Locality and universality of quantum memory effects
B.-H. Liu, S. Wißmann, X.-M. Hu, C. Zhang, Y.-F. Huang, C.-F. Li, G.-C. Guo, A. Karlsson, J. Piilo, H.-P. Breuer
LLocality and universality of quantum memory effects
B.-H. Liu, ∗ S. Wißmann, ∗ X.-M. Hu, C. Zhang, Y.-F. Huang, C.-F. Li, † G.-C. Guo, A. Karlsson, J. Piilo, and H.-P. Breuer ‡ Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China Physikalisches Institut, Universit¨at Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Turku Centre for Quantum Physics, Department of Physics and Astronomy,University of Turku, FI-20014 Turun yliopisto, Finland (Dated: September 26, 2018)Recently, a series of different measures quantifying memory effects in the quantum dynamicsof open systems has been proposed. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and itsenvironment which substantially simplifies its numerical and experimental determination, and fullyreveals the locality and universality of non-Markovianity in the quantum state space. We furtherillustrate the application of this representation by means of an all-optical experiment which allowsthe measurement of the degree of memory effects in a photonic quantum process with high accuracy.
PACS numbers: 03.65.Yz, 42.50.-p, 03.67.-a
In recent years the problem of characterizing non-Markovian dynamics in the quantum regime has initiatedan intense debate. A series of diverse definitions alongwith measures of quantum memory effects have been pro-posed, invoking many different mathematical and physi-cal concepts and techniques. Examples are characteriza-tions of non-Markovianity in terms of deviations from aLindblad semigroup [1], of the divisibility of the dynam-ical map [2], of the dynamics of entanglement [2] andcorrelations [3] with an ancilla system, and of the Fisherinformation [4].In this work we focus on the measure of non-Markovianity introduced in Refs. [5, 6] which defines non-Markovianity through the backflow of information fromthe environment to the open system. This informationbackflow is characterized by an increase of the distin-guishability of time-evolved quantum states. The distin-guishability of two quantum states ρ and ρ is quan-tified by their trace distance D ( ρ , ρ ) = Tr | ρ − ρ | [7–9]. We assume that the open system Hilbert spaces H is finite dimensional. The corresponding space of physi-cal states, represented by the convex set of positive op-erators with unit trace, will be denoted by S ( H ). Wefurther assume that the time evolution of the open sys-tem can be described by a 1-parameter family Φ [10] ofcompletely positive and trace preserving dynamical mapsΦ t , i.e. Φ = { Φ t | ≤ t ≤ T, Φ = I } . The non-Markovianity measure can then be defined as N (Φ) = max ρ ⊥ ρ (cid:90) σ> dt σ ( t, ρ , ρ ) , (1)where σ ( t, ρ , ρ ) ≡ ddt D (Φ t ( ρ ) , Φ t ( ρ )) (2)denotes the time derivative of the trace distance betweenthe pair of states at time t . In Eq. (1) the time integral is extended over all time intervals in which this derivativeis positive, and the maximum is taken over all pairs oforthogonal initial states ρ ⊥ ρ . This measure for non-Markovianity was originally defined in [5] in terms of amaximization over all pairs of quantum states in S ( H ).However, as demonstrated in Ref. [11] the maximizationcan be restricted to pairs of orthogonal initial states. Werecall that two quantum states ρ and ρ are said to be or-thogonal if their supports, i.e. the subspaces spanned bytheir nonzero eigenvalues are orthogonal which is equiva-lent to D ( ρ , ρ ) = 1 [9]. This implies that optimal statepairs exhibiting a maximal backflow of information dur-ing their time evolution are initially distinguishable withcertainty, and thus represent a maximal initial informa-tion content.Although the orthogonality of optimal states greatlysimplifies the mathematical representation of the non-Markovianity measure, its determination still requiresthe maximization over pairs of quantum states. Here,we derive a much simpler representation for the measurewhich is particularly relevant for its experimental real-ization since it only requires a local maximization oversingle quantum states, the second state being an arbi-trary fixed reference state taken from the interior of thestate space. This representation will further be employedin an all-optical experimental setup for the measurementof the non-Markovianity of a photonic process.To formulate our main theoretical result we first define˚ S ( H ) to be the interior of the state space, i.e. the set ofall quantum states ρ for which there is an ε > ρ with unit trace satisfying D ( ρ, ρ ) ≤ ε belong to S ( H ). We further define E ( H ) = { A | A (cid:54) = 0 , A = A † , Tr A = 0 } (3)to be the set of all nonzero, Hermitian and tracelessoperators on H . Considering any fixed reference state ρ ∈ ˚ S ( H ) we can now introduce a particular class of a r X i v : . [ qu a n t - ph ] M a r subsets of the state space: A set ∂U ( ρ ) ⊂ S ( H ) notcontaining ρ is called an enclosing surface of ρ if andonly if for any operator A ∈ E ( H ) there exists a realnumber λ > ρ + λA ∈ ∂U ( ρ ) . (4)Note that by definition ρ itself is not contained in ∂U ( ρ ) and that the full set ∂U ( ρ ) is part of the statespace. It can be easily seen that any state from the in-terior of the state space has an enclosing surface. Forexample, since ρ is an interior point of the state spacethere is an ε > ρ defined by D ( ρ, ρ ) = ε represents a spherical enclosing surface withcenter ρ . However, an enclosing surface ∂U ( ρ ) can havean arbitrary geometrical shape, the only requirement be-ing that it encloses the reference state in all directionsof state space. An example is shown in Fig. 1(a). Usingthese definitions, we can now state our central result. Theorem.
Let ρ ∈ ˚ S ( H ) be any fixed state of the in-terior of the state space and ∂U ( ρ ) an arbitrary enclos-ing surface of ρ . For any dynamical process Φ, the mea-sure for quantum non-Markovianity defined by Eq. (1) isthen given by N (Φ) = max ρ ∈ ∂U ( ρ ) (cid:90) ¯ σ> dt ¯ σ ( t, ρ, ρ ) , (5)where ¯ σ ( t, ρ, ρ ) ≡ ddt D (Φ t ( ρ ) , Φ t ( ρ )) D ( ρ, ρ ) (6)is the derivative of the trace distance at time t dividedby the initial trace distance. Proof.
Let ρ ∈ ∂U ( ρ ). Applying the Jordan-Hahndecomposition [9] to the operator ρ − ρ one concludesthat there exists an orthogonal pair of states ρ and ρ such that ρ − ρ = ρ − ρ D ( ρ, ρ ) , (7)and, hence, we have D (Φ t ( ρ ) , Φ t ( ρ )) = D (Φ t ( ρ ) , Φ t ( ρ )) D ( ρ, ρ ) , (8)by the linearity of the dynamical maps and the ho-mogeneity of the trace distance. This shows that σ ( t, ρ , ρ ) = ¯ σ ( t, ρ, ρ ). It follows that the right-handside of Eq. (5) is smaller than or equal to N (Φ) as definedby Eq. (1). Conversely, suppose ρ , ρ are two orthogonalstates. Since ρ − ρ ∈ E ( H ), there exists λ > ρ ≡ ρ + λ ( ρ − ρ ) ∈ ∂U ( ρ ), by definition of anenclosing surface. Thus, one obtains ρ − ρ = ( ρ − ρ ) /λ .Since ρ ⊥ ρ we find D ( ρ, ρ ) /λ = D ( ρ , ρ ) = 1 and,hence, λ = D ( ρ, ρ ). Thus, we are again led to Eq. (7)and to σ ( t, ρ , ρ ) = ¯ σ ( t, ρ, ρ ). This shows that the mea-sure N (Φ) as defined by Eq. (1) is smaller than or equal FIG. 1: (Color online) Illustration of an enclosing sur-face ∂U ( ρ ) (a) and of a hemispherical enclosing sur-face ∂ ˜ U ( ρ ) with disconnected boundary (b) for an in-terior point ρ of the state space S ( H ).to the right-hand side of Eq. (5) which thus concludesthe proof.The theorem bears several important mathematicaland physical consequences. First, it demonstrates thatthe non-Markovianity measure can be determined bymaximization over single quantum states ρ taken froman arbitrary neighborhood of a fixed state ρ in the in-terior of the state space. Thus, Eq. (5) provides a local representation of non-Markovianity, showing that quan-tum memory effects can be detected locally by samplingsingle states from an arbitrary enclosing surface of a fixedreference state. Note that the theorem cannot be appliedto infinite dimensional Hilbert spaces since ˚ S ( H ) is emptyin this case.Second, the choice of the fixed reference state ρ iscompletely arbitrary, the only condition being that it be-longs to the interior of the state space. Thus, the non-Markovianity of a dynamical process is indeed a universal feature which appears everywhere in state space: The in-formation about non-Markovian behavior is contained inany part of the state space which supports the intuitiveidea that quantum memory effects represent an intrinsicproperty of the dynamical process. This fact is particu-larly relevant when dealing with a dynamical process thathas an invariant state in the interior of the state space.It is then of great advantage to choose ρ as this invari-ant state such that only the sampled states ρ ∈ ∂U ( ρ )evolve nontrivially in time.Third, the theorem greatly simplifies the analytical,numerical or experimental determination of the non-Markovianity measure. In particular, it shows that itis not necessary to scan the whole state space in order tofind an optimal pair of quantum states but rather samplethe states of an enclosing surface of a fixed interior pointof the state space. From the proof of the theorem wealso see that it suffices if the enclosing surface containsall directions emanating from the fixed reference state ρ exactly once, i.e. if Eq. (4) holds for exactly one λ > A or − A . Therefore, the theorem is also valid if ∂U ( ρ ) is re-placed by a hemispherical enclosing surface ∂ ˜ U ( ρ ) whichwe define as follows. A set ∂ ˜ U ( ρ ) ⊂ S ( H ) is said to be ahemispherical enclosing surface of ρ if and only if for any A ∈ E ( H ) there exists exactly one real number λ > ρ + λA ∈ ∂ ˜ U ( ρ ) or ρ − λA ∈ ∂ ˜ U ( ρ ).A hemispherical enclosing surface thus contains all direc-tions, given by operators A ∈ E ( H ), only once. More-over, it needs neither be smooth nor connected (see Fig.1(b) for an example) which makes this characterizationparticularly useful for noisy experiments.We have applied the above theorem to determine thedegree of non-Markovianity in a photonic process. Theopen quantum system is provided by the polarizationdegree of freedom of photons coupled to the frequencydegree representing the environment. The experimentalsetup is depicted in Fig. 2. With the help of a frequencydoubler a mode-locked Ti:sapphire laser (central wave-length 780 nm) is used to pump two 1 mm thick BBOcrystals to generate the maximally entangled two-qubitstate ( | H, V (cid:105)−|
V, H (cid:105) ) / √ | H (cid:105) and | V (cid:105) denoting thehorizontal and vertical polarization states, respectively[12]. A fused silica plate (0 . σ = 7 . × Hzeach which are separated by ∆ ω = 7 . × Hz. Therelative amplitude A α of the two peaks depends stronglyon the tilt angle α whereas the other quantities are al-most constant. A polarizing beamsplitter (PBS) togetherwith a half-wave plate (HWP) and a quarter-wave plate(QWP) are used as a photon state analyzer [14].Photon 1 is directly detected in a single photon detec-tor at the end of arm 1 as a trigger for photon 2. Theoptical setup in part a , b and c (see Fig. 2) is used to pre-pare arbitrary quantum states of photon 2 needed for thesampling process [15]. This set-up conveniently allows toprepare any single pure photon polarization state (in arm2 c ) and reference states (2 a along with 2 b ) together witharbitrary enclosing surfaces which can be controlled bychanging the relative amplitudes of the attenuators builtin in each arm. The path difference between each armis about 25 mm to ensure that the mixture of the threeparts is classical.After the preparation photon 2 passes through birefrin-gent quartz plates of variable thickness which couple thepolarization and frequency degree of freedom and lead tothe decoherence of superpositions of polarization states.The birefringence is given by ∆ n = 8 . × − at 780 nm.The thickness of the quartz plates simulating differentevolution times ranges from 75 λ to 318 λ in units of thecentral wavelength of the FP cavity.Employing the Bloch vector representation, the set of E n t a ng l e m e n t s ou r ce HWP QWP FP IF attenuator QP BS mirror SPDPBS
FIG. 2: (Color online) Experimental setup. Key to thecomponents: HWP – half-wave plate, QWP – quarter-wave plate, FP – Fabry-P´erot cavity, IF – interferencefilter, QP – quartz plate, (P)BS – (polarizing) beam-splitter, SPD – single photon detector.polarization states can be conveniently parametrized bymeans of spherical coordinates r = ( r, θ, φ ). We applythe local representation to two reference states to deter-mine experimentally the degree of non-Markovianity forthree dynamics characterized by the relative amplitudes A α = 0 .
64, 0 .
22 and 0 .
01, ranging from non-Markovianto Markovian evolutions, and compare the results withthe outcome for pairs of orthogonal initial states. Thereference states ρ and ρ used in the experiment aregiven by r = (cid:0) . , π, π (cid:1) , r = (cid:0) . , π, π (cid:1) . (9)Reference state ρ is thus located inside the equato-rial plane, whereas the second reference state lies in thenorthern hemisphere close to the boundary. The enclos-ing surfaces are determined by the convex combination0 . · ρ a,b + 0 . · ρ of the reference states and any pure stateprepared in arm 2 c . These sets thus contain only mixedstates. We measured a total of 5000 states on the sur-face for each reference state which are characterized bythe azimuthal and polar angles of the pure states. Theassociated angles θ and φ are located on a lattice withequal spacing of 2 π/ λ and 318 λ for any state on the enclosing sur-face for the two reference states is shown in Figs. 3(a)-5(a) and 3(b)-5(b) using color coding. Note, that the col-ored surfaces in these figures are non-spherical and notcentered at the origin. By contrast, the ordinary Blochspheres depicted in Figs. 3(c)-5(c) show the measurementoutcomes for pairs of orthogonal initial states.Defining spherical coordinates r loc = ( r loc , θ loc , φ loc )with respect to local coordinate systems centered at theposition of the two reference states ρ and ρ , one recov-ers the polar symmetry present for pairs of orthogonal (a) (b) (c)(d) (e) (f) FIG. 3: (Color online) Experimental results for the in-crease of the trace distance between 175 λ and 318 λ for A α = 0 .
64 for states on the enclosing surface of refer-ence state ρ (a), ρ (b) and pairs of orthogonal states(c). The corresponding φ loc -averaged increase with re-spect to local spherical coordinates is shown in (d), (e)and (f). Error bars show the standard deviations. (a) (b) (c)(d) (e) (f) FIG. 4: (Color online) The same as Fig. 3 for A α =0 . φ loc along lines of lati-tude. To this end, we introduced an appropriate binningon the z -axis and determined the average increase whichwe then assigned to the azimuthal angle θ loc associatedto the mean z -value in the bin. In addition, we allo-cated the standard deviation to each of the averaged out-comes. The resulting data are depicted in Figs. 3(d)-5(d)and 3(e)-5(e) and show the same characteristics as the φ -averaged increase of pairs of orthogonal states displayedin Figs. 3(f)-5(f). Note that the directional dependenceof the trace distance originating from its property of de-pending only on the difference of two states can be nicelyseen for example in Figs. 3(d)-(f).The maximal increase of the trace distance for the tworeference states obtained from the φ loc -averaged data aswell as for pairs of orthogonal states are given in Tab. I.The experimentally determined values are in very good (a) (b) (c)(d) (e) (f) FIG. 5: (Color online) The same as Fig. 3 for A α =0 . A α N theo N (a) N (b) N (c) .
64 0 .
59 0 . ± .
01 0 . ± .
02 0 . ± . .
22 0 .
21 0 . ± .
01 0 . ± .
02 0 . ± . .
01 0 0 . ± . − . ± . − . ± . TABLE I: The quantum non-Markovianity measurefor the three dynamics obtained from the experimentaldata in comparison to the theoretical value.This work was supported by the National Basic Re-search Program of China (2011CB921200), the CAS,the National Natural Science Foundation of China(11274289, 11325419,11374288, 11104261, 61327901), theNational Science Fund for Distinguished Young Schol-ars (61225025), the Fundamental Research Funds for theCentral Universities (WK2470000011), the Academy ofFinland (Project 259827), the Jenny and Antti WihuriFoundation, the Magnus Ehrnrooth Foundation, and theGerman Academic Exchange Service (DAAD). S. W.thanks the German National Academic Foundation forsupport. ∗ These authors contributed equally to this work. † cfl[email protected] ‡ [email protected][1] M. M. Wolf, J. Eisert, T. S. Cubitt, J. I. Cirac, Phys.Rev. Lett. , 150402 (2008).[2] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev.Lett. , 050403 (2010).[3] S. Luo, S. Fu, and H. Song, Phys. Rev. A , 044101(2012).[4] X.-M. Lu, X. Wang, C.P. Sun, Phys. Rev. A , 042103(2010).[5] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. , 210401 (2009).[6] E.-M. Laine, J. Piilo, and H.-P. Breuer, Phys. Rev. A ,062115 (2010).[7] C. A. Fuchs and J. van de Graaf, IEEE Transactions on Information Theory , 1216 (1999).[8] M. Hayashi, Quantum Information (Springer-Verlag,Berlin, 2006).[9] M. A. Nielsen and I. L. Chuang,
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