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Convex geometry of quantum resource quantification

Bartosz Regula

arXiv:1707.06298v6quant-phmath-phmath.MP

TL;DR

Quantifying quantum resources is important for distinguishing useful states, but defining and characterizing measures has often required resource-specific work, with few general results linking them. This paper develops a convex-geometric gauge framework that brings common measures together and establishes criteria and relations among them. It reports that many gauge-based quantifiers have desirable properties and that quantification often simplifies for pure states.

  • Problem

    Defining and verifying quantum-resource measures is usually resource-dependent, and few general results connect the measures and their properties.

    Source: §1 · p. 2

  • Method

    The paper formulates resource quantifiers as gauge functions grounded in the convex geometry of free-state sets, encompassing norm-, robustness-, convex-roof-, and witness-based measures.

    Source: §1.1 · p. 2

  • Results

    Gauge-based measures satisfy properties including faithfulness, convexity, and strong monotonicity under relevant operations, and several quantifiers simplify to ΓV for pure states.

    Source: §1.1 · pp. 3–4

  • Takeaways & Limitations

    The framework supplies general criteria for checking quantifier properties and enables simpler pure-state forms, often easier to compute.

    Source: §1.1 · pp. 3–4

    The best free approximation assigns every resourceful pure state a value of at least 1, so it does not distinguish pure states with different resourcefulness.

    Source: §4.2 · p. 25

Abstract

from arXiv · show

We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures and show that several of them are equal on pure states. Further, we investigate the dual formulation of resource quantifiers, characterising sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the resources and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof-extended negativity, a measure of k-coherence which generalises the L1 norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.

1. Introduction

Quantifying quantum resources matters for identifying useful states, yet constructing and checking measures has generally been resource-dependent. The paper uses convex geometry and gauge functions to unify common quantifiers, derive general criteria and relations, and simplify pure-state quantification.

1. Introduction

  • The paper addresses the resource-dependent difficulty of defining and verifying faithful, monotone measures and the lack of a common framework relating them.Quantification helps discriminate which states are most useful for a physical task.Source: §1 · p. 2

1.1. Summary of the results

  • Gauge functions provide a common convex-geometric framework for matrix-norm, robustness, convex-roof, and witness-based resource measures.The framework applies to finite-dimensional resource theories with convex free-state sets.Source: §1.1 · pp. 2–3
  • For any considered quantum resource, gauge-based measures satisfy properties including faithfulness, convexity, and strong monotonicity under relevant operations, with verifiable criteria for other gauges.The criteria also apply to some witness-based measures.Source: §1.1 · p. 3
  • The framework establishes general relations and bounds among measures, including dual-gauge bounds and connections to distance- and witness-based quantifiers.These results generalize relations previously known for entanglement and coherence.Source: §1.1 · p. 3
  • For pure states, several quantifiers simplify to the gauge ΓV, which is often easier to compute and can yield analytical forms.The paper also reports that several measures coincide on pure states.Source: §1.1 · pp. 3–4
  • Applications to entanglement, coherence, and magic states yield new bounds, analytical formulas, and quantifiers, including measures extending the convex roof–extended negativity and greatest cross norm.Examples include measures of Schmidt-number-k entanglement, k-coherence, and k-partite entanglement.Source: §1.1 · p. 4

2. Gauges and norms

Atomic gauges associate a convex set with a function that can quantify its elements, while norms are a special case. The section also defines convex-roof extensions, which extend values on extreme points to a convex set.

2.1. Atomic gauges

  • Norms are gauges of convex, compact, centrally symmetric sets containing the origin in their interior; general gauges need not be symmetric or finite everywhere.For an atomic gauge to be a norm, its atoms must span the whole space and be centrally symmetric.Source: §2.1 · pp. 5–6
  • An atomic gauge is the gauge of the convex hull of a compact atom set and can be expressed through optimization over conic combinations of atoms.A gauge's unit ball identifies its set when the set is closed and convex and contains the origin.Source: §2.1 · pp. 5–6

2.1.1. Complex vector spaces

  • In complex vector spaces, balanced sets generalize central symmetry, and convex, balanced, compact sets spanning the space define norms.The polar is defined using the real inner product in the complex setting.Source: §2.1.1 · p. 7

2.2. Convex roof

  • A convex roof extends a function on a set's extreme points by minimizing its average over extremal convex decompositions.When it exists, the convex roof is the largest convex extension and the smallest roof extension; the convex envelope instead optimizes over decompositions using points from the whole set.Source: §2.2 · pp. 7–8

3. Atomic gauges for sets of quantum states

This section develops resource quantifiers from atomic gauges over free-state sets, comparing gauge extensions and relating them to robustness measures. It examines pure-state simplifications and conditional equalities in a hierarchy of quantifiers, and uses the dual formulation of generalized robustness to discuss a tighter bound on D-gauge robustness.

3. Atomic gauges for sets of quantum states

  • The resource-free pure-state set V is assumed non-empty, compact, and balanced; ΓV is a norm when 0 lies in the interior of conv(V).Compactness ensures finite infima and suprema are attained; the norm condition is desirable but not assumed universally.Source: §3 · p. 10

3.1. Selection of gauges

  • For the set of all pure states, symmetrizing the density-matrix set yields the trace norm on Hermitian matrices, while a complex operator-set extension gives its full-domain form.Different gauge extensions can coincide in this example but need not do so for general resources.Source: §3.1 · p. 11
  • The convex roof of ΓV squared extends the pure-state gauge to density matrices and is finite for every density matrix when span(V)=Cd.The construction uses an optimization over normalized pure states and can be expressed as a gauge over rank-one decompositions.Source: §3.1 · pp. 12–13

3.2. Base norms and robustness

  • Standard robustness is affine in the base gauge ΓS+∪(−S+), while generalized robustness is affine in ΓS+∪(−D).Robustness with respect to X measures the minimum mixing with a state in conv(X) needed to bring the renormalized mixture into conv(S+).Source: §3.2 · pp. 15–16
  • Generalized robustness reduces on pure states to the vector atomic gauge ΓV, so its evaluation requires optimization over the underlying Hilbert space.This provides computable formulas for pure-state robustness measures.Source: §3.2 · pp. 17–18
  • The pure-state equivalence between standard robustness and the vector atomic gauge can fail for some resources, and characterizing exactly when it holds remains open.The passage cites magic states as an explicit counterexample and notes the known entanglement result.Source: §3.2 · p. 18
  • When standard robustness equals the vector atomic gauge on pure states, the extended hierarchy of quantifiers is also equal on pure states.The passage presents this as a conditional result; it does not hold for every quantum resource, with magic states given as an example.Source: §3.2 · p. 18
  • A tighter bound on D-gauge robustness is obtained using the dual formulation of generalized robustness.The passages first note a bound from Proposition 12, then state that a tighter bound follows from the generalized-robustness dual formulation.Source: §3.2 · p. 19

4. Atomic gauges as resource quantifiers

This section examines atomic-gauge resource quantifiers, characterizing their faithfulness, convexity, and monotonicity and relating their dual witness formulations to distance-based measures. It also presents connections and bounds involving modified trace distance, fidelity-based distances, relative entropy, and resource monotones.

4.1. Properties of atomic gauges

  • All considered atomic-gauge measures are faithful and convex.Faithfulness means that a state is free exactly when the measures take their free-state value; convexity follows from convexity of the underlying gauge functions.Source: §4.1 · pp. 20–21
  • Robustness measures are strong monotones under resource-non-generating operations, whereas Γ_S and Γ_union(S+) are monotones for operation sets contained in stochastically resource-non-generating operations.The latter gauges are not, in general, monotonic under all resource-non-generating operations.Source: §4.1 · p. 23

4.2. Dual characterisation and resource witnesses

  • Resource witnesses are dual-cone elements related to the polar set of free states, and separating hyperplanes guarantee a detecting witness for every resourceful state.Witness-based measures quantify violation of witness constraints and can support resource detection and quantification without full state tomography.Source: §4.2 · p. 24
  • The best free approximation is a faithful strong monotone under resource-non-generating operations, but assigns every resourceful pure state value 1 and therefore misses differences in their resourcefulness.It generalises the best separable approximation.Source: §4.2 · p. 25

4.3. Relations with distance-based measures

  • Distance-to-free-state quantifiers built from contractive quasi-metrics are weak monotones under resource-non-generating operations, but are often difficult to evaluate and have few known strong-monotonicity results.The section develops relations between these distance-based and gauge-based measures despite distance quantifiers not fitting directly into the gauge formalism.Source: §4.3 · p. 26

4.3.1. Trace distance and other gauge-based distances

  • Gauge-based distances are not themselves gauges; replacing the free-state set with the cone it generates gives a gauge-function construction.The trace distance is treated as an example of a gauge-based distance.Source: §4.3.1 · pp. 26–27
  • The modified trace distance is a gauge, while trace distance can fail strong monotonicity under relevant stochastically resource-non-generating operations.This makes distances defined with respect to the unnormalised cone S+ important when strong monotonicity is required.Source: §4.3.1 · p. 27

4.3.2. Geometric measures

  • The geometric measure, based on the largest overlap with a free state, equals the fidelity-based distance measure and reduces its convex-roof quantification to a potentially simpler convex optimisation.The overlap reaches its maximum value of 1 only on free states.Source: §4.3.2 · p. 28

4.3.3. Relative entropy

  • The relative-entropy results establish quantitative relations with atomic gauges and non-convex resource monotones, and the Dmax-based measure is strongly monotonic under resource-non-generating operations.Whether the derived relation can be tightened remains open.Source: §4.3.3 · p. 29

5. Applications

The applications demonstrate the framework across coherence, entanglement, and magic states, showing that known measures and new gauge-based quantifiers can be treated within one formalism. The authors also connect the measures’ properties and relations to the general results established earlier.

5. Applications

  • The formalism is applied to entanglement, coherence, and magic states, introducing gauge-based quantifiers and placing many known measures in a common framework.The authors state that faithfulness, strong monotonicity under relevant operations, and quantitative relations follow from earlier sections.Source: §5 · pp. 29–30
  • The coherence measures introduced here also form a basis for the paper’s investigation of bipartite entanglement.Source: §5 · p. 30

5.1. k-coherence

The section develops k-coherence quantifiers, connecting familiar coherence measures to atomic gauges and extending them to intermediate coherence levels. It introduces k-support and matrix-norm constructions, discusses their computational properties, and identifies measures that are strongly monotonic under k-incoherent operations.

5.1. k-coherence

  • When not every coherent state is useful for a task, k-coherence offers finer-grained quantification, with coherence rank counting nonzero coefficients in the fixed basis.Source: §5.1 · p. 30

5.1.1

  • For k=1 and k=d, the framework recovers familiar gauges and norms for Hermitian matrices, including vector ℓ2, entrywise ℓ1, and trace norms.The listed measures also include the numerical radius and robustness measures of coherence.Source: §5.1.1 · p. 31
  • The measure RC1+ is not a valid quantifier of 2-coherence because it is infinite for states outside the span of C1+.Source: §5.1.1 · p. 31

5.1.2

  • For general k, the section considers the geometric measure and generalized robustness of k-coherence alongside the other quantifiers.Source: §5.1.2 · p. 31
  • The k-support norm is exactly computable and interpolates between the ℓ1 and ℓ2 norms, with its dual norm providing an alternative route to its formula.The section also notes that the dual norm is easier to compute explicitly.Source: §5.1.2 · p. 32
  • The (k,k)-trace norm interpolates between entrywise ℓ1 and trace norms, generalizing the ℓ1 coherence measure, but exact computation is NP-hard in general.A numerical approximation algorithm is noted.Source: §5.1.2 · p. 33
  • The convex roof-extended k-support norm and the (k,k)-trace norm are strongly monotonic under k-incoherent operations.Source: §5.1.2 · p. 33

5.2. Bipartite entanglement

The section applies the convex-gauge framework to bipartite entanglement, including hierarchies based on Schmidt rank and a related resource theory based on positive partial transpose. It identifies familiar measures within these hierarchies and establishes relations and monotonicity properties.

5.2. Bipartite entanglement

  • For Schmidt-rank-k entanglement, the measures are strong monotones under LOCC and also under operations that do not generate Schmidt rank k + 1 entanglement.The relevant resource-nongenerating and stochastic resource-nongenerating operations include LOCC.Source: §5.2 · p. 34

5.2.1. k = 1

  • For k = 1, the vector gauge ΓV1 is the ℓ1 norm of the Schmidt vector, a convex relaxation of pure-state Schmidt rank.The atomic gauge ΓS1 is the greatest cross norm, while Γ∪S1 is twice the convex roof–extended negativity and is faithful and strongly monotonic under separable operations.Source: §5.2.1 · p. 35
  • For k = 1, convex gauges and robustness measures give convex lower bounds on Schmidt number, with negativity itself lower-bounding the gauges.The framework gives a geometric relaxation of Schmidt rank through ΓV1(|ψ⟩) = 2N(|ψ⟩⟨ψ|) + 1.Source: §5.2.1 · p. 35
  • For general k, ΓVk is the k-support norm of the Schmidt vector, extending the convex roof–extended negativity to Schmidt-rank-k entanglement.The robustness reduces on pure states to the k-support norm, and the polar gauge Γ◦Sk can be computed exactly in small dimensions or bounded by semidefinite programs.Source: §5.2.2 · p. 36

5.2.3. Remarks about the resource theory of negative partial transpose

  • In the negative-partial-transpose resource theory, separable operations are contained in PPT operations, which are contained in operations that do not generate NPT resource.PPT and separable states coincide when dA dB ≤ 6; the gauge formalism also applies to alternative PPT free-state sets.Source: §5.2.3 · p. 37

5.3. k-partite entanglement

The section extends the gauge framework to hierarchies of multipartite entanglement, including k-producible states and genuine multipartite entanglement. It relates these gauges to known norms and negativity measures while noting computational difficulty for general multipartite states.

5.3. k-partite entanglement

  • The framework defines k-producible mixed states as convex hulls of states composed of at most k-party states, and its measures are faithful and strongly monotonic under free operations.The considered state sets span the space, giving the measures full domain.Source: §5.3 · pp. 37, 39
  • The ΓM1 norm generalizes the greatest cross norm, and ΓP1 and ΓM1 are projective tensor norms defined on different spaces.The measures are generally difficult to compute without tools such as the bipartite Schmidt decomposition.Source: §5.3 · p. 38
  • For pure states, genuine n-partite entanglement can be quantified by minimizing the bipartite atomic gauge across all 2^n − 1 bipartitions.The resulting Γ²Pn−1 − 1 equals twice the renormalised genuine multipartite negativity; its convex roof extends this to mixed states.Source: §5.3 · pp. 38–39
  • The renormalised genuine multipartite negativity is computable by semidefinite programming and provides lower bounds, while generalized robustness equals twice this negativity on pure states.This yields an efficient quantification for pure states.Source: §5.3 · p. 39

5.4. Magic states

The section applies atomic gauges to magic states, whose free pure states are stabilizer states. It gives gauge characterizations of magic quantifiers and compares robustness measures, while noting that the standard robustness can differ from the other measures on pure states.

5.4. Magic states

  • Because stabilizer states form a finite set, their convex hull is a polytope, so many magic quantifiers reduce to linear or semidefinite programming.The number of pure stabilizer states nevertheless scales superexponentially with Hilbert-space dimension.Source: §5.4 · p. 39
  • The atomic gauge for stabilizer states is a convex relaxation of stabilizer rank, and the nuclear gauge defines a faithful monotone of magic.The set of stabilizer states spans the space, making the corresponding ΓW a norm for all complex matrices.Source: §5.4 · pp. 39–40
  • For |T⟩⟨T|, standard robustness is strictly larger than the other measures, while the difference appears less pronounced for the two-qubit |T ⊗ T⟩ state.For the two-qubit state, standard and generalized robustness are approximately 0.616 and 0.607, respectively.Source: §5.4 · p. 40
  • For a finite free-state set, the framework also applies to quantum coherence by taking the reference-basis vectors as the free pure states.The effective domains of the measures require care.Source: §5.4 · p. 40

6. Conclusions

The conclusion presents atomic gauge functions as a framework for quantifying convex quantum resources. It highlights applications to coherence, entanglement, and magic states, and identifies extensions and further comparisons as open directions.

6. Conclusions

  • Atomic gauge functions of free-state sets provide a framework for describing many resource quantifiers and checking faithfulness and strong monotonicity under free operations.The framework includes matrix-norm, convex-roof, robustness, and more general quantifiers.Source: §6 · p. 41
  • The framework is presented as extendable to infinite-dimensional spaces and arbitrary convex sets, while relations to distance-based and polynomial-invariant quantifiers remain directions for further investigation.The paper applies its results to coherence, entanglement, and magic states, characterizing known monotones and introducing novel measures.Source: §6 · p. 41
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