Catalytic Transformations of Pure Entangled States

Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.

Figure 1

Catalytic LOCC transformation from ${\rho }^S$ to ${\sigma }^S$ with a sequence of catalyst states $\{{\tau }_n^C\}$. The state of the catalyst does not change in the procedure, and the system becomes decoupled from the catalyst for $n\to \infty$. If ${\rho }^S$ and ${\sigma }^S$ are bipartite pure states, the transition is fully characterized by entanglement entropy of the states.

Figure 2

Catalytic quantum state merging. Alice, Bob, and Referee share a single copy of $|\psi {⟩}^{RAB}$. Alice aims to send her part of the state to Bob by using catalytic LOCC, and the Referee can apply local unitary transformations. The process is completely characterized by the quantum conditional entropy $H(A|B)$; see the main text for more details.