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arXiv:quant-ph/0109060 · Physical Review A

Quantum operations, state transformations and probabilities

Anthony Chefles

Published May 3, 2002• 41 citations
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Abstract

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.

DOI
10.1103/physreva.65.052314

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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 pure\ensuremath→pure\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}pure\ensuremath→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 pure\ensuremath→pure\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}pure\ensuremath→pure transformations, and discuss the relationship between these transformations and deterministic local operations with classical communication entanglement transformations.
We give a unified treatment of quantum operations which transform one set of pure states into a another pure set. Extremal cases are deterministic transformations, which have unit success probability, and unambiguous state discrimination, which reveals the state and has the lowest success probability for a fixed initial ensemble. We obtain necessary and, for linearly independent initial states, sufficient conditions for the realisability of a given transformation with a fixed probability of success for each state. We then examine the relationship between deterministic transformations, majorization and distinguishability.
We propose that the entanglement of mixed states is characterised properly in terms of a probability density function P(E)\mathcal{P}(\mathcal{E})P(E). There is a need for such a measure since the prevalent measures (such as \textit{concurrence} and \textit{negativity}) are rough benchmarks, and not monotones of each other. Considering the specific case of two qubit mixed states, we provide an explicit construction of P(E)\mathcal{P}(\mathcal{E})P(E) and show that it is characterised by a set of parameters, of which concurrence is but one particular combination. P(E)\mathcal{P}(\mathcal{E})P(E) is manifestly invariant under SU(2)×SU(2)SU(2) \times SU(2)SU(2)×SU(2) transformations. It can, in fact, reconstruct the state up to local operations - with the specification of at most four additional parameters. Finally the new measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.
Abstract Resource theory of imaginarity provides a useful framework to understand the role of complex numbers, which are essential in the formulation of quantum mechanics, in a mathematically rigorous way. In the first part of this article, we study the properties of ‘real’ (quantum) operations both in single-party and bipartite settings. As a consequence, we provide necessary and sufficient conditions for state transformations under real operations and show the existence of ‘real entanglement’ monotones. In the second part of this article, we focus on the problem of single copy state transformation via real quantum operations. When starting from pure initial states, we completely solve this problem by finding an analytical expression for the optimal fidelity of transformation, for a given probability of transformation and vice versa. Moreover, for state transformations involving arbitrary initial states and pure final states, we provide a semidefinite program to compute the optimal achievable fidelity, for a given probability of transformation.
The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.