just a side note: a pair of particles can be entangled on one basis and not entangled on another. I could have know the spin but not momentum, or vice versa. So they could be in a superposition (and indeterminate) in some bases and not others.