Deterministic quantum state transformations

Gökhan Torun, Ali Kerem Yildiz

Anthony Chefles

In quantum operations, probabilities characterize both the degree of the success of a state transformation and, as density operator eigenvalues, the degree of mixedness of the final state. We give a unified treatment of $\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}$ state transformations, covering both probabilistic and deterministic cases. We then discuss the role of majorization in describing the dynamics of mixing in quantum operations. The conditions for mixing enhancement for all initial states are derived. We show that mixing is monotonically decreasing for deterministic $\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}$ transformations, and discuss the relationship between these transformations and deterministic local operations with classical communication entanglement transformations.

W. A. Majewski

Chefles, A

We derive a necessary condition for the existence of a completely-positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another. This condition is also sufficient for linearly-independent initial states. We also examine the issue of quantum coherence, that is, when such operations maintain the purity of superpositions. If, in any deterministic transformation from one linearly-independent set to another, even a single, complete superposition of the initial states maintains its purity, the initial and final states are related by a unitary transformation.

Runyao Duan, Yuan Feng, Zhengfeng Ji +1 more

We prove that sufficiently many copies of a bipartite entangled pure state can always be transformed into some copies of another one with certainty by local quantum operations and classical communication. The efficiency of such a transformation is characterized by deterministic entanglement exchange rate, and it is proved to be always positive and bounded from top by the infimum of the ratios of Renyi's entropies of source state and target state. A careful analysis shows that the deterministic entanglement exchange rate cannot be increased even in the presence of catalysts. As an application, we show that there can be two incomparable states with deterministic entanglement exchange rate strictly exceeding 1.