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arXiv:2005.01083 · Europhysics Letters (EPL)

The reduction of the number of incoherent Kraus operations for qutrit systems

Jiahuan Qiao, Lingyun Sun, Jing Wang, Ming Li, Shu‐Qian Shen, Lei Li, Shao-Ming Fei

Published Apr 27, 2020• 0 citations
arXivHTMLPDF

Abstract

Quantum coherence is a fundamental property that can emerge within any quantum system. Incoherent operations, defined in terms of the Kraus decomposition, play an important role in state transformation. The maximum number of incoherent Kraus operators has been presented in Streltsov A., Rana S., Boes P. and Eisert J., Phys. Rev. Lett. , 119 (2017) 140402. In this work, we show that the number of incoherent Kraus operators for a single qubit can be reduced from 5 to 4 by constructing a proper unitary matrix. For qutrit systems we further obtain 32 incoherent Kraus operators, while the upper bound in the research of Sterltsov gives 39 Kraus operators. Besides, we reduce the number of strictly incoherent Kraus operators from more than 15 to 13. And we consider the state transformation problem for these two types of operations in single qutrit systems.

DOI
10.1209/0295-5075/129/60006

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Quantum coherence is a fundamental property that can emerge within any quantum system. Incoherent operations, defined in terms of the Kraus decomposition, play an important role in state transformation. The maximum number of incoherent Kraus operators has been presented in Streltsov A., Rana S., Boes P. and Eisert J., Phys. Rev. Lett. , 119 (2017) 140402. In this work, we show that the number of incoherent Kraus operators for a single qubit can be reduced from 5 to 4 by constructing a proper unitary matrix. For qutrit systems we further obtain 32 incoherent Kraus operators, while the upper bound in the research of Sterltsov gives 39 Kraus operators. Besides, we reduce the number of strictly incoherent Kraus operators from more than 15 to 13. And we consider the state transformation problem for these two types of operations in single qutrit systems.
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.
We investigate the single qubit transformations under several typical coherence-free operations, such as, incoherent operation (IO), strictly incoherent operation (SIO), physically incoherent operation (PIO), and coherence-preserving operation (CPO). Quantitative connection has been built between IO and SIO in single qubit systems. Moreover, these coherence-free operations have a clear hierarchical relationship in single qubit systems: CPO ⊂ PIO ⊂ SIO=IO. A new and explicit proof for the necessary and sufficient condition of single qubit transformation via IO or SIO has been provided, which indicates that SIO with only two Kraus operators are enough to realize this transformation. The transformation regions of single qubits via CPO and PIO are also given. Our method provides a geometric illustration to analyze single qubit coherence transformations by introducing the Bloch sphere depiction of the transformation regions, and tells us how to construct the corresponding coherence-free 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.
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.
We introduce the unitary-accessible coherence-erasure distance, which measures the minimum unitary-control cost required to transform a quantum state into an incoherent state without changing its spectrum. The target states are restricted to incoherent states on the same unitary orbit, leading to a geometry different from conventional state-space proximity. We establish inequalities relating orbit-restricted quantum transport, Bures geometry, and the unitary-control distance. These quantities coincide for pure states but can differ for mixed states. For qubits, we derive an exact expression and show that, in the nondegenerate case, the coherence-erasure cost depends on the eigenbasis orientation rather than on the eigenvalues. For nondegenerate qutrits, we reduce the problem to an optimization over the monomial unitary group. We derive the tight universal bound arccos⁡[(22−1)/4]\arccos[(2\sqrt2-1)/4]arccos[(22​−1)/4] and construct an explicit eigenbasis that attains it. The resulting distance also determines the minimum coherence-erasure time under bounded unitary driving, giving it a direct operational meaning.