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Resource Destroying Maps

Zi-Wen Liu, Xueyuan Hu, Seth Lloyd

arXiv:1606.03723v4quant-ph

TL;DR

The paper addresses difficulties in characterizing free operations and resource measures by developing a framework based on resource destroying maps. It shows how these maps connect free states, operations, and measures, including simple distance-based measures monotone under commuting operations, and applies the framework to coherence and discord. The framework also identifies robustness differences among resource-free conditions and constraints imposed by nonconvex free-state sets.

  • Problem

    Characterizing free operations and resource measures is difficult, especially for nonmaximal theories and resources with nonconvex free-state sets.

  • Method

    The paper analyzes resource theories through maps that preserve free states while sending resourceful states to free states, and studies operations and measures relative to those maps.

  • Results

    The framework relates free states, operations, and measures, proves that nonconvex free-state sets preclude linear resource destroying maps, and establishes strong monotonicity for a distance-based measure under selective commuting operations.

  • Takeaways & Limitations

    The theory provides a general resource-free framework applied to coherence and quantum correlations, including discord, while distinguishing robust nongenerating conditions from map-dependent nonactivating and commuting conditions.

  • Takeaways & Limitations

    Resource nonactivating and commuting conditions can be fragile under changes in the resource destroying map and input restrictions, motivating further restrictions or classes of maps.

Abstract

from arXiv · show

Resource theory is a widely-applicable framework for analyzing the physical resources required for given tasks, such as computation, communication, and energy extraction. In this paper, we propose a general scheme for analyzing resource theories based on resource destroying maps, which leave resource-free states unchanged but erase the resource stored in all other states. We introduce a group of general conditions that determine whether a quantum operation exhibits typical resource-free properties in relation to a given resource destroying map. Our theory reveals fundamental connections among basic elements of resource theories, in particular, free states, free operations, and resource measures. In particular, we define a class of simple resource measures that can be calculated without optimization, and that are monotone nonincreasing under operations that commute with the resource destroying map. We apply our theory to the resources of coherence and quantum correlations (e.g., discord), two prominent features of nonclassicality.

GENERAL ASPECTS OF RESOURCE DESTROYING MAPS

The paper develops general properties of resource destroying maps, including convexity constraints on their linearity and robustness of resource-free conditions.

  • GENERAL ASPECTS OF RESOURCE DESTROYING MAPS: Convexity of the free-state set is necessary for an associated resource destroying map to be linear.The section also analyzes the robustness of resource-free conditions.

Linearity of resource destroying map

Nonconvex state sets cannot be stabilized exactly by a linear map, so nonconvex resource theories admit no resource destroying channels.

  • Linearity of resource destroying map: Nonconvexity of a state set rules out any linear map that fixes exactly its members and changes every state outside it.Linearity would preserve a convex combination of two fixed states, contradicting the requirement that the combination outside the set not be fixed.
  • Linearity of resource destroying map: For a nonconvex free-state set, no linear map can satisfy both resource-destroying requirements on all inputs.Therefore, nonconvex resource theories do not admit resource destroying channels.

Robustness of resource-free conditions

The nongenerating condition is robust to the choice of resource destroying map, whereas nonactivating and commuting conditions can vary with that choice and remain meaningful for physically motivated maps.

  • Robustness of resource-free conditions: Resource nongenerating operations are independent of the chosen resource destroying map because the map is surjective onto the free-state set.This condition selects operations under which the free states are closed.
  • Robustness of resource-free conditions: Resource nonactivating and commuting conditions can depend on the resource destroying map, especially when one free state receives all nonfree inputs.Under such a map, operations that fail to stabilize every free state may need to map all states to one free state.
  • Robustness of resource-free conditions: Only the nongenerating condition remains robust under resource destroying map choices, including its selective version.The supplied discussion contrasts this robustness with the fragility of nonactivating and commuting conditions.
  • Robustness of resource-free conditions: Nonactivation under varying maps remains an open study direction, although nonactivating and commuting conditions remain meaningful for physically motivated definitions.Possible restrictions include excluding orphan states or considering a class of resource destroying maps.

Selective monotonicity

The paper establishes selective monotonicity for a simple distance-based resource measure under selective operations that commute with the resource destroying map.

  • Selective monotonicity: Selective monotonicity is defined as average nonincrease of a resource measure over selective measurement outcomes.The outcome probabilities are given by p_µ = tr E_µ(ρ).
  • Selective monotonicity: A distance between a state and its resource-destroyed counterpart obeys strong monotonicity under selective commuting operations.The result applies when the distance satisfies the stated average monotonicity property, including quantum relative entropy.

COHERENCE

The theory distinguishes several coherence-free operation classes derived from the dephasing map Π, whose relationships are tested through Kraus-operator conditions and explicit counterexamples.

  • COHERENCE: The framework identifies existing coherence-operation classes with familiar notions: X̄_s(Π) corresponds to IO, X_s(Π) to SIO, and X(Π) to DIO.The paper also notes that X̄(Π) is the coherence-nongenerating class studied previously.
  • COHERENCE: An operation with Kraus operators K1=|0⟩⟨+| and K2=|1⟩⟨−| belongs to X̄_s(Π) but not X(Π).Its dephased input and output differ in order, demonstrating failure of the commuting condition.
  • COHERENCE: A qutrit operation E2 satisfies X(Π) but not X̄_s(Π), because no Kraus decomposition can make all its Kraus operators incoherent.The given Kraus operators satisfy the X(Π) condition, while unitary relations preserve the failure of incoherent Kraus representations.
  • COHERENCE: The classes X̄_s(Π) and X(Π) are incomparable but overlap because X_s(Π) is contained in both.Thus neither class universally contains the other, although their intersection is nonempty.
  • COHERENCE: Measure-and-prepare operations with Kraus operators Ki=|fi⟩⟨i| belong to X̄*_s(Π), but coherent prepared states place them outside the coherence-nongenerating class.The measurement destroys input coherence, while choosing a coherent |fi⟩ allows coherence generation from the incoherent state |i⟩.

Other definitions of coherence-free operations

Translationally invariant and genuinely incoherent operations impose coherence restrictions from different constructions, and their relation to the Π-based classes depends on Hamiltonian degeneracy and state invariance.

  • Other definitions of coherence-free operations: For a nondegenerate Hamiltonian, TIO* is contained in X_s(Π), whereas general TIO can generate coherence within degenerate decoherence-free subspaces.Therefore, general TIO is not necessarily contained in the maximal Π-based coherence-nongenerating class.
  • Other definitions of coherence-free operations: GIO requires every Kraus operator to be diagonal in the incoherent basis and consequently leaves every incoherent state invariant.This imposes constraints beyond merely preventing coherence generation.
  • Other definitions of coherence-free operations: The Π-based framework separates diagonal and off-diagonal components, with the latter erased by a subsequent dephasing map.This provides the mechanism behind the inclusion of GIO in X_s(Π).
  • Other definitions of coherence-free operations: GIO is a proper subset of X_s(Π): the erasure channel to |0⟩⟨0| belongs to X_s(Π) but is not GIO.It fails GIO’s invariance requirement because it does not preserve incoherent states other than |0⟩⟨0|.

DISCORD

Discord-free analysis is harder than coherence analysis because the local discord-destroying map depends on the input state and is ambiguous within degenerate eigenspaces.

  • DISCORD: The discord-destroying map π_A is a local measurement in an eigenbasis of the reduced state ρ_A, whose spectral projectors define the relevant basis.The basis depends on the input state, creating the central difficulty for classifying discord-free operations.

Unitary-isotropic channels are in XA(πA) and Xs,A(πA)

Unitary-isotropic channels satisfy both ordinary and selective π_A-commutation conditions, while projective measurements and higher-dimensional mixed-unitary channels reveal strict separations among the discord-related classes.

  • Unitary-isotropic channels are in XA(πA) and Xs,A(πA): Unitary-isotropic channels belong to both X_A(π_A) and X_s,A(π_A), including unitary and depolarizing channels as special cases.They have the form ũ_γ(ρ)=(1−γ)UρU†+γI/d and admit a unitary Kraus decomposition through Heisenberg-Weyl twirling.
  • Unitary-isotropic channels are in XA(πA) and Xs,A(πA): A local projective measurement Ψ is in X̄_s,A(π_A) but not X_A(π_A) unless its basis matches the eigenbasis of ρ_A.Its output is always diagonal in the measurement basis, but it generally fails to commute with the input-dependent π_A.
  • Unitary-isotropic channels are in XA(πA) and Xs,A(πA): Selective classes need not be contained in their original counterparts for nonlinear π_A: certain mixed-unitary channels lie outside X̄_A(π_A).This contrasts with the usual strengthening relation expected for selective operation classes.
  • Unitary-isotropic channels are in XA(πA) and Xs,A(πA): For qudits with d>2, the discord-related classes form a strict hierarchy, unlike the qubit case where unital and mixed-unitary channels coincide.The higher-dimensional distinction reflects the absence of a general quantum Birkhoff theorem and the narrower isotropic structure.

A measure-and-prepare map

The paper defines a measure-and-prepare operation that can generate discord but cannot activate it, establishing its nonactivating property through composition with a local resource-destroying map.

  • The measure-and-prepare map ξ can generate discord because it is not commutativity-preserving, yet it cannot activate discord.Its Kraus operators are K_i = |g_i⟩⟨i|, where {|i⟩} diagonalizes ρ_A and some output states are nonorthogonal.
  • The identity ξ_A ⊗ I_B = (ξ_A ⊗ I_B) ◦ π_A shows that ξ is nonactivating.The protocol is not linear, and whether quantum channels can share this property remains open.
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