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Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
William Matthews, Stephanie Wehner, Andreas Winter
TL;DR
The paper asks how restricted measurement families compare with unrestricted measurements for distinguishing quantum states. It formalizes this comparison through domination constants, analyzes single POVMs, designs, and LOCC, and finds near-optimal bipartite data hiding with LOCC bias Ω(1/d).
Problem
The paper studies how to compute or bound domination constants for statistical norms induced by restricted measurements relative to the trace norm.
Method
It develops norm, duality, and convex-geometric tools and applies them to single POVMs, 2-designs, 4-designs, and multipartite LOCC measurements.
Results
For bipartite systems, LOCC achieves bias Ω(1/d) for distinguishing orthogonal states on d x d systems, matching the known tight scaling up to constants.
Takeaways & Limitations
The results establish near-optimality of bipartite data hiding and yield certainty relations and lower bounds on locally accessible information.
Takeaways & Limitations
The analysis assumes finite dimension and, for domination-constant bounds relevant to discrimination, focuses on traceless operators.
Abstract
from arXiv · showhide
Every sufficiently rich set of measurements on a fixed quantum system defines a statistical norm on the states of that system via the optimal bias that can be achieved in distinguishing the states using measurements from that set (assuming equal priors). The Holevo-Helstrom theorem says that for the set of all measurements this norm is the trace norm. For finite dimension any norm is lower and upper bounded by constant (though dimension dependent) multiples of the trace norm, so we set ourselves the task of computing or bounding the best possible "constants of domination" for the norms corresponding to various restricted sets of measurements, thereby determining the worst case and best case performance of these sets relative to the set of all measurements. We look at the case where the allowed set consists of a single measurement, namely the uniformly random continuous POVM and its approximations by 2-designs and 4-designs respectively. Here we find asymptotically tight bounds for the constants of domination. Furthermore, we analyse the multipartite setting with any LOCC measurement allowed. In the case of two parties, we show that the lower domination constant is the same as that of a tensor product of local uniformly random POVMs (up to a constant). This answers in the affirmative an open question about the (near-)optimality of bipartite data hiding: The bias that can be achieved by LOCC in discriminating two orthogonal states of a d x d bipartite system is Omega(1/d), which is known to be tight. Finally, we use our analysis to derive certainty relations (in the sense of Sanchez-Ruiz) for any such measurements and to lower bound the locally accessible information for bipartite systems.
INTRODUCTION
The paper defines restricted-measurement distinguishability through statistical norms and studies their relation to unrestricted state discrimination. It develops results for single POVMs, designs, multipartite measurements, data hiding, certainty relations, and locally accessible information.
- Measurement-restricted distinguishability: POVMs map quantum states to probability vectors, whose statistical distance cannot exceed the trace-norm distance.The measurement map is completely positive and trace preserving, so it is contractive for the trace norm.
- Measurement-restricted distinguishability: The spectral measurement of ρ−σ achieves the optimal discrimination bias, yielding the Helstrom result for unrestricted measurements.For equal priors, the optimal decision rule compares the measurement outcome probabilities.
- Scope and contributions: The paper studies domination constants for restricted measurement norms, including single POVMs, 2-designs, and 4-designs.These constants quantify the performance of restricted measurements relative to the trace norm.
- Scope and contributions: For bipartite systems, the paper analyzes measurements respecting a multipartite partition, including local measurements and LOCC.The analysis compares LOCC distinguishability with tensor products of local isotropic POVMs.
- Scope and contributions: The paper connects restricted measurements to Sanchez-Ruiz certainty relations and lower bounds locally accessible information for bipartite systems.The certainty-relation results apply to 2-design POVMs and strengthen for 4-designs.
1. FIRST OBSERVATIONS ON NORMS AND DUAL NORMS
This section formalizes restricted-measurement distinguishability as a norm, relates it to convex geometry and state discrimination, and defines domination constants for comparing restricted and trace norms.
- Convex-geometric formulation: Norms correspond to full-dimensional symmetric convex bodies, with dual norms represented by polar bodies.The polar operation preserves the relevant convexity, symmetry, and closedness properties.
- Norm construction: ∥·∥M is a norm exactly when the measurement set is separating, and it always satisfies ∥·∥M ≤ ∥·∥1.Separating means every nonzero operator is detected by some POVM outcome.
- Binary reduction: Two-outcome POVMs preserve the optimal bias of any separating measurement set.Grouping outcomes according to the sign of the relevant operator reduces the measurement without increasing the bias elsewhere.
- Norm construction: Any norm bounded above by the trace norm can be represented as ∥·∥M for a suitable set of POVMs.The correspondence uses full-dimensional symmetric closed convex bodies contained in the operator interval.
- State discrimination: The minimum error probability for equal-prior states is 1/2 − 1/4∥ρ−σ∥M, so ∥ρ−σ∥M determines the restricted-measurement bias.This establishes the operational meaning of the measurement norm.
- Domination constants: The constants λ1 and µ1 are the largest and smallest trace-norm comparison factors, and µ1(M)=1 for positive operators.For state discrimination, the analysis focuses on traceless operators and the corresponding λ(M), µ(M).
- Convex-geometric formulation: Unitary invariance preserves the domination constants of the associated convex bodies and measurement norms.The result follows from invariance of the operator interval under unitary conjugation.
2. SINGLE POVMS
The paper analyzes domination constants when the allowed measurement set consists of a single informationally complete POVM.
- Single POVMs: The single-POVM case concerns one informationally complete POVM and its associated CPTP map.The domination constants and measurement norm are written using the same symbol as the POVM map.
A. Uniform POVM
The uniform unitary-invariant POVM provides extremal domination constants among dimension-d POVMs, with asymptotically characterized performance and a conjectured balanced-rank minimizer.
- The uniform POVM attains the supremum of λ(M) and the infimum of μ(M) over all POVMs in dimension d.
- Unitary invariance reduces the analysis of the uniform POVM to traceless operators supported on complementary subspaces of ranks a and b=d−a.
- The authors have not proved that the balanced-rank choice is the true minimum.
- The asymptotic analysis uses independent rescaled χ2 variables, the central limit theorem, Berry–Esseen convergence, and Stirling’s formula.
B. Almost optimal performance of 4-designs
The paper introduces spherical t-design POVMs as finite approximations to the uniform random POVM and shows that 4-designs essentially match its bias.
- t-design POVMs approximate the full random POVM increasingly well as t grows, motivating comparisons of their domination constants.
- A weighted spherical t-design is an ensemble of one-dimensional projectors with probabilities satisfying the t-design condition.
- A 4-design already achieves essentially the same bias as the isotropic POVM.
- The 4-design analysis applies Berger’s moment inequality to a random variable whose second and fourth moments follow from the design property.
- Efficient general constructions of spherical 4-designs are unknown, although weighted designs exist with bounded cardinality and approximate constructions perform nearly as well.
C. Performance of 2-designs
Proper and weighted 2-design POVMs have a universal lower bound on distinguishability, and explicit constructions show this scaling is essentially optimal.
- No bias bounds are provided for 3-design POVMs in this analysis.
- 2-design POVMs are analyzed through their associated CPTP maps, including mutually unbiased bases and conjectured SIC-POVM constructions.
- For any proper 2-design POVM, λ(M) is at least 1/(d+1).
- The proof writes a traceless operator as the difference of orthogonal density operators and applies the 2-design second-moment identity.
- The same lower bound holds for weighted 2-design POVMs by refining their probabilities into approximately equal weights.
- The factor 1/(d+1) is essentially best possible: mutually unbiased bases and SIC-POVM examples give matching upper bounds up to constants.
3. LOCAL POVMS
The paper compares locality-restricted measurement classes through domination constants, finding that bipartite tensor products of isotropic local POVMs nearly match separable and LOCC performance. This establishes Θ(1/d) bias for distinguishing orthogonal states on d × d systems and shows that data-hiding states are essentially optimal.
- PPT and data hiding: 2/(d+1) bounds the PPT domination constant from above, and the bound is not far from tight.The comparison uses the symmetric and antisymmetric subspace states as data-hiding examples.
- Bipartite measurements: A tensor product of isotropic local POVMs is already almost as effective as all separable POVMs for bipartite domination.The construction randomizes independently over the two local systems, and local 4-designs suffice for the analysis.
- Locality-restricted measurements: LOCC, separable, and other locality-restricted POVMs are compared through constants of domination relative to the trace norm.These constants bound the bias achievable when distinguishing states using restricted measurements.
- Bipartite measurements: Θ(1/d) is the optimal-order bias for LOCC discrimination of orthogonal states on a d × d system.The original symmetric and antisymmetric data-hiding states achieve the best available bias up to a constant factor.
- Data-hiding constraints: If one data-hiding state has rank at most r, LOCC measurements achieve bias at least 1/(13r).The result implies that data-hiding states must be highly mixed, consistent with the considerable entropy in known constructions.
4. CERTAINTY RELATIONS
The paper derives certainty relations for restricted measurements and uses the same distinguishability bounds to lower-bound accessible information. For bipartite ensembles, LOCC-accessible information is bounded using independently applied locally invariant continuous POVMs.
- Mutually unbiased bases: A complete set of d + 1 mutually unbiased bases obeys a certainty relation constraining the entropies of measurements on any pure state.The relation expresses that no pure state produces maximum entropy for all such measurements simultaneously.
- 2-designs: Any proper 2-design POVM with n outcomes satisfies a more general, though weaker, certainty bound.The derivation uses the domination result for 2-designs and Pinsker’s inequality.
- 4-designs: 4-design measurements yield stronger certainty bounds through the Hilbert-Schmidt norm.The improvement applies to uni- and bipartite 4-designs, including isotropic POVMs.
- Accessible information: A positive lower bound on accessible information follows for pure-state ensembles, even though their individual linear entropies vanish.The bound is expressed using linear entropies and the measurement-induced distinguishability estimates.
- Bipartite accessible information: For bipartite systems, the mutual information from independently applied locally invariant continuous POVMs lower-bounds LOCC-accessible information.This protocol uses local measurements followed by sharing the outcomes, and the resulting quantity depends on the ensemble beyond its average in general.
5. CONCLUSION
The paper develops measurement-induced norms to study restricted distinguishability and domination constants, connecting the resulting bounds to data hiding and accessible information. It also identifies unresolved questions about accessible information, domination constants, and the geometry of the associated convex bodies.
- Contributions: The paper introduces norms linked to pairwise state distinguishability under restricted classes of measurements.It studies these norms geometrically through their relation to the trace and Hilbert-Schmidt norms.
- Contributions: For single measurements and locality-restricted classes including LOCC and PPT, the paper investigates domination constants relative to the trace norm.The single-measurement cases include isotropic POVMs, 4-designs, and 2-designs.
- Data hiding: Bipartite data-hiding states achieve the best possible bias up to a constant factor.This connects the domination analysis directly to the performance of restricted measurements.
- Open questions: The eventual determination of locally accessible information and sharper domination bounds remains open.The paper also leaves the broader geometry of the convex bodies associated with restricted measurements unresolved.
APPENDIX A: AN ℓ1-INEQUALITY FOR PROBABILITY VECTORS AND DENSITY OPERATORS
The appendix proves an ℓ1 inequality for probability vectors and extends it to density operators, relating trace norm to Hilbert-Schmidt overlap. It also identifies a limitation of one inequality when a factor below one is introduced.
- Classical inequality: The appendix establishes an inequality for probability vectors relating statistical distance to their Euclidean inner product.The statistical distance is the ℓ1 norm of the difference, while the comparison term is the usual Euclidean inner product.
- Quantum generalization: The same inequality extends to density operators, replacing statistical distance with trace norm and Euclidean overlap with Hilbert-Schmidt inner product.The quantum statement follows by diagonalizing one state and dephasing the other in the same basis.
- Proof strategy: The proof uses the relation between trace distance and fidelity, with the substitution t_i = √(p_iq_i).This reduces the quantum result to the corresponding classical inequality.
- Limitation: For sufficiently large n, introducing a factor c < 1 on the inequality’s left-hand side makes it false.The appendix gives a class of counterexamples and an asymptotic comparison showing the failure.
APPENDIX B: AN INTEGRAL OVER THE UNIT SPHERE
The appendix evaluates an integral over uniformly random unit vectors by representing the vector with Gaussian components and reducing the resulting quantities to χ2-distributions. Auxiliary lemmas then complete the calculation using symmetry and an addition formula.
- Projector decomposition: For mutually orthogonal projectors of ranks a and b, the projector overlaps are parameterized by p = a/(a + b).This parameter enters the probability calculation for the random-vector integral.
- Gaussian representation: A random complex Gaussian vector is introduced to represent the uniformly distributed unit vector.Its independent real and imaginary components are Gaussian, and the expected squared norm is specified.
- Distributional reduction: The relevant quadratic forms are expressed using sums of squared Gaussian components, identified as χ2-distributions.The appendix supplies the corresponding probability densities for the variables with parameters a and b.
- Integral evaluation: The resulting probabilities under the integrals are evaluated directly from the χ2 densities.Indicator functions are used to write the relevant events before integration.
- Auxiliary lemma: An auxiliary sequence lemma, together with the addition formula and the base case S0 = 1, completes the appendix calculation.The proof invokes symmetry and the addition formula to establish the final equality.
APPENDIX C: UPPER BOUNDS ON CERTAIN TRACES
The appendix bounds traces arising from fourth-order tensor expressions by grouping permutation terms into R-conjugacy classes and bounding each class. Tensor diagrams encode contractions, while positivity, Cauchy–Schwarz, and partial-transpose symmetry control the class contributions.
- Proof assembly: The projection operators are written as averages over subsystem-permuting unitaries, after which terms are collected according to the multiplicities in Figure 1.All types in a given box share the same upper bound for each pair of permutation conjugacy classes.
- R-conjugacy classes: R-conjugacy classes partition the 24!2 permutation terms more finely than ordinary conjugacy classes because simultaneous conjugation preserves each term.The proof bounds one representative per class and multiplies by the class size.
- Tensor diagrams: Tensor diagrams represent components of ξ and encode index contractions by joining terminals with wires.Grey-line identifications compactly represent parallel matching-wire contractions and are distinct from symmetrization bars.
- Class organization: Figure 1 lists each R-conjugacy class with its size and an upper bound, marking terms containing Tr(ξ) = 0 as identically zero.The lighter diagrams correspond to vanishing terms caused by the tracelessness condition.
- Trace inequalities: The class bounds repeatedly use Cauchy–Schwarz and positivity inequalities, including Tr(PQ) ≤ (Tr(P))(Tr(Q)) for positive semidefinite P and Q.The same strategy bounds traces such as Tr(ξ4) by (Tr(ξ2))2.
- Partial-transpose symmetry: Partial transpose preserves the quantities t, a, and b used in the bounds, so corresponding estimates transfer under ξ → ξΓ.This symmetry also supplies bounds for diagrams obtained by exchanging the parties.