Experimental observation of weak non-Markovianity
Nadja K. Bernardes, Alvaro Cuevas, Adeline Orieux, C. H. Monken, Paolo Mataloni, Fabio Sciarrino, Marcelo F. Santos
EExperimental observation of weak non-Markovianity
Nadja K. Bernardes , Alvaro Cuevas , Adeline Orieux , , C. H. Monken , PaoloMataloni , Fabio Sciarrino , and Marcelo F. Santos Departamento de F´ısica, Universidade Federal de Minas Gerais, Belo Horizonte, Caixa Postal 702,30161-970, Brazil Dipartimento di Fisica, Sapienza Universit`a di Roma, Roma 00185, ItalyE-mail: [email protected]
Abstract.
Non-Markovianity has recently attracted large interest due to significant advances in itscharacterization and its exploitation for quantum information processing. However, up to now, only non-Markovian regimes featuring environment to system backflow of information (strong non-Markovianity)have been experimentally simulated. In this work, using an all-optical setup we simulate and observe the so-called weak non-Markovian dynamics. Through full process tomography, we experimentally demonstratethat the dynamics of a qubit can be non-Markovian despite an always increasing correlation betweenthe system and its environment which, in our case, denotes no information backflow. We also show thetransition from the weak to the strong regime by changing a single parameter in the environmental state,leading us to a better understanding of the fundamental features of non-Markovianity.
Introduction
The development of quantum technologies for information processing, communication and highresolution metrology among other applications has renewed the interest in a better understanding ofthe dynamics of open quantum systems. The most typical description of an open system evolution isthat of a Markovian dynamics caused by the memoryless interaction of a given quantum system with itsenvironment [1]. On the other hand, strong system-environment interaction, environment correlationsor initial system-environment correlations may cause memory effects rendering the dynamics non-Markovian. Recently, non-Markovian dynamics has become a very trendy topic mainly due to thedevelopment of new experimental techniques for controlling and manipulating solid state and manybody systems [2, 3, 4, 5, 6] and also due to its possible applications in information protection andprocessing [7, 8, 9, 10, 11, 12].Commonly, the evolution of a system is defined as Markovian if the corresponding quantum mapis divisible in other completely positive maps (from now on CP maps), i.e. Λ t , = Λ t , t Λ t , for all t ≥ t ≥ Present address: T´el´ecom ParisTech, CNRS LTCI, 46 rue Barrault, F-75634 Paris Cedex 13, France a r X i v : . [ qu a n t - ph ] D ec arkovianity [17] as opposed to a weak non-Markovianity that requires full process tomography and,therefore, is much more difficult to detect.It is important to understand the effects caused by such reservoirs in quantum computational systems,such as qubits, and to observe these effects in controlled experimental setups in order to pinpoint theessential mechanisms behind the different time evolutions generated by them. Hence, very recently, non-Markovianity has been investigated in different setups and contexts such as the control of the initial statesof the environment [18, 19], of its interaction with the system [20, 21] or combinations of them [22], aswell as the observation of non-Markovian effects in simulated many-body physics [23] or in the recoveryof quantum correlations [24, 25]. All these experiments are restricted to detecting and/or exploring thestrong non-Markovianity.In this work we carry out an experimental characterization of the transition between weak andstrong non-Markovianity. More specifically, the term weak non-Markovian will be used for dynamicsrepresented by maps that are divisible in positive maps, but not in completely-positive maps. On theother hand, strong non-Markovian will be used to maps that are not even divisible in positive maps.In particular we adopted full process tomography to observe the weak non-Markovianity dynamics ofa qubit subjected to the interaction with a correlated environment. Furthermore, through the carefulpreparation of the environment state, the transition is induced by changing a single experimentalparameter. Theoretical model
We consider a qubit ρ s that interacts with an environment from which it is initially decoupled. Theinteraction consists of consecutive collisions each of which can produce three different effects on thesystem: either nothing happens, in which case the identity is applied to ρ s or the system is rotated by π / X or Z axis, undergoing a X s ≡ σ sx or Z s ≡ σ sz flip. The map that describes the evolution byone collision is Λ ( · ) = p ( · ) + p x X ( · ) X + p z Z ( · ) Z and { p , p x , p z } are probabilities that add to one.This is a special case of a random unitary qubit evolution. It is also a unital map.All the effects we are interested in can be observed with only two collisions described, in our case,by the general map Λ ( · ) = ∑ mn p mn O n O m ( · ) O m O n where 0 ≤ p mn ≤ ∑ mn p mn = O = , O x = X s and O z = Z s . Note that if the collisions are fully independent (hence non-correlated) thismodel is Markovian by construction and can be easily generalized for any number of collisions. Thestate of the system after n collisions is obtained by the concatenation of single collision CP maps: ρ s ( n ) = Λ n ρ s ( ) = ( Λ ) n ρ s ( ) . In this case, the joint probabilities of two consecutive flips need torespect relations such as p i j = p i p j where i , j = { x , z } .The dynamics becomes more interesting if the collisions are correlated, i.e. when p i j (cid:54) = p i p j . Inparticular, it is shown in Ref. [26] that for any correlation factor Q = p xx + p zz − p xz − p zx p xx + p zz + p xz + p zx larger than zero, thetwo-collision map Λ represents a non-Markovian evolution, i.e. Λ (cid:54) = Λ Λ or, equivalently, ρ s ( ) cannot be obtained by applying a CP map on ρ s ( ) . Naturally, larger values of Q produce a more intensenon-Markovian effect. There is, however, a transition in the type of non-Markovianity that dependson the probabilities of the flips. If correlated flips are very likely, i.e. p ii ( i = { x , z } ) is of the sameorder of p i , p i and p , then the non-Markovianity is strong (sometimes referred to as “essential” inthe literature [17]) which means it can be witnessed by quantities such as the trace distance betweendifferent states of the system or the entanglement between the system and an ancilla. This is related tothe fact that the reconstructed map Λ = Λ Λ − is not even positive, let alone CP, i.e. it does not mapthe Bloch ball onto a set contained in it. The extreme scenario has p xx = p zz = / ρ s afterthe first collision will be given by ρ s ( ) = X ρ s ( ) X + Z ρ s ( ) Z and after the second collision it goesback to ρ s ( ) (since X = Z = ).As the flip probabilities decrease this effect becomes smaller and at some point the reconstructedmap Λ becomes positive, albeit still non-CP. For random unitary maps, as it is the case here and asit is shown in Ref. [27], a map is divisible in positive maps if and only if the von Neumann entropyand the trace distance present a monotonic decay in time, establishing a strict relation between backflow igure 1. (a) Conceptual scheme of the experiment. A maximally entangled state between the qubitof interest ( s ) and an ancillary state ( a ) is initially prepared at time t . The interaction between qubit s and the environment is simulated by a sequence of two channels, performing each a mixture of , X and Z operations. The state after the interaction channel CH ( CH ) is measured at t ( t ). In theactual experiment the two channels are simulated each one by two liquid crystal modulators. Qubit a will not suffer any change, since it is isolated from the environment. (b) Sequence of probabilitiescorresponding to the action of the two channels, p = ( − ε ) , p x = p z = p z = p x = ( − ε ) ε ,and p xx = p zz = ε . X i ( Z i ) is the X ( Z ) operation occurring in the i th-collision. Here we adopted thespecific value of ε = . X or Z ) are chosen to maximize the effect but, in fact,any pair of non-commuting flips will also produce a non-Markovian map for Q > CH and CH are the interaction channels acting on thesystem. Fig. 1b) shows the probabilities associated to the sequence of operations performed by CH and CH . Experimental setup
In the experiment, the system s is the polarization state of an initial maximally entangled photonpair, | ψ as (cid:105) = ( | HV (cid:105) + e i α | V H (cid:105) ) / √ | H (cid:105) ( | V (cid:105) ) represents the horizontal (vertical) polarization (see Fig. 2 for details)[28]. Note that | i (cid:105) a ⊗ | j (cid:105) s will be simply represented as | i j (cid:105) . The environment is simulated by a sequence igure 2. Detailed scheme of the experiment. Twin photons are created by a polarization entanglementsource. One photon (system s ) is sent through a correlated liquid crystal environment, while the other(ancilla a ) is let to go free. Then, the bipartite state is measured by complete state tomography at times t , t and t . Liquid crystals (LCs) (two for CH and two for CH ) act as phase retarders, with the relativephase between the ordinary and extraordinary radiation components depending on the applied voltage.of four voltage controlled liquid crystal cells (LC) lying on the path of photon s . By a suitable controlof the applied voltage on each of them, the four LCs were set to operate either as the identity or ashalf-wave plates. In particular the first and third LCs were oriented with the slow axis along the verticaldirection, thus acting either as or Z , while the slow axis of the second and fourth LCs were orientedalong the diagonal direction (45 ◦ ), thus acting either as or X . According to the collision model,the first environment (giving ρ as ( ) as output result) derives from the actuation of the first two LCs(1 and 2) only, leaving the other LCs (3 and 4) in the identity regime. Finally, the second collisionenvironment (giving ρ as ( ) as output result) corresponds to the actuation of the four LCs in the designedway. The parameter ε gives the probability that either X or Z occurred in the first collision ( p x = ε and p z = ε ) and it is proportional to the time of application of the voltage to the liquid crystal. In theexperimental setup it is defined by the ratio of the width of an applied voltage pulse to the width of ameasurement cycle. Since the photons in our source are generated at random, the action of the liquidcrystal cells driven by the voltage pulses will in fact simulate random collisions. By controlling thetime duration of the applied voltage on each LC intercepting photon s , it was possible to choose theprobability corresponding respectively to the , X and Z operations. This is given by the parameter ε in the following way: p = ( − ε ) , p x = p z = p z = p x = ( − ε ) ε , and p xx = p zz = ε , asshown in Fig. 1b). Note also that p xz = p zx = Q =
1) are implemented. In this case, the theory predicts an always non-Markovian dynamics for anyvalue of ε >
0. In order to verify the dynamical behavior, the measurements were performed after thefirst collision ( t ) (in that case LCs 3 and 4 were acting as the identity and only LCs 1 and 2 were varied)and the second collision ( t ) (with all four LCs varied). As described below, in the case of strong non-Markovianity, we need just to measure an entanglement witness. However, for weak non-Markovianity,quantum state tomography should be realized. igure 3. (a) The negative eigenvalue of H as a function of ε . The inset is the same curve for small valuesof ε , ε < .
1. For qualitative reasons, the weak non-Markovian regime, where the intermediate map ispositive, but not completely positive, is represented by the blue region and the strong non-Markovianregime, where the intermediate map is not even positive, by the red region. The experimental errorbars are estimated from propagation of the Poissonian statistics of photon coincidence countings in thetomographic reconstruction of the process matrix. (b)
The difference between the concurrences of systemand ancilla after two collisions C (2) and one collision C (1) versus ε . The inset shows C (2) and C (1) versus ε plotted separately. For qualitative reasons, the weak non-Markovian regime is represented by the blueregion and the strong non-Markovian regime by the red region. The transition from one region to theother is represented by the dashed line, which can vary its position depending on the imperfections inthe experiment. The experimental error bars are estimated as explained before. The uncertainties, about1%-3% of the concurrence values, are within the size of the symbols.In the experiment the open system dynamics is obtained by temporally mixing the three possiblesettings of CH ( , X , or Z ), giving ρ as ( ) , and by temporally mixing the seven possible settings ofthe action of CH and CH (namely , X1 , , Z1 , XX , ZZ , ), giving ρ as ( ) . Note that onlycorrelated rotations are implemented; this is done in order to maximize the non-Markovian effect as it isbetter explained in Ref. [26]. The map Λ ( Λ ) is obtained from the full tomographic reconstructionof ρ as ( ) ( ρ as ( ) ) and the intermediate map Λ that tells us about the character of the dynamics iscalculated from Λ = Λ Λ − . Non-Markovian analysis
The density matrix of a qubit state can be represented by ρ = ( + (cid:126) r · (cid:126) σ ) /
2, where (cid:126) r is the Bloch vector( r i = Tr ( ρσ i ) ). The action of a map Λ on ρ can be described, in general, by Λ : (cid:126) r (cid:55)→ (cid:126) r (cid:48) = M (cid:126) r + (cid:126) t , where M is a matrix responsible for changing the norm and rotating the Bloch vector while (cid:126) t = ( t x , t y , t z ) shifts itsorigin. For unital maps ( (cid:126) t = × H = (cid:0) + ∑ µ , ν = x , y , z M µ ν σ µ ⊗ σ ∗ ν (cid:1) / Λ is completely positive iff H ≥ λ of H for the map Λ obtained from theexperimental tomographic reconstruction of the system state after one and two collisions ( ρ s ( ) and ρ s ( ) respectively). The fact that λ is always negative for ε > Λ is non-CP. As a consequence, the dynamics of the system is non-Markovian for any value in this range. Thequestion remains whether the measured non-Markovianity is weak or strong. A linear map is positiveiff the corresponding dynamical matrix H is block-positive [30]. For the type of map implemented inour experiment, a simple calculation (see Methods) shows that the condition that guarantees the strongnon-Markovian regime, which means that H is not block-positive, necessarily implies the recoveryf entanglement between the system affected by the environment and an ancilla used to monitor thedynamics. Therefore, in order to search for an eventual transition from weak to strong non-Markovianregime in our case, we have also measured the variation of the entanglement between the system s and an ancilla qubit a after the first and the second collision. This entanglement decreases when thesystem becomes more correlated with the reservoir and vice-versa, therefore, it identifies properly anybackflow of information from the latter to the former. We quantify entanglement by measuring theconcurrence C [32] between system and ancilla and in Fig. 3b) we plot its difference after one andtwo collisions, C ( ) − C ( ) , as a function of ε . The values of C ( ) and C ( ) are obtained from thetomographic reconstruction of the two-qubit density matrices and are ploted in the inset of Fig. 3b).Fig. 3b) shows a transition from a negative to a positive difference at around ε = .
3. Note that positivedifference ( C ( ) > C ( ) ) means that system and ancilla are more entangled after two collisions thanafter one which, in our model, identifies strong non-Markovianity. As ε decreases, the evolution of thesystem will mimic a system that gets more correlated to the reservoir after the second collision (andtherefore less entangled with the ancilla) which defines the regime of weak non-Markovianity where Λ ’s non-CP character can only be evidenced by full tomography of the map itself. Also note thatthe theoretical curve predicts a discontinuity in the derivative of C ( ) − C ( ) and a plateau for a rangeof values of ε . Both behaviors are easily explained if we look at the individual behaviors of C ( , ) plotted in the inset of Fig. 3b). Both entanglements suddenly die [33], C ( ) faster than C ( ) , but C ( ) remains zero for any larger value of ε while C ( ) eventually recovers due to the environmentalcorrelations. The regions of weak and strong non-Markovianity (blue and red regions, respectively) arealso presented in Fig. 3a); however, here there is no clear sign of the transition between one region tothe other, so, if just the divisibility of the maps is calculated, there is no essential difference betweenthese two types of non-Markovianity. The theoretical curves are computed assuming imperfectionsin the preparation of the initial state ( C ( ) ∼ . X exp ( ρ ) = F X ρ X + ( − F ) / Y ρ Y + ( − F ) / Z ρ Z and Z exp ( ρ ) = F Z ρ Z + ( − F ) / Y ρ Y + ( − F ) / X ρ X , and we considered F = . Discussion
We have realized experimentally the collisional model proposed in Ref. [26] to investigate the non-Markovian dynamics of an open quantum system. We showed how the evolution of the same photonicsystem can transit from strong to weak non-Markovian evolution by varying only one parameter.This effect is caused by simply modulating the probability of the photon to undergo a rotation on itspolarization state. As a result, a particular kind of non-Markovianity which is normally not spotted inother experiments is observed here. All non-Markovianity is caused solely by simulating a correlatedreservoir. Finally, the fact that both regimes are produced by the same underlying physical mechanismexplicitly shows that there is nothing necessarily fundamental about strong non-Markovian evolutions.Besides its intrinsic relevance on the fundamental side, the weak non Markovianity experimentallydemonstrated in this work allows to envisage future important applications regarding for instancequantum control techniques and resolution enhancement in quantum metrology [9, 10].
Methods
The positivity and completely positive character of our map can be identified by the dynamical matrix H . For the intermediate map Λ and error models previously explained, the reconstruction of H gives H = h h h h h h h h , (1)here h ( ε , F ) = ( F ( F − ) + ) ε + ( F − ) ε + ( F − ) ε + , h ( ε , F ) = − ε ( ( F ( F − ) + ) ε + F − ) ( F − ) ε + , h ( ε , F ) = ( F − ) ε ( ( F − ) ε + ) ( F − ) ε + , h ( ε , F ) = ε ( (( F − ) F + ) ε + F − ) + ( F − ) ε + . Its eigenvalues are λ = λ = ( F + ) ε , (2) λ = − ε (cid:0) F ε − F ε + F + ε − (cid:1) F ε − ε + , λ = F ε − F ε + F ε + ε − ε + F ε − ε + . First, notice that, considering that our operations are almost perfect ( F = . λ < Λ is not completely positive. However, we would like to identify for which values of ε the map becomes positive. A map is positive if the dynamical matrix is block positive, i.e. λ i + λ j ≥ λ i are the eigenvalues of H [30]. It is possible to show that the only inequality that does not satisfythis condition is λ + λ . For the case of perfect operations ( F = ( − ε ) ≥ C ( ) = Max [ , − ε ] , (3) C ( ) = Max [ , + ε (( F ( F − ) + ) ε − )] . (4)For perfect operations ( F = ( − ε ) ≥
0) will definitely imply C ( ) − C ( ) <
0. A monotonic decay of the concurrence will identifythen when the map is positive, and on the other hand, non-positivity ( ( − ε ) ≤
0) implies an increasein the concurrence. However, when the operations are not perfect, there is a range of values of ε forwhich C ( ) = C ( ) , and this witness will fail to identify the exact point of transition from weak tostrong non-Markovianity. Nevertheless, it will still be true that if C ( ) < C ( ) the map is positive (weaknon-Markovian) and if C ( ) > C ( ) the map is not positive (strong non-Markovian). Acknowledgements
N.K.B, C.H.M. and M.F.S. would like to thank the support from the Brazilian agencies CNPq andCAPES. M.F.S. would like to thank the support of FAPEMIG, project PPM IV. This work is part ofthe INCT-IQ from CNPq. A.C. would like to thank the support of Comision Nacional de Investigaci´onCient´ıfica y Tecnol´ogica (CONICYT). F.S. acknowledges support from CNR Programa ProfessorVisitante do Exterior (PVE). We acknowledge support from the European Research Council (ERC)Starting Grant 3D-QUEST (grant agreement no. 307783).
Author contributions statement
N.K.B., C.H.Monken, and M.F.Santos proposed the original idea, F.S. and P.M. designed theexperimental setup. A.C. and A.O. performed the experiment, N.K.B., A.C., and A.O. analyzed andinterpreted the data. All authors contributed to the writing of the manuscript. dditional informationCompeting financial interests:
The authors declare no competing financial interests.
License:
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