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Mixed-state entanglement from local randomized measurements
Andreas Elben, Richard Kueng, Hsin-Yuan Huang, Rick van Bijnen, Christian Kokail, Marcello Dalmonte, Pasquale Calabrese, Barbara Kraus, John Preskill, Peter Zoller, Benoît Vermersch
TL;DR
The paper addresses bipartite-entanglement detection in many-body mixed states through moments of the partially transposed density matrix. It proposes using local randomized measurements with classical-shadow post-processing, and reports that the p3-PPT condition is sound and can detect entanglement in highly mixed states, although it can fail for close-to-completely mixed small partitions.
Problem
Detecting bipartite entanglement in many-body mixed states requires tests based on moments of the partially transposed density matrix.
Method
The method estimates partially transposed density-matrix moments using local randomized measurements followed by classical-shadow post-processing.
Results
The p3-PPT condition is sound, detects negative eigenvalues in many unit-trace matrices, and detects entanglement in highly mixed states, but can miss close-to-completely mixed states for small partitions.
Takeaways & Limitations
Low-order partially transposed moments can provide an entanglement test, while stronger tests may require combining moments up to a chosen maximum order.
Takeaways & Limitations
Direct higher-moment extensions can be weaker than p3-PPT, and p3-PPT can fail for close-to-completely mixed states in small partitions.
Abstract
from arXiv · showhide
We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by post-processing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al, Science 364, 260 (2019)].
Appendix A: The p3-PPT condition
The p3-PPT condition is derived as the contrapositive of a moment inequality for positive semidefinite unit-trace matrices. Unlike Rényi-entropy monotonicity, it remains applicable when the tested matrix has negative eigenvalues.
- The p3-PPT condition is the contrapositive of a statement about moments of positive semidefinite matrices with unit trace.
- Rényi-entropy monotonicity applies only to density matrices, whereas the p3-PPT condition tests whether a matrix is not positive semidefinite.
- The p3-PPT condition avoids logarithms of non-positive numbers, which can invalidate arguments based on Rényi-entropy monotonicity.
1. Proof of the p3-PPT condition
The proof develops Schatten-norm relations for Hermitian matrices and applies them to establish the p3-PPT condition. Positive semidefinite matrices satisfy the resulting condition because their eigenvalues are nonnegative.
- Norm preliminaries: Schatten-p norms are defined from the eigenvalues of a Hermitian matrix and include trace, Hilbert-Schmidt/Frobenius, and operator norms.These matrix norms inherit properties from corresponding vector ℓp norms.
- Norm preliminaries: Hölder’s inequality and Cauchy-Schwarz yield the Schatten-norm relation needed for the p3-PPT proof.The proof applies Hölder’s inequality with p = 3 and q = 3/2, followed by Cauchy-Schwarz.
- Proof of the p3-PPT condition: The resulting norm inequality establishes the p3-PPT condition from moments of orders p = 1, 2, 3.The proof’s central ingredient is the relation among Schatten-p norms of these three orders.
- Proof of the p3-PPT condition: For positive semidefinite X, every eigenvalue is nonnegative, so |X| = X and ∥X∥p = Tr(Xp)1/p for p ≥1.These properties connect the general Schatten-norm relation to matrix moments.
2. Discussion and potential generalizations
The p3-PPT condition is interpreted as a polynomial approximation to negativity, with soundness and nonvacuity properties for detecting negative eigenvalues. Higher-order extensions can be more expressive, but direct neighboring-moment tests may become weaker.
- 2. Discussion and potential generalizations: Direct higher-moment extensions can be weaker than p3-PPT because neighboring high-order moments suppress small eigenvalues, including negative ones.For partially transposed states, eigenvalues lie in [−1/2, 1] and sum to one, so negative eigenvalues cannot dominate the spectrum.
- 2. Discussion and potential generalizations: Using all moments up to an order pmax is suggested as a potentially stronger strategy for testing negative eigenvalues.This reframes the goal from comparing only neighboring matrix moments to combining multiple moments.
- 2. Discussion and potential generalizations: F3(X) acts as a polynomial approximation to the non-analytic negativity function by applying f3 to the eigenvalues of X.The comparison is made with the negated rectifier function r(−x).
- 2. Discussion and potential generalizations: For positive semidefinite X, F3(X) ≤ 0, so the p3-PPT condition is sound and has no false positives.The polynomial f3(x) is nonpositive for positive x, preventing positive eigenvalues from increasing F3(X).
- 2. Discussion and potential generalizations: Negative eigenvalues can make F3(X) positive, allowing the p3-PPT condition to detect negativity in many, but not all, unit-trace matrices.Thus, the condition is not vacuous, although it is not universally detecting.
- 2. Discussion and potential generalizations: a♯=tr(X2)/tr(X) optimizes the polynomial test for fixed X, reducing to a♯=p2 when X has unit trace.This parameter choice produces the p3-PPT condition.
- 2. Discussion and potential generalizations: Higher-degree polynomials can approximate the negated rectifier more accurately, but choosing their coefficients optimally remains unclear.Some softplus-based approximations are unsuitable because they remain positive for x > 0.
Appendix B: p3-PPT condition for Werner States
For Werner states, the p3-PPT condition matches the full PPT condition and therefore exactly characterizes bipartite entanglement, but related logarithmic quantities have important domain and monotonicity limitations.
- For any local dimension d, Werner states are entangled for 0 ≤ α < 1/2 because their partially transposed density matrix has a negative eigenvalue.The relevant eigenvalue is λ0 = (2α −1)/d.
- The p3-PPT condition is equivalent to the full PPT condition for Werner states and is necessary and sufficient for bipartite entanglement.The same equivalence holds for the closely related isotropic states described in the passage.
- Werner states can have non-positive p3, so logarithms such as R3 = −log2(p3/Tr[ρ3]) are not always defined.For d > 3, an interval [0, α∗) has p3 < 0.
- R3 is not an entanglement monotone because it can be positive for separable Werner states, although it vanishes for all product states.The stated separable range is 1/2 ≤ α < 1/2 + 1/(2d).
Appendix C: Comparison of entanglement conditions for quench dynamics
The appendix compares negativity, the p3-PPT condition, and a nested-subsystem purity condition for entanglement generated by long-range XY-model quenches. Negativity detects entanglement broadly, while p3-PPT performs well for highly mixed states but fails near complete mixing.
- The comparison concerns PPT-based and purity-based diagnostics for mixed states generated by quenches in a long-range interacting XY model.The purity condition was used in earlier experiments to reveal entanglement of weakly mixed states.
- Negativity detects bipartite entanglement for all partition sizes and all times after the quench.
- The p3-PPT condition detects entanglement in highly mixed states whose purity falls to 0.3 for the panel (b) partition at late times.
- The p3-PPT condition fails for close-to-completely mixed four-qubit partitions at late times because it relies only on low-order PT-moments.
- The nested-subsystem purity condition detects entanglement only for large eight-qubit partitions, which remain weakly mixed throughout the evolution.The total system contains N = 10 spins and begins in a separable Néel product state.
Appendix D: Error bars for PT moment predictions
Classical-shadow estimates of subsystem observables converge to their target values through empirical averaging, with variance bounds determining the repetitions needed for a chosen accuracy.
- Random single-qubit rotations from a unitary 3-design followed by computational-basis measurements produce classical snapshots of the unknown N-qubit state.The snapshots reproduce the underlying state in expectation and can be reduced to subsystems.
- For a linear subsystem property Tr(OρAB), the empirical average of independent snapshots converges to the target value, with Chebyshev’s inequality converting variance into an error guarantee.
- Approximately M ≈ 2^|AB|Tr(O^2)/ϵ^2 repetitions are needed to predict an observable to accuracy ϵ.The bound uses subsystem dimension dA = 2^|AB| and the squared Hilbert-Schmidt norm of O.
1. Predicting quadratic properties (p2)
Quadratic and cubic properties are estimated by applying classical-shadow snapshots to tensor-product representations and averaging U-statistics. The resulting p2 and p3 error bounds have regime-dependent sampling behavior.
- 1. Predicting quadratic properties (p2): Quadratic functions of ρ are represented as linear functions on ρ ⊗ρ and estimated using symmetric products of distinct snapshots.
- 1. Predicting quadratic properties (p2): The p2 estimator is unbiased, and its variance decomposes into linear and quadratic contributions controlled by the snapshot-index overlap structure.
- 1. Predicting quadratic properties (p2): For p2, an explicit snapshot bound guarantees |p̂2 − p2| ≤ ϵ with probability at least 1 − δ.
- 1. Predicting quadratic properties (p2): Asymptotically, p2 approximation error decays proportionally to 1/√M, while finite-sample behavior can be dominated by a next-to-leading term when p2 is small.
- 1. Predicting quadratic properties (p2): Median-of-means estimation improves the δ dependence from 1/δ to log(1/δ), increases robustness to outliers, and supports simultaneous purity predictions for many subsystems.
- 1. Predicting quadratic properties (p2): Cubic properties, including p3, are estimated with U-statistics over symmetric products of three distinct snapshots, yielding linear, quadratic, and cubic variance contributions.
- 1. Predicting quadratic properties (p2): For p3, the estimator’s error is bounded with probability at least 1 − δ, with asymptotic 1/√M decay and a small-sample regime proportional to 1/M^3/2.
3. Additional numerical simulations
Additional simulations test statistical-error predictions for p2 and p3 in the transverse-field Ising ground state. They observe the same overall scaling as for the GHZ state, while finite system sizes obscure one early-regime decay rate.
- 3. Additional numerical simulations: The transverse-field Ising ground state exhibits the same statistical-error scaling as the GHZ-state simulations.The simulations are performed at criticality and examine estimates of p2 and p3.
- 3. Additional numerical simulations: p2 displays two decay regimes with rates 1/M and 1/√M.
- 3. Additional numerical simulations: p3 also shows decay rates 1/M and 1/√M, although its early-regime rate is less pronounced.
- 3. Additional numerical simulations: The missing early-regime behavior for smaller systems is attributed to limited system sizes.For the largest system, 1/M^3/2 captures the decay of the red data points, but this rate is absent for smaller systems.
- 3. Additional numerical simulations: Estimating p2 and p3 with accuracy 0.1 requires measurements on the order of 100 × 2^|AB|.
Appendix E: Auxiliary results and wiring diagrams
The appendix introduces wiring diagrams as a pictorial calculus for manipulating traces and partial transposes of bipartite operators. It uses index contractions, tensor-product layouts, and diagrammatic rearrangements to derive canonical expressions needed for variance bounds.
- Appendix E: Auxiliary results and wiring diagrams: Wiring diagrams provide a pictorial calculus for identities involving traces of partial transposes of bipartite operators.The formalism represents operators as boxes with indices and supports partial operations such as partial transpose.
- Appendix E: Auxiliary results and wiring diagrams: Operator multiplication is represented by contracting an outgoing index of one operator with an incoming index of another.The displayed contraction yields the matrix product in index notation.
- Appendix E: Auxiliary results and wiring diagrams: Transposition exchanges outgoing and incoming indices, while tracing pairs indices and sums over them.
- Appendix E: Auxiliary results and wiring diagrams: For bipartite operators, parallel lines represent the tensor factors, with upper lines for A and lower lines for B; identity and swap are key examples.
- Appendix E: Auxiliary results and wiring diagrams: Table I rearranges wiring diagrams into equivalent canonical formulas required by bilinear and trilinear variance bounds.The canonical forms are Tr(O X_AB ⊗ Y_AB) and Tr(O′ X_AB ⊗ Y_AB ⊗ Z_AB).