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General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

Ryuji Takagi, Bartosz Regula

arXiv:1901.08127v4quant-phmath-ph

TL;DR

The paper addresses the lack of a unified operational characterization of resource quantification and manipulation across states, measurements, and channels, including beyond quantum mechanics. It develops discrimination-based robustness characterizations in convex resource theories and GPTs, showing exact links between resource measures, task advantages, and free transformations. The results establish a common operational description for these resources across the stated physical theories.

  • Problem

    A unified account of how resource measures and free transformations relate to operational tasks has remained incomplete across states, measurements, channels, and theories beyond quantum mechanics.

  • Method

    The paper analyzes state and channel discrimination in general convex resource theories, defining robustness-based quantifiers for states, measurements, and channels in GPTs.

  • Results

    Robustness measures exactly characterize maximum discrimination advantages, while classes of discrimination tasks fully characterize selected state and measurement transformations under free operations.

  • Takeaways & Limitations

    Discrimination tasks provide a unified operational description of resource quantification and manipulation for states, measurements, and channels across quantum theory and GPTs.

Abstract

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We establish an operational characterization of general convex resource theories -- describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -- in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory. Specifically, we introduce quantifiers of resourcefulness of states, measurements, and channels in any GPT based on the generalized robustness, and show that they exactly characterize the maximum advantage that a given resource provides over all free states, measurements, or channels in a class of state or channel discrimination tasks. In quantum mechanics, we show that the robustness of measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We furthermore endow the standard robustness of a state with an operational meaning as the quantifier of the maximum advantage in binary channel discrimination tasks. Finally, we show that several classes of channel and state discrimination tasks can form complete families of monotones fully characterizing the transformations of states and measurements under any chosen class of free operations. Our results establish a fundamental connection between operational tasks of discrimination and core concepts of resource theories, valid for all physical theories with no additional assumptions about the structure of the GPT required.

I. INTRODUCTION

The paper seeks a unified operational account of resource quantification and manipulation across states, measurements, and channels in quantum theory and GPTs. It uses state and channel discrimination to connect robustness measures with operational advantages and characterize free transformations.

  • Motivation: Resource theories aim to quantify and manipulate states, measurements, and channels under physically allowed transformations.The paper extends resource-theoretic analysis beyond states toward a unified treatment of dynamic and measurement resources.
  • Motivation: State and channel discrimination provide a general class of operational tasks for assessing resourcefulness across physical settings.These tasks are relevant in quantum and GPT frameworks and can avoid resource-specific operational approaches.
  • Manipulation: The framework uses discrimination tasks not only for quantification but also to fully characterize conversions between states or measurements under free operations.The paper presents this as an operational characterization of resource manipulation in general GPTs.
  • States: Generalized robustness of states in convex resource theories and GPTs exactly quantifies the advantage of resourceful states in channel discrimination.The result extends an earlier quantum-state connection beyond quantum mechanics.
  • Measurements: Measurement robustness exactly quantifies the maximum advantage of a measurement over free measurements in state discrimination tasks.In quantum theory, it also quantifies the increase in min-accessible information produced by resourceful measurements.
  • Channels: Channel resources are treated through resource generating power and generalized channel robustness, with discrimination tasks providing operational interpretations for both.Resource generating power concerns resource creation from free states, while channel robustness belongs to convex resource theories of channels.
  • States: Standard robustness of states quantifies the advantage over free states in balanced binary channel discrimination tasks.This gives standard robustness an operational meaning in general convex resource theories.

C. Resources and their quantification

The section defines resource theories through objects and free transformations, emphasizing convexity and robustness as a quantifier of resource content. It then connects generalized robustness of states to maximal advantage in channel discrimination tasks across GPTs.

  • Resource theories consist of states, measurements, or channels together with transformations designated free because they incur no resource cost.
  • Convexity permits randomization of free objects and is treated as a natural foundation for resource theories in GPTs.Mixing free objects remains free under the convexity assumption.
  • Robustness measures quantify the least mixing needed to turn an object into a free one, distinguishing standard robustness from generalized robustness by the allowed noise.Standard robustness restricts the added object to be free, whereas generalized robustness allows any admissible object.
  • For any GPT, generalized robustness of a state equals its maximal advantage over free states in channel discrimination with a fixed measurement.The operational ratio is psucc({pi, Λi}, {Mi}, ω) / maxσ∈F psucc({pi, Λi}, {Mi}, σ) = 1 + RF(ω).
  • This state-discrimination characterization requires only that the allowed channel set contain the identity map, so it applies regardless of other operational restrictions.
  • The same relation extends from channel discrimination to the broader class of subchannel discrimination tasks.
  • Generalized robustness is directly observable through a single effect and can therefore be bounded from experimental measurement data.

IV. GENERALIZED ROBUSTNESS OF MEASUREMENTS

This section defines generalized robustness for measurements in convex GPT resource theories and establishes its exact operational meaning in state discrimination. The result also connects measurement robustness to data hiding ratios for arbitrary-length ensembles.

  • Measurement resource theories specify a convex, closed cone of free effects, encompassing settings such as local, separable, simulable, trivial, and PPT measurements.
  • Generalized measurement robustness quantifies the resource content of a measurement relative to the chosen free-measurement set.
  • The measurement robustness is faithful, convex, and monotone under effect-cone-preserving maps that preserve the free-effect cone.
  • For any GPT, measurement robustness exactly quantifies the maximum advantage over free measurements in state discrimination across finite state ensembles.
  • Maximizing measurement robustness yields a data hiding ratio generalized from binary ensembles to state ensembles of arbitrary length.

A. Connections with single-shot information theory

The section links measurement robustness to one-shot information theory in quantum mechanics. Specifically, the resource-induced increase in min-accessible information over free measurements is exactly characterized by generalized robustness.

  • The connection between measurement robustness and one-shot accessible information is established specifically within quantum mechanics, not general GPTs.
  • Min-accessible information is used as a single-shot alternative to asymptotic accessible information for state ensembles.It is defined through min-entropy and conditional min-entropy rather than von Neumann or Shannon entropy.
  • A measurement is represented as a measure-and-prepare channel that maps input states to classical output registers.
  • The maximal increase in min-accessible information caused by a resourceful measurement over free measurements equals the generalized robustness of that measurement.
  • Together with the state-discrimination result, this theorem connects measurement robustness, discrimination tasks, and single-shot information theory.

V. ROBUSTNESS MEASURES FOR CHANNELS

The paper gives two operational approaches to channel resources: resource-generating power relative to free states, and robustness relative to a chosen set of free channels. Robustness generating power exactly equals the maximum discrimination advantage achievable by applying a channel to free-state ensembles.

  • A. Robustness generating power of channels: Channel resource can be quantified by how much resource it creates from free states, called the resource generating power.Alternatively, channel resource can be defined directly through robustness relative to an arbitrary set of free transformations.
  • A. Robustness generating power of channels: Theorem 4 identifies robustness generating power with the optimal advantage in state discrimination tasks on free-state ensembles after applying the channel.The comparison is against the identity map acting on the same ensembles and measurements.
  • A. Robustness generating power of channels: The optimal discrimination ratio equals 1 + P_F(Λ), where P_F(Λ) is the robustness generating power.The result holds for any resource theory and any chosen class of free operations.
  • A. Robustness generating power of channels: The proof uses the robustness decomposition of each transformed free state and the dual formulation of robustness to bound discrimination performance.An optimal decomposition supplies free output states, while the dual witness establishes the converse bound.
  • A. Robustness generating power of channels: A separate trace-norm resource-generating-power interpretation for binary channel discrimination extends from quantum mechanics to GPTs because the proof uses only base-norm properties.This is a related operational characterization rather than the robustness identity established in Theorem 4.

B. Generalized robustness of channels

The paper defines generalized robustness directly for channels relative to a convex closed set of free channels and operationalizes it through channel discrimination. In quantum theory, it equals the maximal advantage of a resourceful channel over all free channels, while channel ensembles admit a corresponding discrimination characterization.

  • B. Generalized robustness of channels: Generalized channel robustness directly measures a channel’s intrinsic resourcefulness relative to a chosen set of free channels, without requiring a state resource theory.The free-channel set is assumed convex and closed.
  • B. Generalized robustness of channels: The operational characterization relies on the Choi–Jamiołkowski representation and is restricted here to quantum theory because general GPTs need not satisfy the required properties.The Choi representation encodes complete positivity and trace preservation through conditions on the Choi matrix.
  • B. Generalized robustness of channels: For a single channel, generalized robustness characterizes the advantage over free channels in state discrimination with fixed input states and measurements.The channel acts on one subsystem before the discrimination measurement.
  • B. Generalized robustness of channels: Theorem 5 states that the maximum state-discrimination advantage of Λ over free channels equals 1 + R_O_F(Λ).The maximization ranges over all bipartite state ensembles and measurements.
  • B. Generalized robustness of channels: For channel ensembles, the framework connects maximum discrimination advantage with the maximum robustness among the ensemble’s channels.The task samples a channel from a prior distribution, applies it to part of a bipartite state, and performs a collective measurement, allowing inconclusive outcomes.
  • B. Generalized robustness of channels: The two channel-resource approaches coincide for quantum coherence under maximally incoherent operations, but broader conditions for equality remain open.The established equality is P_F(Λ) = R_O_F(Λ) in that specific resource theory and operation class.

VI. STANDARD ROBUSTNESS OF STATES

This section gives the standard robustness of states a universal operational meaning through balanced binary channel discrimination in convex resource theories. It also discusses the assumptions, measurement restrictions, divergent cases, and connections to other robustness applications.

  • The standard robustness quantifies the maximum advantage a resource state provides over free states in balanced binary channel discrimination.This operational interpretation is established for convex resource theories of states and extends the role of robustness measures in discrimination tasks.
  • Even when standard robustness diverges, the operational result implies that some channel pair cannot be performed better than random guessing using any free state.This covers resource theories such as coherence and asymmetry without requiring finite standard robustness.
  • Balanced binary channel discrimination compares two equiprobable channels by the increase in success probability achieved through measurement-based inference.The output space must contain at least two distinct states, because otherwise the discrimination task is trivial.
  • The result remains valid when measurements are restricted to any informationally complete closed free measurement set.The proof replaces the general norm with the corresponding distinguishability norm.
  • The quantity 1 + 2R_F^F(ω) corresponds to the base norm induced by the free-state set and connects to classical-simulation overhead in magic theories.The paper identifies the deeper relationship between channel discrimination and classical simulation as an open question.

VII. COMPLETE SETS OF MONOTONES

The paper frames complete sets of monotones as necessary-and-sufficient tests for free transformations. It shows that discrimination-task performance supplies such a complete operational characterization in general GPT resource theories.

  • A single monotone usually gives only necessary conditions for free transformations and does not completely characterize convertibility.If one object has more resource according to a monotone, a free operation cannot transform the less resourceful object into it.
  • A complete set of monotones is a possibly infinite family that fully characterizes the necessary and sufficient conditions for a free transformation.Previously known complete families were tied to specific settings rather than general resource theories.
  • State- and measurement-performance in channel or state discrimination tasks forms complete monotone families for general resource theories in any GPT.Together with the robustness results, this provides an operational characterization of resource quantification and exact transformations.

A. Complete sets of monotones for states

For states, discrimination performance under a chosen class of operations characterizes exactly which transformations are possible. The paper derives equivalent formulations using general, binary, and effect-based discrimination tests.

  • A convex, closed operation set containing the identity and closed under concatenation defines the transformation framework for state convertibility.These assumptions are natural for free operations but the results are stated more generally.
  • Optimized channel discrimination performance provides a complete set of monotones for state transformations under the chosen operations.This holds for tasks allowing inconclusive outcomes and for variants with prior operations applied to channel ensembles.
  • Binary channel discrimination already suffices to characterize state transformations when the probability distribution is freely chosen.This substantially simplifies the operational test compared with arbitrary channel ensembles.
  • There exists an allowed transformation from ω to ω′ if and only if the corresponding three-outcome discrimination inequalities hold for every measurement.This criterion greatly reduces the difficulty of deciding whether a free state transformation exists.
  • Equivalent characterizations use all positive effects or success-probability comparisons for two-element distributions and two-outcome measurements.These formulations include noise-detection interpretations, where one state can dominate another across detection strategies.
  • The same framework can be extended from channel discrimination to subchannel discrimination using normalization non-increasing maps.The corresponding proofs proceed analogously.

B. Complete set of monotones for measurements

The measurement-side dual perspective treats channels as transformations of measurement effects. Under suitable free effect operations, discrimination success probabilities completely characterize measurement transformations.

  • Channels can be viewed dually as operations transforming measurements rather than states, placing states, channels, and measurements in one information-processing framework.This motivates analyzing measurement transformations through state-discrimination tasks.
  • A convex, closed set of effect-cone-preserving unital maps containing the identity and closed under concatenation defines the allowed measurement operations.These conditions mirror the operation-set assumptions used for state transformations.
  • Success probability in state discrimination with prior free operations on measurement effects forms a complete set of monotones for measurements.The task is equivalently expressible as state discrimination with dual transformations applied to the states.
  • There exists a free operation mapping M to M′ if and only if M performs at least as well as M′ on every state ensemble.This gives a necessary-and-sufficient operational criterion for measurement convertibility.
  • The dual framework also supplies complete monotones for transformations between state ensembles under the chosen operations.The paper presents equivalent conditions involving measurements, probability distributions, and ensemble transformations.

VIII. CONCLUSIONS

The paper gives a general operational characterization of resource quantification and manipulation through state and channel discrimination, covering states, measurements, and channels in convex resource theories and GPTs. It also connects these formulations to data hiding, experimentally accessible bounds, and future extensions.

  • General scope: Discrimination tasks operationally characterize both the quantification and manipulation of resources across general convex resource theories.The framework covers multiple resource objects and remains applicable beyond quantum mechanics.
  • Applications: The formulations apply to states, measurements, and channels in quantum resource theories including coherence, entanglement, magic, athermality, and asymmetry.The paper also states that multipartite and multi-level generalizations fit the framework.
  • Data hiding: The framework encompasses generalized data-hiding scenarios involving restricted measurements, resource states, channels, and more intricate discrimination tasks.Its flexibility includes varied encoding strategies and physical settings.
  • GPT implications: Generalized robustness and discrimination advantage coincide in the considered tasks for GPTs, so these quantities do not separate a theory from quantum mechanics.The result follows from convexity of the underlying cones and requires no additional GPT structure.
  • Operational consequences: Discrimination tasks provide experimentally accessible bounds on geometric resource measures and characterize resource transformations in GPTs.The conclusion links operational tasks to both resource quantification and transformation criteria.
  • Open problems: A remaining problem is to give standard robustness an operational meaning for measurements and channels and determine whether broader robustness families admit similar characterizations.The paper also identifies channel transformations under superchannels as an avenue for future work.

NOTE ADDED

The authors note independent related results concerning the connection between state discrimination and measurement-resource quantification in quantum mechanics.

  • Related work: Independent works by R. Uola et al. and M. Oszmaniec and T. Biswas obtained results similar to Theorem 2.The related results concern state discrimination tasks and measurement-resource quantification within quantum mechanics.

Appendix A: Proofs of results in Sec. VII

The appendix proves that discrimination-task inequalities characterize allowed transformations of states and collections of states under free operations. The proofs use separating effects, measurement constructions, and convexity-based minimax reasoning.

  • State transformations: Theorem 12 characterizes when a free operation Λ exists such that ω′ = Λ(ω) through alternative discrimination-task conditions.The theorem states an if-and-only-if transformation criterion.
  • State transformations: Transformation existence is tested using state-discrimination success probabilities over measurements and probability distributions.The proof considers N-outcome measurements and distributions, including separate cases for N ≥2 and N ≥1.
  • Channel discrimination: For channel-based conditions, the proof compares success probabilities over channel ensembles and measurements, with one case fixing an identity channel.The equivalence is established for all channel ensembles and measurement choices.
  • Proof strategy: The minimax step relies on convexity and compactness of channels, measurements, and free operations, together with linearity of the objective.These conditions justify exchanging optimization orders in the proof.
  • Proof strategy: The reverse implication constructs a separating effect E whenever the target state is unreachable, then embeds it into a binary measurement.The construction uses N = 2 and the measurement {E, U − E}.
  • Collections of states: The appendix extends the transformation statement to collections of states, requiring corresponding discrimination inequalities for all indexed states.The criterion is expressed as σ′_i = Λ(σ_i) for every i.

Appendix B: Duality in conic optimization

The appendix formulates the paper’s robustness optimizations as conic programs and derives their dual forms using Lagrange multipliers, cone duality, and strong duality conditions.

  • Conic formulation: The considered optimization problems fit a general conic form with a linear objective, linear map, affine constraint, and closed convex cone.This template encompasses the resource-measure optimizations used throughout the paper.
  • Dual construction: The primal problem is paired with a dual problem obtained by introducing Lagrange multipliers and interchanging minimization with maximization.Separating hyperplanes and the dual cone enforce feasibility constraints.
  • Duality conditions: Weak duality gives p ≥ d, while Slater’s condition yields equality between the primal and dual optima.Strong duality requires a strictly feasible point in the relative interior of the cone.
  • Robustness measures: The generalized robustness of states is converted into a dual optimization by writing its Lagrangian and optimizing over the associated multipliers.The appendix identifies this dual as the form used in the main text.
  • Robustness measures: The standard robustness of states changes the primal constraint from membership in the cone C to membership in cone(F).This substitution produces the corresponding standard-robustness dual formulation.
  • Robustness measures: Analogous Lagrangian derivations produce dual programs for measurement and channel robustness.The channel derivation uses the Choi representation, trace constraints, and multipliers X, Y, and Z.
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