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Scientific Reports

Transformations of superpositions by means of incoherent operations

Marcelo Losada, G. M. Bosyk, Héctor Freytes, Giuseppe Sergioli

Published May 19, 2020• 2 citations
Publisher

Abstract

In this paper we study how the coherence of a superposition of pure states is related with the coherence of its components. We consider two pure initial states and two pure final coherent states, such that the former ones cannot be transformed into the latter ones by means of incoherent transformations. In this situation, we analyze conditions for the existence of superpositions of the initial states that can be transformed into superpositions of the final states. In particular, we consider superpositions formed by quantum states belonging to orthogonal subspaces. By appealing to the majorization theory, we obtain necessary and sufficient conditions for such transformations to be possible. Finally, we provide some examples that illustrate the difference between the obtained conditions and the necessary criterion based on the relative entropy of coherence.

DOI
10.1038/s41598-020-63661-w

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In this paper we study how the coherence of a superposition of pure states is related with the coherence of its components. We consider two pure initial states and two pure final coherent states, such that the former ones cannot be transformed into the latter ones by means of incoherent transformations. In this situation, we analyze conditions for the existence of superpositions of the initial states that can be transformed into superpositions of the final states. In particular, we consider superpositions formed by quantum states belonging to orthogonal subspaces. By appealing to the majorization theory, we obtain necessary and sufficient conditions for such transformations to be possible. Finally, we provide some examples that illustrate the difference between the obtained conditions and the necessary criterion based on the relative entropy of coherence.
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.
The concept of block coherence encompasses the case where experimental capabilities are not so delicate to perform arbitrary refined measurements on individual atoms. We develop a framework which facilitates further investigation of this resource theory in several respects. Using this framework, we investigate the problem of state conversion by incoherent operations and show that a majorization condition is the necessary and sufficient condition for state transformation by block-incoherent operations. We also determine the form of the maximally coherent state from which all other states and all unitary gates can be constructed by incoherent operations. Thereafter, we define the concept of block-cohering and block-decohering powers of quantum channels and determine these powers for several types of channels. Finally, we explore the relation between block coherence and a previous extension of coherence, known as kkk coherence.
Catalytic coherence transformations allow the otherwise impossible state transformations using only incoherent operations with the aid of an auxiliary system with finite coherence that is not being consumed in any way. Here we find the necessary and sufficient conditions for the deterministic and stochastic catalytic coherence transformations between a pair of pure quantum states. In particular, we show that the simultaneous decrease of a family of R\'enyi entropies of the diagonal parts of the states under consideration is a necessary and sufficient condition for the deterministic catalytic coherence transformations. Similarly, for stochastic catalytic coherence transformations we find the necessary and sufficient conditions for achieving a higher optimal probability of conversion. We thus completely characterize the coherence transformations among pure quantum states under incoherent operations. We give numerous examples to elaborate our results. We also explore the possibility of the same system acting as a catalyst for itself and find that indeed self-catalysis is possible. Further, for the cases where no catalytic coherence transformation is possible we provide entanglement-assisted coherence transformations and find the necessary and sufficient conditions for such transformations.
It is well known that the majorization condition is the necessary and sufficient condition for the deterministic transformations of both pure bipartite entangled states by local operations and coherent states under incoherent operations. In this paper, we present two explicit protocols for these transformations. We first present a permutation-based protocol which provides a method for the single-step transformation of ddd-dimensional coherent states. We also obtain generalized solutions of this protocol for some special cases of ddd-level systems. Then, we present an alternative protocol where we use ${d}^{\ensuremath{'}}$-level (${d}^{\ensuremath{'}}<d$) subspace solutions of the permutation-based protocol to achieve the complete transformation as a sequence of coherent-state transformations. We show that these two protocols also provide solutions for deterministic transformations of pure bipartite entangled states.
We study partial coherence and its connection with entanglement. First, we provide a sufficient and necessary condition for bipartite pure-state transformation under partially incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by use of symmetric concave functions. Third, we establish the relationship between partial coherence and entanglement. We show that the minimal partial coherence under local unitary transformations is also a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof extension. Conversely, every entanglement measure can be constructed as a partial coherence measure, and we further show that partial coherence measures are induced through maximal entanglement under partially incoherent operations for bipartite pure states.
Block coherence provides a natural generalization of standard quantum coherence by treating superpositions across different subspaces as a resource. This work studies deterministic pure-state conversion under three free operations: physically block incoherent operations (PBIO), strictly block incoherent operations (SBIO), and block dephasing covariant incoherent operations (BDCO). For PBIO, we prove that, under a natural nondegeneracy condition on the active Kraus branches, any deterministic conversion from one pure state to another must be implemented by a block incoherent unitary. When the nondegeneracy requirement is removed, the condition becomes more general. It demands that the blockwise action of every active branch reproduce the target block structure with a common proportionality factor across all output blocks. For SBIO and BDCO, we show that deterministic pure-state transformation is completely characterized by the majorization relation between the input and output block probability vectors. The converse proof is constructive, yielding an explicit Kraus representation for every admissible BDCO transformation. In the rank-one limit, these conditions reduce to the known pure-state transformation criteria for physically incoherent operations (PIO), strictly incoherent operations (SIO), and dephasing covariant incoherent operations (DIO) in the standard resource theory of coherence. Using the majorization condition, a maximally block-coherent state with uniform block weights is also identified as a universal pure-state resource under BDCO and SBIO. We have also provided geometric numerical illustrations comparing the state transformation power of BDCO and DIO for a fixed input state, fixed output state and mutual convertibility scenarios.