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Quantum-coherence quantifiers based on the Tsallis relative $α$-entropies

Alexey E. Rastegin

arXiv:1512.06652v3quant-ph

TL;DR

The paper addresses how to quantify quantum coherence relative to a prescribed basis while preserving crucial monotonicity properties. It constructs measures from Tsallis quantum divergences, derives their basic properties and coherence–mixedness trade-offs, and finds that selective-measurement monotonicity requires a parametric reformulation, while two quadratic measures behave differently.

  • Problem

    Quantum coherence measures should satisfy suitable properties, especially monotonicity, but an intuitive squared Hilbert–Schmidt-based measure does not satisfy it.

  • Method

    The paper studies coherence quantifiers induced by Tsallis quantum relative entropies and compares two quadratic measures based on matrix-element moduli.

  • Results

    The proposed measures obey a reformulated monotonicity relation for incoherent selective measurements, while the compared quadratic measures differ in their monotonicity behavior.

  • Takeaways & Limitations

    Monotonicity under incoherent selective measurements is naturally treated through a parametric extension of the standard formulation.

Abstract

from arXiv · show

The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum level. Various measures could be proposed to quantify coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. Basic properties of the considered coherence quantifiers are derived. Trade-off relations between coherence and mixedness are examined. The property of monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.

I. INTRODUCTION

Quantum coherence is studied as a physical resource connected to thermodynamics, entanglement, interference, and quantum algorithms. The paper focuses on constructing coherence quantifiers whose monotonicity properties are valid, especially for Tsallis-relative-entropy-based measures.

  • Quantum coherence is relevant to low-temperature thermodynamics, multipartite entanglement, decoherence, and quantum-information processes.
  • Monotonicity is central because the squared Hilbert–Schmidt-based coherence measure fails to satisfy valid coherence monotonicity.
  • The paper studies coherence quantifiers induced by Tsallis relative entropies and derives their basic properties.
  • The study examines coherence–mixedness trade-offs, reformulates monotonicity under incoherent selective measurements, and compares two quadratic coherence measures.

II. PRELIMINARIES

The preliminaries define coherence relative to a fixed orthonormal basis and review candidate quantifiers, quantum states, norms, and quantum operations. They emphasize that a natural squared ℓ2-based measure fails the monotonicity requirement.

  • The framework restricts the subsequent treatment to finite-dimensional Hilbert spaces and uses operator norms to characterize distances.
  • Quantum states are represented by normalized positive operators, while physical processes are modeled using completely positive maps and Kraus operators.
  • Trace-preserving completely positive maps are identified as quantum channels, supplying the operation framework for coherence properties.
  • A coherence measure assigns a non-negative real number to a quantum state relative to a prescribed orthonormal basis, with diagonal states treated as incoherent.
  • The squared ℓ2-norm provides a natural coherence candidate, but it does not obey the monotonicity requirement.

III. QUANTUM DIVERGENCES OF THE TSALLIS TYPE

This section introduces Tsallis quantum relative entropies and establishes positivity, monotonicity, convexity, and selective-measurement inequalities needed to analyze induced coherence measures. The quantum α-divergence is monotone under TPCP maps for α in (0,2].

  • The Tsallis relative α-entropy generalizes quantum relative entropy and reduces to the standard relative entropy as α approaches 1.
  • Quantum α-divergences of the Tsallis type are monotone under TPCP maps for α ∈ (0,2] and jointly convex in that range.
  • The classical Tsallis relative entropy is monotone for all α ≥ 0, whereas the quantum α-divergence has the narrower established range α ∈ (0,2].
  • For α > 0, the quantum α-divergence is non-negative, with equality conditions analyzed through the associated operator inequalities.
  • The divergence framework extends a standard relative-entropy inequality to Tsallis quantum divergences, producing a form used for incoherent selective measurements.

IV. COHERENCE QUANTIFIERS BASED ON THE TSALLIS DIVERGENCES

The paper constructs coherence quantifiers from Tsallis α-divergences, derives their basic properties, and relates them to purity and mixedness. It also gives an explicit minimizing incoherent state and compares the resulting bounds with earlier coherence relations.

  • The minimizing incoherent state is determined by probabilities proportional to the diagonal elements of ρ^α, yielding the Tsallis-based coherence measure in terms of matrix elements of ρ^α.For fixed ρ and α, the minimum is attained by choosing the diagonal state with δ_j proportional to ⟨e_j|ρ^α|e_j⟩.
  • At α = 1 the construction reduces to the relative-entropy coherence, while α = 2 produces a measure depending on squared off-diagonal moduli in a more complicated form.The α = 2 expression is written using ρ_ij = ⟨e_i|ρ|e_j⟩.
  • The Tsallis coherence quantifiers vanish exactly for incoherent states, so they satisfy the corresponding zero-coherence condition.Positivity of the Tsallis α-divergence establishes strict positivity for non-incoherent states.
  • For α ≤ 2, coherence is bounded above by a purity-dependent expression, providing a single-quantifier analogue of earlier complementarity relations.The bound uses Tr(ρ^2) and can be estimated experimentally through schemes related to Brukner–Zeilinger information, although complete MUB sets are not known for all dimensions.
  • For 0 < α ≤ 2, increasing mixedness decreases the upper bound on the corresponding coherence α-quantifier.The result is obtained by upper-bounding the purity-dependent expression with a linear function of normalized purity.

V. ON COHERENCE OF A SINGLE QUBIT

For a single qubit, the Tsallis coherence quantifiers admit explicit ranges and trade-offs with mixedness, with maximal coherence attained by pure states and zero coherence by incoherent states.

  • Single-qubit quantifiers: For fixed diagonal parameter u, C2(ω̂) and Cℓ1(ω̂) cover the same values, reaching their maxima on pure states and vanishing for incoherent states.Pure states satisfy |w|^2 = u(1 − u), while incoherent states have w = 0.
  • Single-qubit quantifiers: The maximal Cα(ω̂) curves for α = 1, 2, 3, and 4 show similar behavior, with larger α appearing more sensitive to coherence.The curves are symmetric about u = 1/2 and span the attainable quantifier values between the abscissa and each curve.
  • Coherence–mixedness trade-offs: For 0 < α ≤2, the general single-qubit trade-off obeys Cα(ω̂) + M(ω̂) ≤1.This follows from the dimension-two specialization of the general bound.
  • Coherence–mixedness trade-offs: The sum C2(ω̂) + M(ω̂) lies between bounds attained by incoherent and pure states, forming a narrow interval and making the upper bound sufficiently tight when diagonal elements are similar.The figure compares the dashed lower bound with the solid upper bound; the sum is analogous to, but distinct from, the quadratic relation for Cℓ1.
  • Coherence–mixedness trade-offs: The single-qubit quantifiers exhibit natural coherence–mixedness trade-offs, supporting the quadratic measure as an alternative whose monotonicity requires further examination.The paper next addresses whether this candidate satisfies the required monotonicity properties.

VI. FORMULATION OF THE MONOTONICITY PROPERTY

The paper establishes monotonicity for Tsallis-based coherence measures under incoherent operations and reformulates selective-measurement monotonicity as a parameter-dependent extension of the standard relation.

  • Proof strategy: The proof uses a common output-space construction for Kraus operators, block-column operators, and joint convexity of the Tsallis divergence.The construction yields orthogonal output blocks and connects the divergence bounds to the coherence measures.
  • Incoherent operations: For α ∈(0, 2], the Tsallis-based coherence quantifiers are monotone under incoherent quantum operations.The result follows from the divergence property and the minimization defining the coherence measure.
  • Selective measurements: Under incoherent selective measurements, the measures satisfy a reformulated monotonicity relation involving probabilities q_n associated with the reference incoherent state.The theorem applies for all α ∈(0, 2] and allows Kraus operators with different output spaces.
  • Selective measurements: This selective-measurement relation reduces to the standard α = 1 statement and serves as its natural parametric extension.Unlike the usual probabilities p_n, q_n depend on the incoherent reference state and indirectly on the input state.
  • Selective measurements: The standard selective-measurement form is difficult to obtain for general coherence measures; the paper identifies Cℓ1 and C1 as the known examples satisfying it directly.Linear combinations inherit the listed linear properties when each component measure satisfies them.

VII. TWO QUADRATIC MEASURES COMPARED

The section compares two quadratic coherence measures under selective measurements. The Tsallis-based quantifier satisfies the reformulated monotonicity condition, whereas the earlier measure fails both the standard and reformulated forms.

  • VII. TWO QUADRATIC MEASURES COMPARED: For the example, the weighted output coherence equals 2 + √/4 ≈0.4571, illustrating the reformulated monotonicity relation.
  • VII. TWO QUADRATIC MEASURES COMPARED: The Tsallis-based quantifier satisfies the reformulated monotonicity condition, while the earlier quadratic measure does not satisfy the standard condition.The reformulated condition is therefore necessary for the Tsallis-based measure's example, rather than an automatic consequence of ordinary relative-entropy monotonicity.
  • VII. TWO QUADRATIC MEASURES COMPARED: The right-hand side of the standard comparison reaches 0.5 at |b| = 1 and exceeds C2(ρ̂) ≈0.4571.
  • VII. TWO QUADRATIC MEASURES COMPARED: The earlier quadratic measure also violates the reformulated condition, so the proposed formulation is a nontrivial extension that reduces to the standard form as α →1.

VIII. CONCLUSIONS

The conclusions examine quantum-coherence measures based on Tsallis-type α-divergences and address trade-off relations involving coherence.

  • VIII. CONCLUSIONS: The paper examines quantum-coherence measures based on α-divergences of the Tsallis type.
  • VIII. CONCLUSIONS: The conclusions address trade-off relations involving coherence.
  • VIII. CONCLUSIONS: The section frames these results as part of the study of quantum-coherence measures.
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