Experimental on-demand recovery of entanglement by local operations within non-Markovian dynamics

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.

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.
Our scope is distributed networks of spatially separated quantum nodes, each subject to a local environment. They model the structure of physically relevant distributed architectures. Remarkably, when environments induce a system dynamics which can be physically unraveled into an ensemble of entangled pure state evolutions, it may happen that quantum correlations of initially entangled states are not destroyed. Indeed even if entanglement appears degraded when measured on the averaged state, it is not lost but rather hidden in the lack of classical knowledge about the elements of the ensemble 22 . In this case, leveraging the existence of such classical information, the initially shared entanglement can be restored at an arbitrary time without resorting to non-local operations.
An operational scenario emerges in which local controls can be used for on-demand restoration of entanglement in distributed architectures. Here we demonstrate this concept by two all-optical experiments. We consider two entangled photons subject to local environments as an instance of distributed entanglement. In the first experiment a local environment produces low-frequency noise 23 , and it is shown how by local open loop control 24 the initial entanglement is recovered, even if in the absence of control it would be degraded to very low values. In the second experiment, decoherence is due to the coupling between a subsystem (photon) and a quantum environment, whose degrees of freedom are experimentally accessible. A measurement on this environment and a subsequent conditional local operation on the system implement a local closed-loop control 25 , succeeding in restoring the initial entanglement. The relevance of these setups to different physical systems is discussed at the end of this work.
The non-Markovian nature of the dynamics 26,27 is a key ingredient for the entanglement recovery in our experiments. Certain physical aspects are common to other phenomena involving non-Markovianity, such as spontaneous entanglement revivals during the system dynamics [28][29][30][31][32][33][34] , or entanglement preservation by dynamical decoupling [35][36][37] . Here, at variance with these examples, we engineer the overal dynamics by a local control, and we demonstrate that distributed entanglement can be fully restored on-demand by suitable local operations even though it would vanish in the absence of active control.

Results
We consider a prototype distributed quantum network implemented by an all-optical setup where the information carriers are two photons, A and B. The information is encoded in the photon polarization, being either horizontal jHae, or vertical jVae. The system AB is prepared in the two-photon entangled Bell state Theoretical framework. Typically the information available at time t on the system AB is encoded in the reduced density matrix r(t), obtained after tracing over the environment. It can be decomposed, in an infinite number of ways, in terms of pure states jn(t)ae, each one occurring with a probability p n (t): where E is some measure, reducing for pure states to the entropy of entanglement 11,12 . In this work we study the entanglement of formation 11,12 E f , but we discuss our results in terms of the concurrence C, an entanglement measure with a simpler and very intuitive form Ref. 38.
Note that E f monotonically depends on the concurrence, and can be readily calculated from it as E f (C), see Methods.
In the absence of interaction with the environment, the density matrix does not evolve r(t) 5 r(0) 5 jY 2 aeAEY 2 j, whereas due to the interaction of the qubits with the environment, r(t) evolves in a statistical mixture. The corresponding amount of entanglement is no longer equal to that of the initial state jY 2 ae.
Here we demonstrate that, acting by suitable local controls, the entanglement initially present in the system AB can be restored. This is possible if r(t) corresponds to a specific physical decomposition Q t ð Þ: p n t ð Þ , n t ð Þ j i À Á È É , and we are somehow able to tag each member of the ensemble Q t ð Þ and to know its state. In this case we can distill the average entanglement of Q by using local operations and classical communication only. For any convex measure E of entanglement, we have that E Q §E r . The inequality has a natural meaning, namely the classical knowledge of the state of each member of the quantum ensemble Q t ð Þ allows for a larger entanglement with respect to situations where this information is not available.
The experiments. In the two experiments we present here, the twophoton state AB is generated by a spontaneous parametric downconversion (SPDC) source of photon pairs based on Ref. 39. In both experiments, the degraded entanglement E r (t), Eq.(1), is restored by a local control, whose effect is to make available the entanglement E Q t ð Þ, Eq. (2). For the sake of simplicity photon A is measured directly whereas photon B is sent through a noisy channel before being measured. We address situations where the evolution of the system AB is non-unitary, due to the interaction with a local environment O inducing a non-Markovian dynamics of AB 30,40 .
In a first experiment we simulate classical non-Markovian noise through a sequence of liquid crystal retarders which add random phases to photon B during propagation (pure dephasing). As a result, the initial entanglement of the system AB decays monotonically as a function of the channel length. Despite we are aware of the physical decomposition in terms of pure states n t ð Þ j iand probabilities p n t ð Þ, we are not able to tag each state n in the ensemble. Therefore the entanglement of the average state is given by Eq. (1). The action of a bit-flip on photon B applied at half-way propagation along the channel restores the initial entanglement (open-loop control). Therefore such a local control is able to retrieve the classical information on each member of the quantum ensemble Q t ð Þ, thus allowing to make available the average entanglement E Q , Eq. (2). The amount of entanglement recovered by this technique depends on the degree of correlations (non-Markovianity) among the environment-added random phases.
In a second experiment, we simulate a quantum environment by a third qubit, O, interacting with qubit B through a controlled-NOT (CNOT) gate. The coherent exchange of information between AB and the environment O roots the emergence of memory effects and of a non-Markovian evolution of the reduced system AB. Measurement of the environment in a given basis makes possible the physical selection of a given quantum ensemble Q t ð Þ, allowing at the same time to tag each state of this ensemble. The average entanglement of Q t ð Þ (which can be equal to the initial entanglement) is then obtained, by applying a local operation on qubit B, which depends on the (classical) information gained on the environmental state (closed-loop control).
Both experiments and their results are described in detail in the following subsections.
Open-loop control. In this experiment, the photon B interacts with a classical environment O described by a stochastic process x(t). The corresponding noisy channel acting on B is designed to induce pure dephasing, according to the Hamiltonian: where s z 5 jHaeAEHj 2 jVaeAEVj and d is the Dirac delta function. Here, the interaction between B and O takes place stroboscopically at times t k (see Fig. 1 a). This Hamiltonian is experimentally realized by sending the photon through a sequence of four liquid crystal retarders (LC k ), each one introducing a phase x k ; x(t k ) between the photon polarization components: The phase x k depends on the voltage V k applied to LC k and can be varied continuously from 0 to p (see Fig. 1 b). To simulate the stochastic process, we generate a set of N random phase sequencesx~x 1 ,x 2 ,x 3 ,x 4 f g , in which the phases x k are Gaussian random variables with the same variance s 2 , and correlations m ; AEx k x k11 ae/s 2 , m g [0, 1] (see Methods). The system dynamics averaging with respect to these sequences is obtained by mixing together the tomographic measurement data obtained with each of the N random phase sequences.
We investigated three different situations: a) the uncontrolled dynamics, where we simply look at the entanglement degradation resulting from the noisy channel; b) the controlled dynamics where we show how accessing classical information on the environment allows to operate corrections which fully restore the entanglement, and c) the echoed dynamics where entanglement is recovered by a simple local operation with no need to have access to the classical information.
Uncontrolled dynamics. The reduced dynamics of AB induced by noise is in this case unambiguously described by the quantum ensemble with Q k~X k j~1 x j the overall phase accumulated up to step k. As a result, each state of the ensemble is maximally entangled, so that also the average entanglement (2) is maximum for any k: E Q k ð Þ~1. However, the entanglement (1) of the average state, E r (k), exhibits quite a different behavior. The system concurrence for the uncontrolled dynamics decays with k: in the case m 5 1 (full correlations), theory gives C unco k ð Þ~e { 1 2 s 2 k 2 . For m , 1 the concurrence, though more involved (see Methods), shows a similarly decaying behavior. In the experiment we measured the entanglement E r (k) obtained for three different values of m with the generated sets of random phase sequencesx. For k , 4, the LC i with i . k are set at a constant phase p 2 instead of x i . The experimental (black symbols) and theoretical results (black lines) are presented in Fig. 2. These results show that the system entanglement decreases as the accumulated phase Q k grows.
Corrected dynamics. In this case, we know the induced noisex (since we generate it), and we can compensate it. Indeed, all that is needed is to insert another LC (LC corr ) after the channel and to apply to it a voltage so as to produce a phase x corr 5 2Q 4 , see Fig. 1. In practice, having only four LCs available, we used LC 4 as the correction step and set it at a phase x 4 5 2Q 3 . For such corrected dynamics the initial state is fully recovered, together with its entanglement: C corr (4) 5 1, for any m, see blue symbols (experiment) and blue lines (theory) in Fig. 2. This demonstrates that the channel-induced degradation of entanglement is only due to a lack of classical knowledge: once Q 3 is known, entanglement is recovered by a local phase-shift operation.
Echoed dynamics (open-loop control). Accessing information on the environment is not possible in practice in real quantum networks, where noise sources correspond to a large number of uncontrollable environmental degrees of freedom. Nevertheless, an open loop scheme operated by local control may still allow to recover the entanglement. To demonstrate this we use a control technique introduced in NMR 41 : just after the step k 5 2, we apply a local bit-flip operation (U echo 5 s x ) on photon B, which flips its polarization: s x jHae 5 jVae and s x jVae 5 jHae, see Fig. 1 a. Experimentally we insert a half-wave plate at 45u between LC 2 and LC 3 when measuring the entanglement at k 5 3, 4, see Fig. 1 b. Our purpose is to induce an entanglement echo in the system dynamics. The state (3) which describes the system in each run of the experiment for k $ 3 now reads: The local pulse is equivalent to a change of sign of the phases acquired at steps k 5 3, 4, and it may tend to cancel the effect of Q 2 5 x 1 1 x 2 if the four x i phases are to a certain extent correlated. In general the echo pulse is expected to favor the recovery of the average state entanglement, especially for an environment with non-Markovian correlations. The plots in Fig. 2 show that, indeed, the entanglement for the echoed dynamics starts to increase at k $ 3, see red symbols (experiment) and red lines (theory). It is also clear that entanglement recovery strongly depends on correlations. For m 5 1 (Fig. 2 a), the characteristic time scale of the environment dynamics is much larger than that of the system dynamics. In this case full entanglement recovery is possible: x 3 1 x 4 5 2Q 2 and we have that (red line), see Methods. Partial recovery 2). In that case, x 3 1 x 4 cancels Q 2 only partially, and we have to deal with a bit more involved expression for the concurrence (red lines), see Methods. Let us stress that, whereas in the corrected dynamics we used the knowledge of the sequencex to cancel the accumulated random phase, in the echoed dynamics the (partial) cancellation of the phase needs no knowledge of the environment. Our experimental data reported in Fig. 2 show a very good entanglement recovery, both for the corrected and the echoed dynamics. Deviations from the ideal expectations are mainly due to the imperfect preparation of the input state: in Fig. 2 we also plot (dashed lines) the expected entanglement for a mixed input state with a (measured) fidelity F Y {~Y { r in j jY { h ĩ0:96 to the ideal input state jY 2 ae, see Eqs. (10) and (11) in Methods. Black and red dashed lines refer respectively to the uncontrolled and echoed entanglement, which are derived from Eqs. (18) and (19) in Methods. For the corrected dynamics (blue squares), the recovered entanglement at step k 5 4 does not quite reach the initial entanglement at k 5 0, this is due to the finite precision we have on setting the phases for each LC. In the same way, for the echoed dynamics, the slight discrepancy between the measured (red triangles) and the theoretical (red dashed line) values for k $ 3 is due to small differences between the phase responses of each LC.
We remark that, although the system dynamics we consider is non-Markovian 26,27 , no spontaneous entanglement revival occurs in the uncontrolled dynamics. Non-Markovianity remains undetected by any measure based on the non-monotonicity of entanglement 40 . The local control triggers a reverse flow of classical information from the environment to the system 42 . This also makes non-Markovianity be phenomenologically evident and detectable by measures which are based on the non-monotonicity of some suitable quantity 27 .  ð Þ is experimentally implemented by a balanced beamsplitter and an attenuation filter in both its output ports, with a ratio p9 5 p/(1 2 p) between their respective intensity transmission coefficients; the CNOT gate G BO is implemented by an half-wave plate at 45u in the ''down'' path of photon B, see Fig. 3 b.
Uncontrolled dynamics. The state of ABO after the BO interaction is: where W { j i~HH j i{ VV j i ð Þ . ffiffi ffi 2 p is another Bell state. After tracing out the environment, the system is described by r out p ð ÞT 2pj. No further operation is applied on B before we measure the entanglement of AB. The corresponding entanglement of formation E f depends on p, exhibiting a monotonous decay in the range [0, 1/2], see the black curve (theory) and points (experiment) in Fig. 4 a.
Controlled dynamics. We show how it is possible to recover entanglement by a ''closed-loop'' like control scheme. It consists of two steps: 1) after the gate G BO , O is projectively measured in the basis , and 2) depending on the outcome k, a unitary operation U B k is performed on B, U B u~1 l B and U B d~s xB , see Fig. 3 a. Experimentally, the selection of the O measurement basis is obtained by a Mach-Zehnder interferometer with a variable phase Q 5 2h which realizes the rotation R' O h ð Þ~e ihsx O , see Fig. 3 b. The unitary U B k is obtained by doing nothing further on the output path jh u ae and inserting a half-wave plate at 45u in the path jh d ae. The entanglement in AB is measured by mixing the polarization tomography data obtained on both output paths. Note that we actually used a folded version of the set-up presented in Fig. 3 b, see Methods and Fig. 3 c.
The fact that we can distinguish the modes k 5 u, d emerging from the second interferometer, allows to associate on a physical basis the ensemble to the state of the system. Moreover, we tag the actual state of the ensemble during each run of the experiment. This classical information enables us to apply some suitable local control which depends on the actual state. The goal is to obtain a new quantum ensemble , whose corresponding density matrix, r' out , exhibits the average entanglement of Q. We investigate two special cases. First we take h 5 0 (the measurement basis of O is thus its natural basis), yielding the quantum ensemble for any value of p. This entanglement is restored by the local control which flips the photon B polarization each time the system is the state jW 2 ae. This produces the output state , whose concurrence C cont (p) 5 1. We plot the corresponding entanglement obtained after the control loop as a function of p in Fig. 4 a, see violet line (theory) and triangles (experiment): the restoration of the initial entanglement is achieved for any value of p when the measurement basis angle is h 5 0. As a second example, we consider the case in which the entanglement for the uncontrolled dynamics vanishes, which happens for p 5 0.5. We show that the amount of entanglement recovery depends on the measurement performed on O, that is to say on the angle h of the rotated measurement basis. This measurement selects the system AB quantum ensemble Þ , since the concurrence of both states Q h k is jcos(2h)j. The O measurement by itself does not produce any effect on the AB entanglement, which is vanishing for any h in that case, see black line (theory) and points (experiment) in Fig. 4 b. Instead application afterwards of the local control on the qubit B may lead to recovery of the average entanglement of the ensemble, Q 2 h ð Þ. Indeed, the resulting output state for AB is whose concurrence is C cont (h) 5 jcos(2h)j, see the pink line (theory) and triangles (experiment) in Fig. 4 b. Here we see that the natural basis of O is the optimal measurement basis to recover the entanglement, whereas for h~p 4 no entanglement is regained.
The experimental data (symbols) plotted in Fig. 4 show some deviation from the ideal expectations (continuous line). To explain them, we take into account the imperfect state preparation by our SPDC source: we also plot on Fig. 4 the expected entanglement for a mixed input state whose fidelity with the ideal input state is F Y {~0 :90 (dotted line) or 0.95 (dashed line), see Methods. Note that in Fig. 4 a, the entanglement measured for p 5 1 is slightly lower than for p 5 0 (violet triangles) mainly because of the noise added by the CNOT on jW 2 ae.

Discussion
In this article we have presented two experiments showing that recovery of entanglement after the interaction of the members of an entangled state with a local non-Markovian environment can be achieved on-demand through local operations only. In the first experiment, the local environment is classical and induces low frequency noise, so that entanglement may be recovered by a local echo pulse. In the second experiment, the environment is a small quantum system (here a qubit) accessible to measurement, whose output controls a subsequent local operation on the system: as a result, entanglement recovery is obtained.
We note that in the second experiment there exists a measurement basis that selects system states with the largest amount of average entanglement. This maximum entanglement, recovered by the subsequent local operation, is known as entanglement of assistance 43,44 , a concept of which we provide here an experimental illustration. Instead the first experiment is a prototype of situations where the environment is not accessible in practice, typically because a large number of uncontrolled degrees of freedom are involved. Nevertheless, the recovery of entaglement by local operations is possible, since entanglement is not actually destroyed, but rather hidden due to a lack of classical information about which element in an ensemble of entangled states we are dealing with 22 .
Interestingly, the scheme of our first experiment is the simplest instance of recently proposed architectures, which should limit the detrimental up-scaling hardware-intrinsic decoherence, namely networks of smaller mutually entangled subsystems. Striking examples are trapped ions where modes in a single trap become dense as the trap becomes larger, acting as stray degrees of freedom, besides exquisitely solid-state systems as electrostatically defined quantum dots, silicon-based implanted impurities and optically active dopants 10 . Very effective on-demand recovery by local operations is possible when noise has strong low-frequency components. We stress that this is a very relevant physical case, 1/f noise being a major drawback for solid state quantum computation 10,23 , and it is likely to be an important source of decoherence in practical computation and communication networks. Direct applications of the scheme of our second experiment can be also envisaged in cavity quantum electrodynamics (QED) 45,46 or in circuit QED systems 47 , where two-level quantum systems are strongly coupled to discrete photon modes in high-quality cavities. Finally we remark that this work, besides experimentally demonstrating novel physics related to non-Markovianity of open quantum systems, namely entanglement recovery by local operations, envisages new applications of quantum control techniques [48][49][50][51][52] to distributed architectures.

Methods
Entanglement of formation. As entanglement measure E r we use the entanglement of formation E f 11,12 , which can be directly obtained from the concurrence C(r) 38 by the following formula: where h(x) 5 2x log 2 x 2 (1 2 x) log 2 (1 2 x) and C(r) 5 max{0, l 1 2 l 2 2 l 3 2 l 4 }.
Here l i (l i $ l i11 ) are the square roots of the eigenvalues of the non-Hermitian matrix r s y 6s y À Á r Ã s y 6s y À Á , r Ã being the conjugate of r.
Open-loop control. Random phases. The random variables x k are Gaussian with a standard deviation s 5 0.6 rad and an average x: x k h i~p 2 , AE?ae being the ensemble average. Since we disposed of half-wave LCs, which could only generate phases in [0, p], this setting provided that 99% of the random phases were inside the experimentally achievable interval. Each sequence of random variables x~x 1 ,x 2 ,x 3 ,x 4 f g were generated (using Scilab random number generation function ''grand'') by keeping x i equal to x i21 with probability m and x i independent from x i21 with probability 1 2 m.
The degree of memory inx may be quantified by m, which corresponds to the correlation coefficient 53 between x k and x k11 inside each sequence: Imperfect initial state. To take into account the effect of the imperfect state generated by our SPDC source, we consider the following partially mixed state as input sate: where input state 1 4 1l 4 is the maximally mixed input state and the mixing parameter g is related to the fidelity Theoretical calculation of the entanglement. We calculate the output state of the twophoton system when the initial state is r in 5 jY 2 aeAEY 2 j.