Measures of coherence generating power for quantum unital operations

Abstract

Given an orthonormal basis in a $d$-dimensional Hilbert space and a unital quantum operation $\mathcal{E}$ acting on it one can define a non-linear mapping that associates to $\mathcal{E}$ a $d\times d$ real-valued matrix that we call the Coherence Matrix of $\mathcal{E}$ with respect to $B$. We show that one can use this coherence matrix to define vast families of measures of the coherence generating power (CGP) of the operation. These measures have a natural geometrical interpretation as separation of $\mathcal{E}$ from the set of incoherent unital operations. The probabilistic approach to CGP discussed in P. Zanardi et al., arXiv:1610.00217 can be reformulated and generalized introducing, alongside the coherence matrix, another $d\times d$ real-valued matrix, the Simplex Correlation Matrix. This matrix describes the relevant statistical correlations in the input ensemble of incoherent states. Contracting these two matrices one finds CGP measures associated with the process of preparing the given incoherent ensemble and processing it with the chosen unital operation. Finally, in the unitary case, we discuss how these concepts can be made compatible with an underlying tensor product structure by defining families of CGP measures that are additive.

References in corpus (4)

• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• Observable measure of quantum coherence in finite dimensional systems
• Quantum coherence, time-translation symmetry and thermodynamics
• Extending Noether's theorem by quantifying the asymmetry of quantum states

Cited by in corpus (31)

• Operational Resource Theory of Quantum Channels
• Resource theory of non-Gaussian operations
• Dynamical Resource Theory of Quantum Coherence
• Estimating coherence measures from limited experimental data available
• Coherifying quantum channels
• Quantum coherence as a signature of chaos
• Experimental quantification of coherence of a tunable quantum detector
• Random Matrix Theory of the Isospectral twirling
• Scrambling of Algebras in Open Quantum Systems
• Quantum coherence and the localization transition
• Enhancing coherence of a state by stochastic strictly incoherent operations
• Coherence generating power of quantum dephasing processes
• Quantum scrambling of observable algebras
• Quantum coherence, quantum Fisher information and teleportation in the Ising-Heisenberg spin chain model of a heterotrimetallic Fe-Mn-Cu coordination polymer with magnetic impurity
• Coherence generating power of unitary transformations via probabilistic average
• Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian Geometry
• Coherence and complementarity based on modified generalized skew information
• Skew information-based coherence generating power of quantum channels
• Operational Quantum Mereology and Minimal Scrambling
• Cohering and decohering power of massive scalar fields under instantaneous interactions
• Coherence generation, symmetry algebras and Hilbert space fragmentation
• Quantifying coherence of quantum channels based on the generalized $\alpha$-$z$-relative Rényi entropy
• Localizable quantum coherence
• Coherence and Imaginarity as Resources in Quantum Circuit Complexity
• Measurement-induced nonlocality in arbitrary dimensions in terms of the inverse approximate joint diagonalization
• Long-time Quantum Scrambling and Generalized Tensor Product Structures
• Leggett-Garg inequality in Markovian quantum dynamics: role of temporal sequencing of coupling to bath
• Creation of quantum coherence with general measurement processes
• Coherence monotones of quantum channels based on two generalized quantum relative entropies
• Dynamics of cohering and decohering power under Markovian channels
• Statistics of Marginal Wavefunctions as a Real-Space Diagnostic of Quantum Entanglement