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All Pure Bipartite Entangled States can be Self-Tested

Andrea Coladangelo, Koon Tong Goh, Valerio Scarani

arXiv:1611.08062v2quant-ph

TL;DR

The paper addresses the open question of whether all bipartite pure entangled states can be self-tested. It constructs explicit self-testing correlations for each such state and shows that the associated ideal measurements can also be characterized, with potential applications to randomness and quantum computing.

  • Problem

    Whether all bipartite pure entangled quantum states can be self-tested remained an outstanding open question.

  • Method

    The paper builds on the framework of Yang and Navascués to construct explicit correlations with corresponding ideal measurements for arbitrary target entangled states.

  • Results

    The construction self-tests every bipartite entangled pure state and, under a local isometry, makes the physical measurements equivalent to the ideal target measurements.

  • Takeaways & Limitations

    The result provides flexibility in choosing self-tested bipartite states and d-outcome measurements for randomness certification and quantum computing.

Abstract

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Device-independent self-testing allows to uniquely characterize the quantum state shared by untrusted parties (up to local isometries) by simply inspecting their correlations, and requiring only minimal assumptions, namely a no-signaling constraint on the untrusted parties and the validity of quantum mechanics. The device-independent approach exploits the fact that certain non-local correlations can be uniquely achieved by measurements on a particular quantum state. We can think of these correlations as a "classical fingerprint" of the self-tested quantum state. In this work, we answer affirmatively the outstanding open question of whether all pure bipartite entangled states can be self-tested, by providing explicit self-testing correlations for each.

I. INTRODUCTION

Device-independent self-testing characterizes shared quantum states from observed correlations under no-signaling and quantum-mechanical assumptions. The paper addresses whether every bipartite pure entangled state can be self-tested, using explicit correlations in a three-setting/four-setting, d-outcome Bell scenario.

  • Motivation: Self-testing lets a classical verifier characterize the joint state shared by potentially untrusted parties up to local isometries.The verifier inspects observed correlations without assuming the devices’ internal workings.
  • Motivation: Non-local correlations enable self-testing because they can arise from measurements on entangled quantum states but not from classical provers.The singlet is a prominent example, self-tested through maximal CHSH violation.
  • Contribution: The paper answers affirmatively whether all bipartite pure entangled states can be self-tested by constructing an explicit family of self-testing correlations.The construction also targets certain ideal measurements.
  • Framework: The proposed correlations use three measurement settings for Alice, four for Bob, and d outcomes per party, where d is the local dimension.Inputs are x ∈ {0, 1, 2} and y ∈ {0, 1, 2, 3}; outputs range from 0 to d − 1.
  • Framework: Correlations are represented by twelve d × d tables, one for each pair of measurement settings in the [{3, d}, {4, d}] scenario.The collection of conditional probabilities P(a, b|x, y) is called a correlation.
  • Framework: The device-independent setting does not require a priori dimensional bounds or purity assumptions, and estimating P(a, b|x, y) is sufficient experimentally.Projective measurements may be assumed via Naimark’s theorem, while mixed-state correlations can be reproduced by pure states of the same dimension.

B. Tilted CHSH inequality

The tilted CHSH inequality generalizes CHSH through a parameter and, at maximal quantum violation, self-tests a partially entangled two-qubit state and its measurements. The paper uses this result as a building block for a sufficient criterion that self-tests general pure bipartite states.

  • Tilted CHSH inequality: The tilted CHSH inequality is a one-parameter generalization of the CHSH inequality for binary observables.
  • Tilted CHSH inequality: Maximal violation self-tests the state |ψ⟩ = cos θ|00⟩ + sin θ|11⟩ together with the corresponding measurements.The ideal observables are expressed using Pauli matrices and an angle μ determined by θ.
  • Self-testing criterion: The sufficient criterion introduces operators constructed from binary observables and projections to establish a local isometry for a general pure bipartite entangled state.The criterion targets states of the form |ψ⟩ = Σ_i c_i|ii⟩ with positive coefficients.
  • Self-testing criterion: The local isometry adds two ancilla qudits and generalizes the qubit SWAP isometry using quantum Fourier transforms and controlled operations.The construction is depicted in Figure 2 and is defined through the operators R and S.
  • Self-testing criterion: The construction supplies the correlations needed to obtain the operators required by the Yang–Navascués criterion, which was not provided in that earlier framework.

III. SELF-TESTING CORRELATIONS

The paper states its main result as an explicit family of correlations for every bipartite entangled qudit state. These correlations imply self-testing of the target state and, additionally, the ideal local measurements.

  • Main result: The paper constructs self-testing correlations using three measurement settings for Alice, four for Bob, and d outcomes per party.
  • Main result: For every bipartite entangled qudit state |ψtarget⟩, there exist correlations in the specified Bell scenario that self-test it.
  • Main result: When reproduced from a joint state ρ, the correlations imply a local isometry mapping ρ to ρextra ⊗ |ψtarget⟩⟨ψtarget|.
  • Main result: Under the same isometry, the unknown local measurements are equivalent to ideal measurements acting trivially on the auxiliary state.

B. The idea behind our correlations

The proposed correlations arrange d-outcome data into 2 × 2 blocks, each corresponding to a two-level component of the target state. The blocks realize suitably chosen tilted CHSH correlations for alternating outcome pairs.

  • Block structure: The construction applies d-outcome measurements to produce block-diagonal correlation tables with 2 × 2 blocks.
  • Block structure: For settings x, y ∈ {0, 1}, each block pairs outcomes (0,1), (2,3), through (d−2,d−1) and self-tests the corresponding even-odd component.
  • Block structure: For settings x ∈ {0, 2} and y ∈ {2, 3}, the red blocks certify the complementary odd-even outcome pairs.
  • Tilted CHSH blocks: The 2 × 2 blocks correspond to ideal tilted CHSH correlations with appropriately chosen angles.
  • Scope of construction: The particular block choice is not essential if alternative correlations can establish the operators required by the self-testing criterion.

C. The correlations

The proposed correlations use selected d-outcome measurement settings whose tables are block-diagonal, with 2 × 2 blocks encoding partially entangled qubit components of the target state. Two families of settings suffice to specify the correlations and self-test both the state and ideal measurements.

  • Selected correlation tables: Only tables for x,y ∈{0,1} and x ∈{0,2}, y ∈{2,3} are needed to self-test the target state.The constraints determine a single quantum correlation because they also self-test the ideal measurements.
  • Unshifted blocks: For x,y ∈{0,1}, the correlation tables are block-diagonal with 2 × 2 blocks.The blocks correspond to consecutive outcome pairs and are specified separately for even and odd local dimensions.
  • Unshifted blocks: Each unshifted 2 × 2 block implements a maximal tilted-CHSH violation that self-tests a partially entangled two-qubit state.The relevant state is cos(θ_m)|00⟩ + sin(θ_m)|11⟩, with parameters defined from the target coefficients.
  • Shifted blocks: For x ∈{0,2} and y ∈{2,3}, the tables are likewise block-diagonal but shifted by one outcome.Their 2 × 2 blocks correspond to outcomes 2m+1 and 2m+2 and are listed for even and odd d.

D. The ideal measurements

The ideal strategy uses computational-basis measurements together with direct sums of two-dimensional Pauli observables, with separate constructions for even and odd d. Bob’s observables use angles determined by the target-state coefficients.

  • Ideal correlation: The ideal correlation is the correlation produced by these ideal measurements on the target state.The target state is the weighted bipartite state |ψ_target⟩=Σ_i c_i|ii⟩.
  • Block observables: The notation [A]_m embeds a single-qubit observable A in the basis {|2m⟩, |2m+1⟩}.The shifted notation [A]'_m acts on {|2m+1⟩, |2m+2⟩}, and direct sums combine these blocks.
  • Alice’s measurements: Alice measures in the computational basis for x=0.For x=1 and x=2, her measurements use eigenbases of direct sums of two-dimensional observables, with separate even- and odd-d constructions.
  • Bob’s measurements: Bob uses eigenbases of direct-sum observables for y=0,1,2,3, with even- and odd-d cases treated separately.For y=0 and y=1, the rotation angles depend on μ_m=arctan(sin(2θ_m)) and θ_m determined by coefficient ratios.

IV. PROOF OF SELF-TESTING

The proof constructs operators satisfying the sufficient self-testing conditions and then shows that the resulting local isometry also certifies the ideal measurements.

  • Proof strategy: The proof’s main task is constructing operators that satisfy the sufficient conditions from Lemma 2.These include suitable projections on Bob’s side.
  • Operator construction: The proof first constructs projections and associated unitary “flip” operators.These objects are developed in subsection IV A.
  • Operator construction: It then builds the required unitaries as alternating products of the flip operators.This step is carried out in subsection IV B.
  • Measurement certification: The same local isometry from Lemma 2 is shown to self-test the ideal measurements as well as the state.Thus the proof extends the certification from the target state to the specified measurement implementation.

A. Constructing the projections and the “flip” operators

The construction derives local projections and unitarized observables from the correlation constraints, then uses tilted-CHSH relations and normalization to establish the required operator identities.

  • Pairwise block operators: Alice’s and Bob’s outcome projectors are combined into ±1 observables on consecutive outcome pairs.For x,y∈{0,1}, these observables act on the two-dimensional blocks indexed by m.
  • Constraint consequences: The correlation constraints enforce norm relations between corresponding Alice and Bob projections.Cauchy–Schwarz is used to relate the block probabilities to the norms of projected states.
  • Block normalization: Each block is normalized to a state |ψ_m⟩ before applying the tilted-CHSH self-testing relation.This normalization addresses the fact that the original block state need not have unit norm.
  • Unitarization: The proof unitarizes the block observables by replacing zero eigenvalues and restricting attention to the relevant support subspaces.The resulting operators act as the required Pauli-like observables on the supported components.
  • Isometry conditions: The resulting projections and flip operators satisfy the identities needed to invoke the self-testing isometry.These identities are derived from the block constraints and the tilted-CHSH implications.
  • Shifted blocks: The same construction is repeated for shifted outcome pairs and yields analogous primed observables and unitaries.These operators support the second family of correlation constraints.

B. Constructing the unitaries

The proof constructs alternating products of local unitary flip operators to satisfy the isometry conditions, extending the self-testing construction to general mixed states and allowing alternative block-diagonal correlations.

  • Constructing the flip operators: The operators X_A,m and Y_A,m, with Bob-side analogues, act as flip operators between neighboring projected subspaces.Their alternating products are chosen to produce the unitary operators required by Lemma 2.
  • Verifying the construction: The constructed unitaries satisfy the required relations, completing the local-isometry construction used in the proof of Theorem 1.The same isometry is then used to self-test the ideal measurements.
  • Defining the unitaries: The operators X_A^(k) and X_B^(k) are defined as alternating products of the X and Y flip operators, with separate forms for even and odd k.Because they are products of unitaries, the resulting operators are unitary.
  • Mixed-state extension: The proof extends from pure to mixed joint states by replacing vector equalities with density-matrix equalities and adapting the inner-product argument.The mixed-state formulation uses the support of the density operator and retains Cauchy–Schwarz where the needed symmetry holds.
  • Alternative correlation choices: Any block-diagonal correlations suffice when each 2 × 2 un-normalized block implies the required local reflections for suitable angles.For maximally entangled qudits, correlations from Wang et al. yield a [{3, d}, {3, d}] Bell scenario after dropping Bob’s fourth setting.

VI. CONCLUSION AND OUTLOOK

The paper answers affirmatively whether all bipartite pure entangled states can be self-tested, using explicit correlations that also certify corresponding d-outcome measurements. It identifies cryptographic and quantum-computing applications while leaving robustness and related game-based characterizations open.

  • Main conclusion: The authors provide explicit correlations that self-test every bipartite entangled pure state and corresponding d-outcome ideal measurements.They frame this as an affirmative answer to the open self-testing question.
  • Applications: The result offers flexibility in choosing the self-tested bipartite state and measurements for randomness-certification and quantum-computing applications.The paper specifically mentions device-independent randomness expansion and certification of randomness.
  • Outlook: The paper leaves open whether Bell-inequality violations or optimal non-local-game values can characterize all bipartite entangled states.It also notes limited awareness of non-local games self-testing non-maximally entangled states.

Appendix A: Proof of Lemma 2

The proof constructs a local isometry mapping the shared state to an auxiliary state tensored with the ideal target state, then extends this equivalence to the measurement operators. It also uses orthogonalization to obtain exactly orthogonal projections and notes that the argument extends to mixed states.

  • State self-testing: The proof explicitly constructs a local isometry Φ that maps |ψ⟩ to |extra⟩⊗|ψtarget⟩.The target state is written as a Schmidt decomposition, with |extra⟩ an auxiliary state.
  • State self-testing: Orthogonal projections on Bob’s side are obtained by applying an orthogonalization lemma to projections that are orthogonal when acting on |ψ⟩.The resulting projections have exact orthogonality while preserving their action on the shared state.
  • Isometry construction: The isometry uses Fourier transforms, controlled X and Z operations, and the corresponding projection and unitary operators on Alice’s and Bob’s systems.The proof evaluates Φ on |ψ⟩ with auxiliary registers initialized to |0⟩.
  • Extension: The proof can also be repeated for a mixed joint state, yielding a corresponding version of the lemma for general mixed states.
  • Measurement self-testing: The same local isometry maps the physical block measurements to the ideal two-qubit measurements on |ψtarget⟩.This follows from maximal tilted-CHSH violation self-testing the ideal single-qubit measurements on each relevant subspace.
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