Entanglement witness

In quantum information theory, an entanglement witness is a functional which distinguishes a specific entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the entangled state is strictly outside the range of possible expectation values of any separable state.

Let a composite quantum system have state space ${\displaystyle H_{A}\otimes H_{B}}$. A mixed state ρ is then a trace-class positive operator on the state space which has trace 1. We can view the family of states as a subset of the real Banach space generated by the Hermitian trace-class operators, with the trace norm. A mixed state ρ is separable if it can be approximated, in the trace norm, by states of the form

where ${\displaystyle \rho _{i}^{A}}$'s and ${\displaystyle \rho _{i}^{B}}$'s are pure states on the subsystems A and B respectively. So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem:

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