Question

Consider a bipartite system, defined on a Hilbert space H=H_A ⊗ H_B. Consider a basis B_j⟩. What is the general form of a density operator such that it predicts a perfect correlations between pairs (A_i,B_i) upon measurement? (i.e. null probability of a joint measurement outcome (A_i,B_j) for i≠j).

If one had a pure state for the bipartite system, then one could appeal to the following Schmidt decomposition:
|ψ⟩=Σ_iα_i|A_i⟩|B_i⟩
But if the quantum state is not pure, no such Schmidt decomposition is avaiable.

In other words, what is the equivalent of a Schmidt decomposition for an impure density operator?

Answer 1

The density operator for a bipartite quantum system predicting perfect correlations between pairs (A_i, B_i) is a mixed state represented as a sum of projectors onto correlated pure states, where each term is a projector onto the state |A_i⟩|B_i⟩, ensuring certain outcomes in measurements of subsystems.

Step-by-step explanation:

For a bipartite quantum system where the Hilbert space is H = H_A ⊗ H_B, the general form of a density operator that predicts perfect correlations between pairs (A_i, B_i) for a measurement, such that there is null probability for joint outcomes (A_i, B_j) when i ≠ j, can be expressed as a mixed state that is a statistical mixture of perfectly correlated pure states. This mixed state is represented by the density operator ρ which is a sum of projectors onto the correlated states:

ρ = Σ_i p_i |A_i
angle ⟨ B_i|⟨A_i| ⟩ B_i|

where p_i are probabilities that sum to 1. In each term |A_i
angle ⟨ B_i|⟨A_i| ⟩ B_i| represents a projector onto the pure correlated state |A_i
angle|B_i
angle. Unlike the Schmidt decomposition for pure states, this expression for ρ does not decompose the state into orthogonal states, but it does maintain the requirement that measurement of subsystem A (or B) will result in a subsequent measurement of subsystem B (or A) with certainty if it is in the corresponding state B_i (or A_i).