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Experimental on-demand recovery of entanglement by local operations within non-Markovian dynamics
Adeline Orieux, Antonio d'Arrigo, Giacomo Ferranti, Rosario Lo Franco, Giuliano Benenti, Elisabetta Paladino, Giuseppe Falci, Fabio Sciarrino, Paolo Mataloni
TL;DR
Noisy distribution can degrade entanglement, while the entanglement of the averaged state may conceal correlations retained across pure-state ensemble members. These experiments use local open-loop correction or echo and environment-conditioned closed-loop control; they demonstrate on-demand recovery, including restoration for every p when θ = 0.
Problem
The averaged state's entanglement can be degraded even when ensemble members remain maximally entangled, leaving correlations hidden without information about the ensemble member.
Source: C. Open-loop control · p. 3, Conclusion · p. 6Method
Two optical experiments use local phase correction or echo, and measurement of an accessible environment to condition a local operation.
Source: C. Open-loop control · pp. 3–4, D. Closed-loop control · p. 5Results
For θ = 0, initial entanglement is restored for every p; at p = 0.5, recovery depends on measurement angle θ.
Source: D. Closed-loop control · p. 5Takeaways & Limitations
The experiments illustrate that local operations can recover entanglement hidden by missing classical information about which ensemble member is present.
Source: Conclusion · p. 6
Abstract
from arXiv · showhide
In many applications entanglement must be distributed through noisy communication channels that unavoidably degrade it. Entanglement cannot be generated by local operations and classical communication (LOCC), implying that once it has been distributed it is not possible to recreate it by LOCC. Recovery of entanglement by purely local control is however not forbidden in the presence of non-Markovian dynamics, and here we demonstrate in two all-optical experiments that such entanglement restoration can even be achieved on-demand. First, we implement an open-loop control scheme based on a purely local operation, without acquiring any information on the environment; then, we use a closed-loop scheme in which the environment is measured, the outcome controling the local operations on the system. The restored entanglement is a manifestation of "hidden" quantum correlations resumed by the local control. Relying on local control, both schemes improve the efficiency of entanglement sharing in distributed quantum networks.
Results
The optical setup encodes qubits A and B in photon polarizations and prepares them in the Bell state |Ψ−⟩. They propagate through free space, where each experiences local environmental interactions.
Results
- Photons A and B encode qubits in their horizontal or vertical polarization, and the system is prepared in |Ψ−⟩.The pair forms the two-photon entangled Bell state.Source: Results · p. 2
- The photons propagate through free space as a communication channel.Source: Results · p. 2
- During propagation, the qubits experience local interactions with different environmental degrees of freedom.Source: Results · p. 2
A. Theoretical framework
The framework distinguishes entanglement in the reduced, averaged state from the average entanglement of a physically selected pure-state ensemble. When local operations can use information identifying ensemble members, they can make that larger average entanglement available.
A. Theoretical framework
- Tracing out the environment yields a reduced density matrix that can be decomposed into pure states in infinitely many ways.The ensemble members occur with probabilities pν(t).Source: A. Theoretical framework · p. 2
- The framework studies entanglement of formation Ef and also uses concurrence C, which monotonically determines Ef.For pure states, the entanglement measure reduces to entropy of entanglement.Source: A. Theoretical framework · p. 2
- Environmental interaction turns the initial Bell state into a statistical mixture whose entanglement quantifier differs from the initial value.Source: A. Theoretical framework · p. 2
- If local control can identify and tag members of a specific physical ensemble, LOCC can access its average entanglement, which is at least that of the reduced state.For any convex entanglement measure E, E_Q ≥ E_ρ.Source: A. Theoretical framework · p. 2
B. The experiments
Two all-optical experiments use local control to restore entanglement degraded by non-Markovian environments. One applies an echo without environment information; the other measures an accessible quantum environment and conditions the local operation on that outcome.
B. The experiments
- Both experiments generate two-photon states with an SPDC source and study non-unitary dynamics from a local environment.Photon A is measured directly, while photon B passes through the noisy channel; the reduced system dynamics are non-Markovian.Source: B. The experiments · p. 2
- In the classical-noise experiment, a local bit-flip halfway through the channel restores initial entanglement without tagging individual ensemble states.The amount recovered depends on correlations among the environment-added random phases.Source: B. The experiments · p. 2
- In the quantum-environment experiment, measuring a third qubit selects and tags a system ensemble, after which a conditional local operation acts on photon B.The ensemble's average entanglement can equal the initial entanglement.Source: B. The experiments · pp. 2–3
C. Open-loop control
The open-loop experiment compares uncontrolled dephasing, phase correction, and a local echo in a four-step liquid-crystal channel. Correction restores the initial entanglement for any phase-correlation level, while echo recovery requires correlated noise and uses no environment information.
C. Open-loop control
- Four liquid-crystal retarders apply random phases to photon B, with the phase correlations set by µ.The photon undergoes stroboscopic pure dephasing; the experiment compares uncontrolled, corrected, and echoed dynamics.Source: C. Open-loop control · p. 3, Figure 1 · p. 3
- Uncontrolled entanglement decreases as accumulated phase grows, although each pure-state ensemble member remains maximally entangled.For µ = 1, the uncontrolled concurrence follows Cunco(k) = e^(−1/2 σ^2k^2).Source: C. Open-loop control · p. 3
- Ccorr(4) = 1 for any µ after a local phase shift cancels the known accumulated noise.The experiment uses LC4 as the correction step, setting its phase to x4 = −ϕ3.Source: C. Open-loop control · p. 3
- The echo pulse flips photon B after step k = 2, changing the sign of later acquired phases.This can cancel the earlier accumulated phase when the random phases are correlated.Source: C. Open-loop control · pp. 3–4
- Figure 2 compares measured entanglement of formation across steps for three µ values and the uncontrolled, corrected, and echoed dynamics.It also shows theoretical Bell-state curves and simulations for an input state with fidelity F = 0.96.Source: Figure 2 · p. 4
- The echo begins recovering entanglement at k ≥ 3; recovery is complete for µ = 1 and partial for 0 < µ < 1, without environment information.The pulse reverses the sign of phases acquired at steps 3 and 4, partially canceling the earlier accumulated phase when the phases are correlated.Source: C. Open-loop control · p. 4
- Experimental data show very good recovery for corrected and echoed dynamics, but corrected recovery at k = 4 falls short of its initial value.The authors attribute the shortfall to finite precision in setting the liquid-crystal phases.Source: C. Open-loop control · p. 4
D. Closed-loop control
In the closed-loop experiment, measuring the environment selects a system-state ensemble and the measurement outcome controls a local operation on photon B. The natural measurement basis restores the initial entanglement across all tested environment rotations, while recovery in a rotated basis depends on the measurement angle.
D. Closed-loop control
- Without control, tracing out O gives concurrence Cunco(p) = |1 − 2p|, and entanglement of formation decays monotonically over p in [0, 1/2].Figure 4a shows the uncontrolled theoretical curve and experimental points.Source: D. Closed-loop control · p. 5
- The closed loop projectively measures O in a rotated basis and applies an outcome-dependent unitary to B; O is encoded in photon B’s path degree of freedom.The environment first interacts with B through a CNOT gate.Source: D. Closed-loop control · pp. 4–5
- For the rotated measurement basis, recovered concurrence is Ccont(θ) = |cos(2θ)|; no entanglement is regained at θ = π/4.The natural basis is optimal for recovery.Source: D. Closed-loop control · p. 5
- At θ = 0, local control restores the initial entanglement for every p; the corrected output has concurrence Ccont(p) = 1.The result is shown by the corrected data in Figure 4a.Source: Figure 4 · p. 5, D. Closed-loop control · p. 5
- Experimental data deviate from ideal expectations, which the authors model using imperfect input-state fidelities F = 0.90 and F = 0.95.The deviations are attributed to imperfect state preparation by the SPDC source.Source: D. Closed-loop control · p. 5
Conclusion
The experiments demonstrate on-demand recovery of entanglement after interaction with local non-Markovian environments, using either a local echo pulse or measurement-conditioned local control. The conclusion connects recovery to hidden ensemble correlations and identifies possible applications in distributed architectures and QED systems.
Conclusion
- Both experiments recover entanglement on-demand through local operations after interaction with a local non-Markovian environment.The first uses a local echo pulse under classical low-frequency noise; the second measures an accessible quantum environment and uses its output to control a subsequent local operation.Source: Conclusion · p. 6
- In the second experiment, a measurement basis selects system states with the largest average entanglement, termed entanglement of assistance.The subsequent local operation recovers this maximum entanglement.Source: Conclusion · p. 6
- The first experiment’s local-recovery scheme is presented as the simplest instance of architectures based on networks of smaller mutually entangled subsystems.Such architectures are proposed to limit hardware-intrinsic decoherence associated with up-scaling.Source: Conclusion · p. 6
- The authors envisage applications of the second scheme in cavity and circuit QED, and of quantum control techniques in distributed architectures.They also frame the experiments as demonstrating entanglement recovery by local operations as a phenomenon related to non-Markovian open-system dynamics.Source: Conclusion · p. 6
E. Entanglement of formation
The section defines entanglement of formation through concurrence and gives the quantities used to calculate concurrence.
E. Entanglement of formation
- Entanglement of formation Ef is obtained directly from concurrence C(ρ).The section presents Ef as the entanglement measure used in the article.Source: E. Entanglement of formation · p. 6
- The function h(x) is defined as −x log2 x − (1 − x) log2(1 − x).It is part of the stated formula relating entanglement of formation to concurrence.Source: E. Entanglement of formation · p. 6
- Concurrence C(ρ) is the maximum of 0 and λ1 − λ2 − λ3 − λ4, where the λi are ordered square roots of specified matrix eigenvalues.The matrix is ρ(σy ⊗ σy)ρ⋆(σy ⊗ σy), with ρ⋆ the conjugate of ρ.Source: E. Entanglement of formation · p. 6
F. Open-loop control
The section models random phase noise and imperfect initial-state preparation, then calculates output-state entanglement for uncontrolled and echoed dynamics.
F. Open-loop control
- Random phases have standard deviation σ = 0.6 rad and average χ̄ = π, while the memory parameter μ sets the probability that each phase equals its predecessor.The passage identifies μ as the correlation coefficient between consecutive phases in a sequence.Source: F. Open-loop control · p. 6
- The input is modeled as a partially mixed state to account for imperfect preparation, with η related to its fidelity to |Ψ−⟩.The calculation also considers the ideal singlet input |Ψ−⟩⟨Ψ−|.Source: F. Open-loop control · p. 6
- For uncontrolled dynamics, phase averaging affects the output-state coherences ρoutbc, which are calculated by averaging the phase factor eiϕk.The passage identifies the noise as affecting only those coherences.Source: F. Open-loop control · p. 7
- For echoed dynamics, the calculation replaces the states used for k = 3, 4 and evaluates concurrence and entanglement of formation for the output state.The section describes concurrence formulas for uncontrolled and echoed dynamics.Source: F. Open-loop control · p. 7
G. Closed-loop control
The closed-loop experiment uses a folded interferometric setup and models its measurement-conditioned ensembles and imperfect state preparation. The section also describes theoretical concurrence calculations and experimental error estimates.
G. Closed-loop control
- The second experiment uses two Sagnac interferometers in place of two Mach–Zehnder interferometers and measures both exit modes on the same exit port.For angle θ, the settings are φ = 2θ for θu and φ = π − 2θ for θd.Source: G. Closed-loop control · p. 7
- The section gives the conditional pure-state ensembles for Q̄, Q̄′, and Q̄2(θ) as functions of θ and p.The listed states are combinations of |Ψ−⟩ and |Φ−⟩.Source: G. Closed-loop control · p. 7
- The initial AB-state fidelity to |Ψ−⟩ was estimated between 0.90 and 0.95 to model imperfect preparation.The passage attributes slightly lower measured entanglement at p = 1 than at p = 0 mainly to noise added by the CNOT on |Φ−⟩.Source: G. Closed-loop control · p. 7
- The section calculates concurrence for the uncontrolled and controlled output states using the same mixed input state.Vertical error bars for entanglement of formation propagate Poissonian errors from 36 tomography coincidence counts.Source: G. Closed-loop control · p. 7