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Physical Review A

Generalized physically incoherent operations

Ping Li, Yongming Li

Published Jun 18, 2024• 2 citations
Publisher

Abstract

Quantum coherence is an intrinsic property within a single quantum system. Many different coherent resource theories have been proposed to characterize this property within the framework of resource theory. This difference is based on the difference in the identification of ``free'' operations. In this article, generalized physically incoherent operations, which are strictly contained in the intersection of incoherent operations and dephasing-covariant operations (dephasing-covariant incoherent incoherent operations), are introduced. We characterize the Kraus operators of dephasing-covariant incoherent incoherent operations and generalized physically incoherent operations, and further study the coherent state transformations under generalized physically incoherent operations. Specifically, we investigate the constraints imposed by the rank of the initial state on the extraction of a pure coherent state. Additionally, we present a sufficient condition for the transformation from a mixed state to a pure state ensemble.

DOI
10.1103/physreva.109.062424

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Quantum coherence is an intrinsic property within a single quantum system. Many different coherent resource theories have been proposed to characterize this property within the framework of resource theory. This difference is based on the difference in the identification of ``free'' operations. In this article, generalized physically incoherent operations, which are strictly contained in the intersection of incoherent operations and dephasing-covariant operations (dephasing-covariant incoherent incoherent operations), are introduced. We characterize the Kraus operators of dephasing-covariant incoherent incoherent operations and generalized physically incoherent operations, and further study the coherent state transformations under generalized physically incoherent operations. Specifically, we investigate the constraints imposed by the rank of the initial state on the extraction of a pure coherent state. Additionally, we present a sufficient condition for the transformation from a mixed state to a pure state ensemble.
Considerable work has recently been directed toward developing resource theories of quantum coherence. In this Letter, we establish a criterion of physical consistency for any resource theory. This criterion requires that all free operations in a given resource theory be implementable by a unitary evolution and projective measurement that are both free operations in an extended resource theory. We show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations as a natural generalization of the physically consistent operations. Necessary and sufficient conditions are derived for the convertibility of qubit states using dephasing-covariant operations, and we show that these conditions also hold for other well-known classes of 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.
A resource theory of quantum coherence attempts to characterize the quantum coherence that exists in a given quantum system. Many different approaches to a resource theory of coherence have recently been proposed, with their differences lying primarily in the identification of ``free'' or ``incoherent'' operations. In this article, we compare a number of these operational classes. In particular, the recently introduced class of dephasing-covariant operations is analyzed, and we characterize the Kraus operators of such maps. A number of coherence measures are introduced based on relative R\'enyi entropies, and we study incoherent state transformations under different operational classes. In particular, we show that the incoherent Schmidt rank can be increased arbitrarily large by certain noncoherence generating operations. The distinction between asymmetry-based versus basis-dependent notions of coherence theory is clarified, and we further develop the resource theory of NNN asymmetry, where NNN is the group of all diagonal incoherent unitaries.
Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between the classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations that arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this Letter, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixed-state conversion problem for a single qubit.
An important problem in the coherent resource theory is the convertibility of coherent states by free operations. In this paper we consider such a problem from a mixed coherent state into a pure one by using both stochastic incoherent operations and incoherent operations. We prove that stochastic incoherent operations can transform a mixed coherent state into a pure coherent state if and only if the density matrix of the initial state contains a singular principal submatrix. Then we provide two sufficient conditions with explicit expressions for the probabilistic transformation and the deterministic transformation. These two sufficient conditions facilitate the construction of concrete operators of stochastic incoherent operations and incoherent operations. These results demonstrate that incoherent operations are strictly more powerful than strictly incoherent operations within the probabilistic transformation as well as the deterministic transformation.
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.