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Genuine quantum coherence

Julio I. de Vicente, Alexander Streltsov

arXiv:1511.08346v2quant-phcond-mat.stat-mechmath-phmath.MPphysics.optics

TL;DR

The paper addresses how to define free operations for coherence when the physical setting does not specify them. It introduces genuinely and fully incoherent operations and characterizes their state transformations. The resulting frameworks have severe limits on deterministic manipulation and lack a golden unit.

  • Problem

    Coherence has no clear physical restriction that uniquely determines which operations should be free, leaving alternative operation sets to be assessed.

    Source: §I · p. 2

  • Method

    The paper defines GIO by preservation of all incoherent states, characterizes GIO and FIO, and analyzes their state transformations.

    Source: §V · pp. 17–18

  • Results

    Neither GIO nor FIO has a golden unit, and GIO also lacks general asymptotic distillation and dilution transformations.

    Source: §3 · p. 16, §V · pp. 17–18

  • Takeaways & Limitations

    A coherence resource theory with a golden unit must include free operations that can create coherence in some experimental realization.

    Source: §V · p. 18

    Under GIO, single-copy transformations require the same diagonal elements, and general distillation and dilution are impossible.

    Source: §3 · p. 16, §V · pp. 17–18

Abstract

from arXiv · show

Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. We propose here the framework of genuine quantum coherence. This approach is based on a simple principle: we demand that a genuinely incoherent operation preserves all incoherent states. This framework captures coherence under additional constrains such as energy preservation and all genuinely incoherent operations are incoherent regardless of their particular experimental realization. We also introduce the full class of operations with this property, which we call fully incoherent. We analyze in detail the mathematical structure of these classes and also study possible state transformations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e., there is no single state from which all other states can be created deterministically with the free operations. This suggests that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.

I. INTRODUCTION

Coherence is a quantum resource, but unlike entanglement its physical setting does not uniquely determine which operations should be free. The paper introduces genuinely and fully incoherent operations, characterizes them, and studies the resulting state transformations.

I. INTRODUCTION

  • Coherence is treated as a resource because it underlies interferometry and may contribute to biological processes, motivating a framework for its quantification and manipulation.Resource theories specify free states and operations, which determine the permitted transformations and the corresponding resource measures.Source: §I · p. 1
  • Because coherence lacks a clear physical restriction on free operations, the paper examines alternative operation sets and their state-manipulation limits.The standard incoherent operations are defined through a Kraus decomposition, although the paper notes that their physical justification is unclear.Source: §I · p. 2
  • GIO allow limited deterministic manipulation and no general distillation or dilution, while stochastic transformations have a richer structure.The paper derives an optimal success probability for stochastic GIO transformations.Source: §I · p. 2, B. Mathematical characterization · p. 4

A. Definition and motivation

  • Genuinely incoherent operations (GIO) preserve every incoherent state, a condition the paper relates to unspeakable coherence and energy-preserving settings.When the incoherent basis consists of eigenstates of a nondegenerate Hamiltonian, GIO correspond to energy-preserving operations.Source: A. Definition and motivation · pp. 2–3, §I · p. 2
  • Unlike standard incoherent operations, GIO remain incoherent under every Kraus decomposition; fully incoherent operations (FIO) are the most general maps with this property.The paper characterizes FIO as a broader class than GIO.Source: §I · p. 2, A. Definition and motivation · p. 3

B. Mathematical characterization

  • GIO are exactly the maps that fix all incoherent states, have diagonal Kraus operators in every representation, and admit a Schur-map characterization with a positive semidefinite matrix whose diagonal entries are 1.The paper uses this characterization to analyze the mathematical structure of GIO and state transformations.Source: B. Mathematical characterization · pp. 3–4
  • Every GIO is unital and a strict subset of standard incoherent operations; qubit and qutrit GIO are mixtures of diagonal unitaries, but this form fails from dimension 4 onward.The paper also characterizes stochastic GIO through a Schur-map condition with diagonal entries between 0 and 1.Source: B. Mathematical characterization · p. 4

C. Coherence rank and coherence set

For pure states under genuine incoherent operations, both the number and identity of occupied basis elements matter. The coherence set captures the latter distinction, which reflects the framework’s unspeakable character.

C. Coherence rank and coherence set

  • Diagonal unitaries relate pure states with equal genuine coherence, so phases can be removed and amplitudes taken nonnegative without loss of generality.An incoherent pure state has only one nonzero amplitude.Source: C. Coherence rank and coherence set · p. 5
  • Coherence rank counts the nonzero basis amplitudes, while the coherence set records which basis elements have nonzero amplitudes.Both properties constrain possible transformations between resource states under genuine incoherent operations.Source: C. Coherence rank and coherence set · p. 5
  • Unlike standard incoherent operations, genuine incoherent operations cannot exchange incoherent basis states, so states with different occupied basis elements are not interchangeable.This makes the identity of the occupied basis elements relevant in this unspeakable-coherence framework.Source: C. Coherence rank and coherence set · p. 5

D. Quantifying genuine coherence

The paper defines genuine coherence measures through monotonicity under genuinely incoherent operations and develops distance-based constructions. It identifies examples and notes that strong monotonicity remains unresolved for some measures.

D. Quantifying genuine coherence

  • A genuine coherence measure is nonnegative, vanishes only on incoherent states, and does not increase under genuinely incoherent operations; stronger conditions include average monotonicity and convexity.Average monotonicity together with convexity implies ordinary monotonicity.Source: D. Quantifying genuine coherence · p. 5
  • Because GIO are a subset of standard incoherent operations, standard coherence monotones remain genuine coherence monotones, and Wigner–Yanase skew information is also a convex measure.Whether the skew information satisfies average monotonicity remains open.Source: D. Quantifying genuine coherence · p. 5
  • The distance from a state to the incoherent states defines a genuine coherence measure, and it is convex when the distance is jointly convex.The result applies to distances that do not increase under unital operations.Source: D. Quantifying genuine coherence · p. 5
  • Distances based on Schatten p-norms yield genuine coherence measures, but strong monotonicity is not established for the simpler distance-to-dephased-state construction.For general Schatten norms, dephasing need not produce the closest incoherent state.Source: D. Quantifying genuine coherence · p. 6

E. State manipulation under GI operations

The section examines how GI operations order states and finds deterministic and probabilistic transformations sharply constrained. Pure-state conversions are trivial deterministically, while stochastic and mixed-state conversions depend on support and basis structure.

E. State manipulation under GI operations

  • The section frames state conversions as an ordering of resource usefulness and surveys pure and mixed, deterministic and stochastic, and single- and many-copy cases.The authors characterize GI state manipulation as rather contrived.Source: E. State manipulation under GI… · pp. 6–7

1. Single-state transformations

  • Deterministic GI transformations between pure states are trivial, and the framework has no single state from which all others can be prepared.Pure states can only be transformed within classes related by genuinely incoherent unitaries.Source: §1 · p. 7
  • Every mixed state can be obtained from some pure state, and a pure state can deterministically produce a mixed state exactly when their basis populations match.GI operations preserve the diagonal entries of the density matrix.Source: §1 · p. 7
  • A mixed state cannot be deterministically transformed into a pure state by GI operations.The stated exception would require all pure components to be related by a genuinely incoherent unitary, which would make the input pure.Source: §1 · p. 7
  • A probabilistic pure-state conversion is possible exactly when the target’s coherence set is contained in the source’s.The theorem also gives an optimal success probability, but the supplied passages do not provide its expression.Source: §1 · p. 8
  • Stochastic conversion probabilities do not define a consistent ordering of pure states, while fixing a target yields a computable genuine-coherence monotone.Different targets produce different monotones, each maximal for a different state in the maximal genuinely coherent set.Source: §1 · p. 8
  • A mixed state can probabilistically yield a pure coherent state only if its projection onto some two-basis-element subspace is pure and coherent.The authors conclude that most mixed states cannot be stochastically transformed to any pure state in the one-copy regime.Source: §1 · p. 8

2. Multiple-state and multiple-copy transformations

The section studies joint GI operations and asymptotic manipulation across multiple states and copies. It finds no activation and rules out non-trivial distillation and dilution under the stated conditions.

2. Multiple-state and multiple-copy transformations

  • The authors identify activation as a multiple-state resource-manipulation possibility and define catalysis as activation where the auxiliary state is returned.Their lemma establishes that no activation phenomenon occurs under GI operations.Source: §2 · p. 9
  • Joint GI operations cannot produce activation or any non-trivial form of distillation or dilution across multiple states and copies.The section defines the joint incoherent basis and requires joint GI maps to preserve incoherent states in that basis.Source: §2 · p. 9
  • GI operations cannot distill σ from ρ at any positive rate when ρ has lower relative entropy of coherence than σ.For a single qubit, no non-equivalent state can distill |+2⟩.Source: §2 · p. 10
  • For any two coherent states of the same dimension, GI operations cannot dilute σ to ρ at any rate R < 1.The result is stated to hold independently of the coherence measure used.Source: §2 · p. 10

F. Relation to entanglement theory of maximally correlated states

The section establishes that a GI conversion between states induces an LOCC conversion between their corresponding maximally correlated states. This extends the coherence–entanglement connection to tasks involving GI operations.

F. Relation to entanglement theory of maximally correlated states

  • If σ is obtained from ρ by a genuinely incoherent operation, then σ’s maximally correlated state is obtainable from ρ’s by LOCC.The theorem proves the conjectured relation for conversions under genuinely incoherent operations.Source: F. Relation to entanglement theory of… · p. 11
  • The proof realizes the corresponding LOCC map by applying the same GI operation to Alice’s subsystem.The proof writes the converted state explicitly and applies the operation to the first subsystem of the maximally correlated state.Source: F. Relation to entanglement theory of… · p. 11
  • For any coherence-based task involving GI operations, the theorem permits corresponding statements about the entanglement-based task involving LOCC.Earlier results equate distillable coherence, coherence cost, and coherence of assistance with the corresponding entanglement quantities for maximally correlated states.Source: F. Relation to entanglement theory of… · p. 11

A. General concept

Fully incoherent (FI) operations are incoherent in every Kraus representation, with a characterization requiring all Kraus operators to be incoherent and share the same form. FI operations permit preparation of pure incoherent states but are non-convex, limiting preparation of mixed incoherent targets.

A. General concept

  • FI operations are exactly those whose Kraus operators are incoherent and have the same form.Their nonzero entries occur at the same row in each column across the Kraus operators.Source: A. General concept · p. 11
  • FI maps include GI maps, but the erasing map is FI and not GI.The erasing map sends every input to |0⟩⟨0|, so other incoherent states are not preserved.Source: A. General concept · pp. 11–12
  • Every pure incoherent state can be prepared from an arbitrary input using an FI erasure map.The map sends every state to the chosen incoherent basis state.Source: A. General concept · p. 12
  • FI maps allow incoherent unitaries that permute basis elements, making coherence rank the relevant measure of coherence.The passage describes this as speakable coherence.Source: A. General concept · p. 12
  • FI maps are non-convex, so mixing free operations with probabilities need not yield a free operation, and mixed incoherent states are not universally reachable.The paper links non-convexity to FI being characterized by the set of Kraus operators; not every input can be transformed to every mixed incoherent target.Source: A. General concept · pp. 12–13

B. Permutations as basic FI operations

Fully incoherent permutations provide unitary relabelings, but deterministic pure-state conversion remains tightly constrained. The qutrit state |+3⟩ has only three possible pure-state outputs, up to FI-unitary equivalence.

B. Permutations as basic FI operations

  • Any general permutation decomposes into swaps of two basis states, and every such permutation is an FI unitary.The permutation section treats these operations as basic fully incoherent transformations.Source: B. Permutations as basic FI operations · p. 13

1. Pure state deterministic transformations

  • Coherence rank cannot increase under deterministic FI transformations; transformations between states of equal rank obey an additional necessary-and-sufficient condition.The supplied theorem passage does not state that condition's formula.Source: §1 · p. 13
  • For a single qubit, pure-state outputs are limited to incoherent states or states FI-unitarily equivalent to the input, unlike general incoherent operations.General incoherent operations can transform |+2⟩ into any single-qubit state, whereas FI operations cannot.Source: §1 · p. 13
  • Deterministic FI operations map |+3⟩ to only three distinct pure states and their FI-unitary equivalents.The theorem identifies the full set of pure-state outputs up to FI unitaries.Source: §1 · p. 14
  • Even when coherence rank decreases, majorization is not sufficient for FI conversion; general incoherent operations can transform |+3⟩ into any qutrit state.Under FI operations, only one coherence-rank-two state, up to FI-unitary equivalence, is obtainable from |+3⟩.Source: §1 · p. 15

2. Pure state stochastic transformations

The section characterizes stochastic conversion under fully incoherent operations by optimizing genuinely incoherent protocols over input permutations. It establishes an optimal probability for equal-rank conversions and lower bounds for rank-decreasing conversions.

2. Pure state stochastic transformations

  • For rank-decreasing conversions, permutation-optimized genuinely incoherent protocols provide lower bounds on the maximal FI conversion probability.The section notes that these bounds can be stronger than the unpermuted protocols.Source: §2 · p. 15
  • A state of higher coherence rank cannot be reached with nonzero probability by FI operations.This restriction accompanies the section's bounds for rank-decreasing transformations.Source: §2 · p. 15
  • For states with equal coherence rank, the optimal FI conversion probability is given by the theorem's bound, which becomes an equality.The bound optimizes over permutations of the input state.Source: §2 · p. 15

3. Many-copy transformations

FI operations permit a two-copy activation example that is impossible with one copy. This counters the cited obstruction for GI operations, while leaving the broader many-copy structure open.

3. Many-copy transformations

  • FI operations invalidate the GI activation obstruction, leaving open the possibility of richer many-copy transformations.The paper presents this as an open possibility rather than an established general characterization.Source: §3 · p. 16
  • Two copies of |+2⟩ can be transformed by FI operations into (2|00⟩ + |01⟩ + |11⟩)/2; tracing out one qubit activates a transformation to ρ.The passage gives this as a two-copy example of activation.Source: §3 · p. 16
  • The one-copy transformation from |+2⟩ to ρ is impossible because the full-rank output requires full-rank Kraus operators, restricting the map to permutations and GI maps with mismatched diagonals.The cited corollary requires the input and output diagonal entries to agree up to permutations.Source: §3 · p. 16

IV. RELATION TO OTHER CONCEPTS OF QUANTUM COHERENCE

The section compares several coherence-operation classes through their defining constraints and inclusion relations. It shows that these classes do not form a single hierarchy, with some relations depending on dimension.

IV. RELATION TO OTHER CONCEPTS OF QUANTUM COHERENCE

  • MIO preserves incoherent states, whereas DIO commutes with dephasing.Source: §IV · p. 16
  • TIO commutes with time translations, SIO has Kraus operators that commute with dephasing, and PIO uses incoherent projections and unitaries.Source: §IV · p. 16
  • FIO is contained in IO, GIO is contained in SIO, and neither FIO nor SIO contains the other.The stated Kraus-operator constraints distinguish the two classes.Source: §IV · pp. 16–17
  • GIO is contained in PIO for qubits and qutrits, but this inclusion fails in higher dimensions.The paper gives a four-dimensional GIO map that is not PIO.Source: §IV · p. 17
  • GIO is a strict subset of TIO, while FIO and TIO are incomparable; the erasing operation is TIO but not GIO.The fully depolarizing qubit map is TIO but not FI.Source: §IV · p. 17

V. CONCLUSIONS

The conclusions characterize genuine and fully incoherent operations and emphasize their limits for coherence manipulation. The authors also connect those limits to the conditions a resource theory would need to have a golden unit.

V. CONCLUSIONS

  • GIO preserves all incoherent states, admits a Schur-map characterization, and allows neither a golden unit nor asymptotic distillation or dilution.Single-copy GIO transformations require states to have the same diagonal elements.Source: §V · p. 17
  • Stochastic GIO transformations can be more general, and Theorem 10 gives the optimal probability for this procedure.Source: §V · p. 18
  • GIO is incoherent for every Kraus decomposition, and FIO is the complete class of operations with that property.FIO strictly contains GIO, but still has no maximally coherent state and permits only restricted deterministic pure-state transformations.Source: §V · p. 18
  • Whether FIO permits distillation or dilution remains open.Source: §V · p. 18
  • The authors suggest that GI and FI results could serve as building blocks for protocols in less restrictive frameworks.Source: §V · p. 18
  • A resource theory with a golden unit must include free operations that can create coherence in some experimental realization.Source: §V · p. 18

Appendix A: Mathematical structure of the set of GI maps

The appendices characterize the extremal structure of GI maps, establish a dimension-dependent mixed-unitary result, and verify a coherence measure. They also state a preparation restriction for FI maps.

Appendix A: Mathematical structure of the set of GI maps

  • A GI map is extremal among GI maps exactly when it is extremal among unital maps.GI maps are unital, and the equivalence lets the authors use unital-map structure to analyze them.Source: Appendix A · p. 18
  • Every GI operation on L(C2) and L(C3) is mixed-unitary, whereas for every d ≥ 4 there exists a GI map that is not mixed-unitary.Source: Appendix A · p. 19

Appendix B: Wigner-Yanase skew information is a convex measure of genuine coherence

  • For nondegenerate diagonal H, Wigner-Yanase skew information is a convex measure of genuine coherence.The appendix establishes the measure conditions using its vanishing criterion, monotonicity under translationally invariant operations, and convexity.Source: Appendix B · pp. 19–20

Appendix C: FI maps do not allow to prepare arbitrary mixed incoherent states from arbitrary input states

  • FI maps cannot prepare arbitrary mixed incoherent states from arbitrary inputs, although pure incoherent states can be obtained from any state.Source: Appendix B · p. 20
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