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One-Shot Operational Quantum Resource Theory

Zi-Wen Liu, Kaifeng Bu, Ryuji Takagi

arXiv:1904.05840v2quant-ph

TL;DR

One-shot quantum resource conversion asks how efficiently a state can be formed from standard resource states or distilled into them when only a single or finite number of copies is available. The paper develops general rate bounds using resource measures and theory-dependent coefficients, and gives examples spanning multiple resource theories.

  • Problem

    The framework seeks a unified account of optimal resource formation and distillation rates when only one or finitely many state copies are available and some error is allowed.

    Source: §I · p. 1

  • Method

    The paper derives general one-shot rate bounds from smoothed max/min-relative entropies, hypothesis-testing relative entropy, and free robustness, using theory-dependent modification coefficients.

    Source: §I · p. 1

  • Results

    The bounds are nearly achievable in many cases, and golden states make several resource measures coincide in generic convex theories.

    Source: §I · p. 1

  • Takeaways & Limitations

    The results give resource measures operational meanings through state conversion and extend resource-specific one-shot results across a wide range of theories.

    Source: §I · p. 1

Abstract

from arXiv · show

A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-non-specific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max/min-relative entropies and hypothesis testing relative entropy, as well as the free robustness measure, providing them with general operational meanings in terms of optimal state conversion. Our results encompass a wide class of resource theories via the theory-dependent coefficients we introduce, and the discussions are solidified by important examples, such as entanglement, coherence, superposition, magic states, asymmetry, and thermal non-equilibrium.

I. INTRODUCTION

The paper develops a resource-independent framework for one-shot formation and distillation, asking how efficiently quantum states can be converted under free operations. It gives general rate bounds using resource measures and shows that the framework applies across several resource theories.

I. INTRODUCTION

  • Resource theory provides a shared framework for quantifying resource features across entanglement, coherence, superposition, magic states, asymmetry, purity, thermal non-equilibrium, and non-Gaussianity.The paper situates its resource-independent approach within this breadth of applications.Source: §I · p. 1
  • The framework bounds one-shot formation and distillation rates using smoothed max/min-relative entropies, hypothesis-testing relative entropy, and free robustness, with theory-dependent modification coefficients.The bounds are nearly achievable in many cases and give the measures operational meanings through state conversion.Source: §I · p. 1
  • The analysis covers maximal free operations and commuting operations characterized through resource-destroying maps.Examples include non-entangling operations, MIO, Gibbs-preserving maps, and DIO.Source: §I · p. 1
  • The framework addresses the one-shot setting, where only one or finitely many copies are available and some error is allowed.The introduction contrasts this with asymptotic results for infinitely many i.i.d. copies and notes that only a few resource-specific one-shot results were known.Source: §I · p. 1
  • Golden states make useful reference currencies: in generic convex theories, several resource measures and their modification coefficients coincide, yielding nearly tight bounds.The paper presents this as a generalization of existing resource-specific results and illustrates it with new examples.Source: §I · p. 2

II. PRELIMINARIES

The preliminaries define the resource measures, free-state assumptions, and resource-destroying maps used to formulate the paper’s one-shot bounds. They also establish the monotonicity and operation classes needed for the later results.

II. PRELIMINARIES

  • The free states are assumed topologically closed, so extrema over the free-state set are well-defined.The paper also assumes the resource measures used throughout are well-defined.Source: §II · pp. 2–3
  • Smoothed max- and min-relative entropies optimize their divergences over an epsilon-neighborhood defined by fidelity, while hypothesis-testing relative entropy uses a test operator with a type-I error constraint.The hypothesis-testing quantity is identified with the definition in Ref..Source: §II · p. 2
  • Data processing makes the max- and min-relative entropies and purified distance monotones under free operations, and the free fidelity is the maximum fidelity with a free state.The max- and min-relative entropies are introduced as ends of the quantum Rényi relative-entropy spectrum.Source: §II · p. 2
  • The free robustness minimizes the noise needed to mix a state with another state until the mixture is free; finite free robustness is imposed as Condition (FFR).For affine free-state sets, resource states have infinite free robustness; generalized robustness instead allows the added state to be arbitrary and yields the max-relative entropy.Source: §II · p. 2
  • A resource-destroying map sends resource states to free states while fixing free states; exact maps output a closest free state, whereas RD channels include dephasing and twirling.The map induces resource-nongenerating operations and commuting operations; contractive distances to the map’s output are monotones under the latter.Source: §II · pp. 2–3

III. RESOURCE CURRENCIES AND MODIFICATION COEFFICIENTS

The paper defines modification coefficients for reference-state currencies and identifies golden states whose normalized resource measures coincide. It then gives examples across several resource theories.

III. RESOURCE CURRENCIES AND MODIFICATION COEFFICIENTS

  • Modification coefficients normalize the resource measures of reference states by log d, and are defined from free fidelity, max/min-relative entropy, and log-robustness.Reference states serve as the currency for forming a target state or distilling standard resource.Source: §III · p. 3
  • The coefficient of a maximum-resource currency can encode features such as the size of the free-state set; the maximum magic state is described as closer to stabilizer states than the coherence counterpart.The paper relates this closeness to a smaller modification coefficient for magic than for coherence.Source: §III · p. 3
  • For each dimension, a pure state can maximize free fidelity and min- and max-relative-entropy measures with a common value g_d under Condition (CH); the equality also holds for exact resource-destroying maps.Condition (CH) requires the free states to be the convex hull of pure free states, a property stated to hold for entanglement, coherence, superposition, and magic-state theories.Source: §III · p. 3
  • In coherence theory, the complete dephasing channel is an exact resource-destroying map and g_d=1 for all valid dimensions.Theorem 1 applies to this example.Source: §III · p. 3
  • For entanglement, the golden state is a maximally entangled state and g_d=1/2 for the stated valid dimensions.The paper notes that constant golden coefficients underpin simple one-shot entanglement and coherence manipulation results.Source: §III · p. 3
  • Golden states also exist in quantum thermodynamics despite failure of Condition (CH), and in the infinite-temperature purity theory g_d=1 for all dimensions.The passage also reports a single-qubit magic-state coefficient of approximately 0.34.Source: §III · p. 3

IV. OPTIMAL RATES OF ONE-SHOT RESOURCE MANIPULATION

The section bounds one-shot formation and distillation rates using resource measures and theory-dependent modification coefficients. Under stated conditions, golden reference states can simplify these bounds, while commuting-operation distillation remains more limited.

IV. OPTIMAL RATES OF ONE-SHOT RESOURCE MANIPULATION

  • Golden states satisfy CT in coherence theory because their overlap with every incoherent state is 1/d.The text also identifies a multi-qubit superposition theory example in Appendix G.Source: §IV · p. 4
  • Formation costs admit general bounds from resource measures and modification coefficients, with distinct choices for resource non-generating and commuting operations.The applicable measures and coefficients depend on the operation class and conditions such as CT or FFR.Source: §IV · p. 4
  • Golden states satisfying CH and CT yield distillation bounds indexed by g_d log d, while golden states also support roughly matching formation bounds.For the distillation corollary, d_0 is the smallest dimension satisfying g_d log d ≥ R_ε(ρ).Source: §IV · p. 4
  • A pure golden state is a root state when the maxima of the specified resource measures coincide and either CT holds or FFR and the stated pure-state equality condition hold.A root state allows any state on the same Hilbert space to be obtained by a free operation.Source: §IV · p. 4
  • For pure reference states, distillation under any free operations is bounded using the hypothesis-testing relative entropy through the largest qualifying reference dimension.The threshold is −log f(Φ_d) ≤ D^ε_H(ρ).Source: §IV · p. 4
  • For pure reference states, an additional distillation bound applies to any free operations, and an FFR condition gives a bound for general reference states using log-robustness.A separate upper bound is given for commuting operations, but only lower bounds are available for some special notions of those operations.Source: §IV · p. 4
  • Magic-state bounds apply to conventional distillation using copies of the T state under stabilizer-preserving operations, complementing results for a different setup.The discussion states that Clifford magic states have coinciding m_f, m_min, and m_max, while m_LR is generically larger and nonconstant.Source: §IV · p. 5

V. CONCLUDING REMARKS

The paper concludes that one-shot formation and distillation rates can be bounded by resource monotones and modification coefficients, which can collapse for max-resource states. It also relates formation and distillation divergences to the extreme orders of quantum Rényi relative entropy.

V. CONCLUDING REMARKS

  • The work establishes theory-independent bounds linking one-shot formation and distillation rates to one-shot divergence and log-robustness resource monotones.Modification coefficients account for the resourcefulness of the currency and can exhibit a collapsing property for a notion of max-resource states.Source: §V · p. 5

Appendix A: Max- and min-relative entropies as quantum R´enyi relative entropies

  • Max- and min-relative entropies correspond to extreme orders of quantum Rényi relative entropy, with formation and distillation bounds using related quantities.The passage characterizes these bounds as normally strongest in their respective forms because Rényi relative entropies are non-decreasing in order.Source: Appendix A · p. 8
  • The max-relative entropy is the sandwiched Rényi divergence limit as α → ∞, whereas the min-relative entropy is the order-zero Rényi relative entropy.The appendix distinguishes another quantity sometimes called min-relative entropy, equivalent to the sandwiched order-1/2 divergence and directly related to fidelity.Source: Appendix A · p. 8

Appendix B: Non-affine theory without free robustness

The appendix shows that non-affinity of the free-state set does not ensure finite free robustness for all states. For convex free sets, finiteness for every state is equivalent to the full state space lying in the span of the free states.

Appendix B: Non-affine theory without free robustness

  • A non-affine free-state set can still leave some states without finite free robustness.The qubit example generated by four real-valued free states excludes states with imaginary computational-basis coefficients.Source: Appendix B · p. 9
  • For general convex theories, every state has finite free robustness if and only if the full state space lies in the span of the free states.When the free set is the convex hull of pure free states, the appendix notes an earlier corresponding criterion using the span of that set.Source: Appendix B · p. 9
  • The example’s free states span only real computational-basis coefficients, so states with imaginary coefficients lack finite free robustness.The free set is the convex hull of |0⟩, |1⟩, |+⟩, and |−⟩ states.Source: Appendix B · p. 9

Appendix C: Exact resource destroying maps

An exact resource-destroying map sends each state to a closest free state under relative entropy, and the appendix considers when such maps have simple forms. It also notes limits to the framework and an important failure of exactness for quantum discord.

Appendix C: Exact resource destroying maps

  • An exact resource-destroying map sends every state to a closest free state measured by relative entropy.Here, relative entropy is D(ρ∥σ) := Tr{ρ log ρ − ρ log σ}.Source: Appendix C · p. 9
  • A simple exact resource-destroying map may be a quantum channel or be characterized by other natural principles; in thermodynamics it outputs the Gibbs state.The thermodynamic map is the constant channel for the given Hamiltonian and temperature.Source: Appendix C · p. 9
  • The paper’s discussion does not directly apply to non-Gaussianity because continuous-variable systems have infinite-dimensional Hilbert spaces, though it remains an example of a simple exact map.Source: Appendix C · p. 9
  • For quantum discord, the natural local-eigenbasis measurement is not exact, and the closest free states do not seem to have any generic feature.Source: Appendix C · p. 10

Appendix D: Free robustness implies nonlinearity of resource destroying maps

The appendix establishes that if free robustness exists for every resource state, resource-destroying maps must be nonlinear. It relates this result to the affine structure required for resource-destroying channels.

Appendix D: Free robustness implies nonlinearity of resource destroying maps

  • If free robustness exists for every resource state, resource-destroying maps must be nonlinear, so resource-destroying channels do not exist.Source: Appendix D · p. 10
  • Assuming a linear resource-destroying map, applying it to the free-robustness decomposition yields ρ = Λ(ρ) ∈ F, contradicting that ρ is a resource state.Source: Appendix D · p. 10
  • Finite free robustness requires a non-affine free-state set, while resource-destroying channels require an affine set [26]; lacking finite robustness is necessary but not sufficient for such channels.Source: Appendix D · p. 10

Appendix E: Collapse of the optimized modification coefficients (part of Theorem 1)

This section establishes one-shot resource-formation cost bounds for resource-non-generating and resource-destroying-map-commuting operations, examining when optimized modification coefficients collapse for common reference states, including golden-state references and quantum thermodynamics. It also surveys relevant conditions and bounds, gives a superposition-theory example of the constant-trace condition, and classifies resource theories by key properties.

Appendix E: Collapse of the optimized modification coefficients (part of Theorem 1)

  • Under Condition (CH), a pure state simultaneously maximizes mf, mmin, and mmax, and the three coefficients equal the golden coefficient gd.The proof first establishes that a minimum-free-fidelity state can be pure; the same state then maximizes the three measures over all states.Source: Appendix E · pp. 10–11
  • In quantum thermodynamics, the highest-energy eigenstate of the Hamiltonian is a pure minimum-free-fidelity state where mf, mmin, and mmax coincide and are maximal.Its mf value is log(1/τd,imax), where τd,imax is the Gibbs-state eigenvalue corresponding to the maximum energy.Source: Appendix E · pp. 11–12

Appendix G: Resource theory of superposition and the constant trace condition

  • For superposition on a two-dimensional system with two pure basis states, the golden state has constant overlap with every free state, satisfying Condition (CT).The proposed multi-qubit extension retains Condition (CT) when the reference state is the tensor power of that golden state.Source: Appendix G · p. 14

Appendix H: Classification of resource theories

  • The classification diagram groups resource theories by convexity, affinity, existence of a resource-destroying channel, and Conditions (FFR), (CH), and (CT), locating important examples.For asymmetry, whether Condition (CT) holds depends on the symmetry and subsystem structure considered.Source: Appendix H · p. 14, Figure 2 · p. 15

Appendix I: One-shot formation cost by resource non-generating operations (part of Theorem 2)

  • For FNG formation, the appendix gives a lower bound from modified smooth max-relative entropy and an upper bound under Condition (CT).With golden reference states and Conditions (CH) and (CT), it also gives two-way bounds using the collapsed coefficient gd.Source: Appendix I · pp. 14–16
  • When free robustness is finite for all states, the appendix also provides a formation-cost lower bound using modified smooth free log-robustness.The bound is stated for reference states whose modified free log-robustness reaches the target state's smoothed free log-robustness.Source: Appendix I · p. 16
  • Under convexity and finite free robustness, formation is bounded using the smallest reference dimension whose mf(Φ_d) log d reaches L_R^ϵ(ρ), with log d_0 achievable.The lower-bound argument rules out smaller dimensions, while the construction at d_0 achieves formation.Source: Appendix I · p. 16

Appendix J: One-shot formation cost by commuting operations (part of Theorem 2)

  • For commuting operations, the general lower bound uses D^ϵ_max,λ(ρ) and the coefficient m_max,λ; with an exact map, the coefficient collapses to m_max.The lower bound follows from commutation with λ and data processing; the exact-map simplification follows from the stated collapse result.Source: Appendix J · p. 17
  • A general upper-bound construction for commuting operations assumes constant overlap Tr{Φ_dσ} across free states, and uses idempotence of λ to ensure the constructed map commutes with λ.The construction achieves formation at the threshold dimension d_0 defined using −log f(Φ_d)=m_f(Φ_d) log d and D^ϵ_max,λ(ρ).Source: Appendix J · pp. 17–18
  • For golden states and an exact resource-destroying map, the coefficient collapses enable two-way formation-cost bounds expected to be tight when the threshold dimension is close to its minimum.The collapse identifies the relevant coefficients with g_d, allowing bounds to be expressed using g_d log d.Source: Appendix J · p. 18

Appendix K: Root states and blind resource formation (Corollary 4 and additional results)

The appendix identifies conditions under which a state that can generate every state in its dimension exists, then shows that such root states simplify one-shot resource formation. It also defines blind formation and establishes its equivalence to formation using any root-state family.

Appendix K: Root states and blind resource formation (Corollary 4 and additional results)

  • A state is a root state when free operations can map it to every state in the same dimension.The definition requires this reachability for all states in D(H_d).Source: Appendix K · p. 18
  • A common maximizer of m_f, m_min, and m_max is a root state if Condition (CT) holds, or if (FFR) holds and m_max equals m_LR on pure states.Condition (CT) requires constant overlap with free states; (FFR) requires finite free robustness for all states.Source: Appendix K · p. 18
  • Root states occur in bipartite entanglement, k-entanglement, k-coherence, purity, and quantum thermodynamics; their absence implies the robustness measures need not coincide on pure states.Fully separable multipartite entanglement is given as an example without a root state.Source: Appendix K · p. 19
  • Using root states as currency ensures the optimal reference-state dimension is no larger than the target dimension in one-shot formation and distillation.The appendix describes these effects as resource dilution and concentration, respectively.Source: Appendix K · p. 19
  • Blind formation minimizes the resource-state dimension needed to produce an ε-approximation without specifying reference states, and its cost is achieved using any root-state family.The cost is defined as the minimum log dimension over input states and free operations; Proposition 22 equates it with formation using root states.Source: Appendix K · p. 19

Appendix L: One-shot distillation yield by resource non-generating operations (Theorem 5 and additional results)

The appendix bounds one-shot distillation under resource-non-generating operations using hypothesis-testing relative entropy. It gives constructions based on free fidelity and depolarizing maps, alongside an upper-bound argument for any distillable reference state.

Appendix L: One-shot distillation yield by resource non-generating operations (Theorem 5 and additional results)

  • A free-fidelity construction sets d_0 by −log f(Φ_d) = m_f(Φ_d) log d ≤ D_H^ε(ρ), and achieves distillation rate log d_0.The constructed operation is resource-non-generating and its output has fidelity at least 1−ε with Φ_d0.Source: Appendix L · pp. 19–20
  • For a reference state Φ_d, any distillable dimension d is bounded by log d_0, where d_0 is set by m_f(Φ_d0) log d_0 ≤ D_H^ε(ρ).The upper-bound argument uses data processing for hypothesis-testing relative entropy and the target-state error condition.Source: Appendix L · pp. 19–20
  • Under Condition (FFR), a second lower-bound approach chooses d_0 using the log-robustness m_LR(φ_d) log d ≤ D_H^ε(ρ).This construction starts from a free state mixed with the reference state according to its free robustness.Source: Appendix L · p. 20
  • A depolarizing-map construction requires a convex free-state set containing the maximally mixed state and a nonempty eligible dimension set D′; it achieves rate log d_0.The constructed map preserves free states by expressing their outputs as convex combinations of free depolarized reference states.Source: Appendix L · pp. 20–21

Appendix M: One-shot distillation yield by commuting operations (Theorem 6 and additional results)

Appendix M bounds one-shot distillation under operations commuting with resource-destroying maps, then extends the constructions to pseudo-resource-destroying maps. The bounds and constructions use conditions on the reference states and operations.

Resource-destroying channels

  • For a resource-destroying channel Λ, Proposition 26 defines d0 through fΛ(Φd)=d^-m_f,Λ(Φd) ≥ 2^-D_H,Λ^ε(ρ)−2√ε, yielding a distillation-yield upper bound log d0.The proof rules out any d>d0 using the hypothesis-testing relative entropy bound.Source: Appendix M · pp. 21–22

Pseudo-resource-destroying maps

  • Pseudo-resource-destroying maps send every state to a free state, and their commuting operations preserve that commutation relation.When the map’s image is a proper subset of the free states, commuting operations need not be free; intersecting them with resource-non-generating operations addresses this issue.Source: Appendix M · p. 22
  • Setting the induced distance to zero on free states restores a faithful monotone, which is non-increasing under pseudo-commuting resource-non-generating operations.The simple distance δ(ρ, λ(ρ)) may be positive on a free state when λ does not fix every free state.Source: Appendix M · p. 22
  • For depolarizing pseudo-resource-destroying maps, Proposition 28 gives a lower-bound construction when the optimal operator P has unit trace and the eligible reference-state set is nonempty.When Tr P=1, the constructed map commutes with the depolarizing map and is resource-non-generating, yielding the same bound as Proposition 25.Source: Appendix M · pp. 22–23

Appendix N: One-shot distillation yield with error tolerance on the input

Appendix N studies distillation when the input may be replaced by an ε-close state, defining the corresponding yield and deriving bounds. It relates this input-error task to standard output-error distillation and gives constructive lower bounds under stated conditions.

Input-error task

  • The input-error distillation yield is the largest reference-state size obtainable from an ε-approximation of the primitive state by an allowed operation.The definition maximizes over both the approximating input state and the operation.Source: Appendix N · p. 23

Relation to output-error distillation

  • Input-error distillation has a stronger error constraint than standard output-error distillation, so its yield is upper-bounded by the standard yield.A protocol that distills from an ε-close input guarantees output within ε of the target when applied to the original input.Source: Appendix N · p. 23

Upper bounds

  • Bounds derived for standard distillation also upper-bound input-error yield; the appendix additionally establishes general bounds using state- and operator-smoothing.The state-smoothed min-relative entropy remains zero for sufficiently small ε when its unsmoothed value is zero.Source: Appendix N · p. 24
  • For resource-non-generating operations, the appendix derives upper bounds using the smoothed min-relative entropy; related bounds are given for commuting operations.For exact resource-destroying maps and golden reference states, the min-relative-entropy quantities collapse under the stated Condition (CH).Source: Appendix N · pp. 24–25

Lower bounds

  • For pure inputs with finite free robustness, Proposition 32 constructs a resource-non-generating distillation map achieving rate log d0 under its stated robustness threshold.The construction chooses an ε-close pure input and maps it to the reference state indexed by d0.Source: Appendix N · p. 25
  • For pure inputs, an isotropic-state construction gives lower bounds when the free states are convex, contain the maximally mixed state, and the eligible reference-state set is nonempty.The construction maps the chosen ε-close pure input to Φd0; the same bound extends to commuting operations for the stated depolarizing map.Source: Appendix N · pp. 25–26

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