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arXiv:2408.02885 · Quantum Information and Computation

State Convertibility under Genuinely Incoherent Operations

Shuanping Du, Zhaofang Bai

Published Oct 1, 2024• 2 citations
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Abstract

Abstract State convertibility is fundamental in the study of resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, we give a complete characterization of state convertibility under genuinely incoherent operations. It is found that convexity of the robustness of coherence plays a central role. Based on this, the majorization condition of determining convertibility from pure states to mixed states under strictly incoherent operations is provided. Moreover, maximally coherent states in the set of all states with fixed diagonal elements are determined. It is somewhat surprising that convexity of the robustness of coherence can also decide conversion between off-diagonal parts of coherent states. This might be a big step to answer completely the question of state convertibility for mixed states under incoherent operations.

DOI
10.2478/qic-2024-0007

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Abstract State convertibility is fundamental in the study of resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, we give a complete characterization of state convertibility under genuinely incoherent operations. It is found that convexity of the robustness of coherence plays a central role. Based on this, the majorization condition of determining convertibility from pure states to mixed states under strictly incoherent operations is provided. Moreover, maximally coherent states in the set of all states with fixed diagonal elements are determined. It is somewhat surprising that convexity of the robustness of coherence can also decide conversion between off-diagonal parts of coherent states. This might be a big step to answer completely the question of state convertibility for mixed states under incoherent operations.
Coherence manipulation is fundamental in the study of the resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, it is shown that a finite number of measure conditions are not sufficient to characterize coherence manipulation on general mixed states. Further, a finite number of measure conditions to classify coherence manipulation on subspace-independent states are found. For the input pure state, we also furnish a structural characterization of coherence manipulation in terms of a finite number of measure conditions.
Quantum state transformation under free operations plays a central role in the resource theory of coherence. In this paper, we investigate the transformation from a mixed coherent state into a pure one by using both incoherent operations and stochastic incoherent operations. We show that contrary to the strictly incoherent operations and the stochastic strictly incoherent operations, both the incoherent operations and the stochastic incoherent operations can increase the dimension of the maximal pure coherent subspace of a state. This means that the incoherent operations are generally stronger than the strictly incoherent operations when we want to transform a mixed coherent state into a pure coherent one. Our findings can also be interpreted as confirming the ability of incoherent operations to enhance the coherence of mixed states relative to certain coherence monotones under strictly incoherent operations.
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.
Coherence distillation is one of the central problems in the resource theory of coherence. In this Letter, we complete the deterministic distillation of quantum coherence for a finite number of coherent states under strictly incoherent operations. Specifically, we find the necessary and sufficient condition for the transformation from a mixed coherent state into a pure state via strictly incoherent operations, which recovers a connection between the resource theory of coherence and the algebraic theory of majorization lattice. With the help of this condition, we present the deterministic coherence distillation scheme and derive the maximum number of maximally coherent states obtained via this scheme.
The coherence resource theory needs to study the operational value and efficiency which can be broadly formulated as the question: when can one coherent state be converted into another under incoherent operations? We answer this question completely for one-mode continuous-variable systems by characterizing conversion of coherent Gaussian states under incoherent Gaussian operations in terms of their first and second moments. The no-go theorem of purification of coherent Gaussian states is also built. The structure of incoherent Gaussian operations of two-mode continuous-variable systems is discussed further and is applied to coherent conversion for pure Gaussian states with standard second moments. The standard second moments are images of all second moments under local linear unitary Bogoliubov operations. As concrete applications, we obtain some peculiarities of a Gaussian system: (1) there does not exist a maximally coherent Gaussian state which can generate all coherent Gaussian states; (2) the conversion between pure Gaussian states is reversible; (3) the coherence of input pure state and the coherence of output pure state are equal.
One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work we advance on this characterization within the quantum coherence resource theory by introducing the generalized coherence vector of an arbitrary quantum state. The generalized coherence vector is a probability vector that can be interpreted as a concave roof extension of the pure states coherence vector. We show that it completely characterizes the notions of being incoherent, as well as being maximally coherent. Moreover, using this notion and the majorization relation, we obtain a necessary condition for the conversion of general quantum states by means of incoherent operations. These results generalize the necessary conditions of conversions for pure states given in the literature, and show that the tools of the majorization lattice are useful also in the general case. Finally, we introduce a family of coherence quantifiers by considering concave and symmetric functions applied to the generalized coherence vector. We compare this proposal with the convex roof measure of coherence and others quantifiers given in the literature.