PRL 115, 010404 (2015)

week ending 3 JULY 2015

PHYSICAL REVIEW LETTERS

Robust Asymptotic Entanglement under Multipartite Collective Dephasing 1

Edoardo G. Carnio,1,2,* Andreas Buchleitner,1,3 and Manuel Gessner1,†

Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany 2 Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 3 Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstraße 19, 79104 Freiburg, Germany (Received 4 February 2015; published 2 July 2015) We derive an analytic solution for the ensemble-averaged collective dephasing dynamics of N noninteracting atoms in a fluctuating homogeneous external field. The obtained Kraus map is used to specify families of states whose entanglement properties are preserved at all times under arbitrary field orientations, even for states undergoing incoherent evolution. Our results apply to arbitrary spectral distributions of the field fluctuations. DOI: 10.1103/PhysRevLett.115.010404

PACS numbers: 03.65.Yz, 03.65.Ud, 03.67.Mn

Control of the coherent evolution of quantum systems in noisy environments [1] is one of the crucial prerequisites for exploiting nontrivial quantum effects in composite systems of increasing complexity. Whether in the context of controlled molecular reactions [2], of many-particle quantum dynamics [3], or of quantum computers and simulators [4], uncontrolled fluctuations and noise are detrimental to most purposes of optimal control. Various strategies may be followed to counteract the harmful influence of the environment: shielding the system degrees of freedom [5], correcting environment-induced errors [6], exploiting basins of attraction in dissipative systems [7], or compensating dissipation— e.g., by coherent dynamics [8], dynamical decoupling [9], or periodic measurements [10]. Such approaches can effectively reduce the environmental effects and can enhance coherence times, but a perfect protection of the quantities of interest is generally not possible. By restricting to superposition states within a decoherence-free subspace, initially entangled states can be shielded completely from collective noise sources, hence protecting their entanglement at all times [11,12]. However, such subspaces are rather fragile to small perturbations, which limit their applicability in the context of dynamical processes [13]. Here, we identify conditions that ensure complete preservation of arbitrary degrees of entanglement, even for states that are not invariant under an incoherent time evolution. Specifically, we consider an important class of environment-induced fluctuations, which are frequently encountered in state-of-the-art experiments [14–16]: they manifest in intensity fluctuations of spatially homogeneous experimental control fields, giving rise to an effective dephasing process. We show how control of the external field’s orientation can lead to the complete preservation of entanglement in bipartite—as well as multipartite— settings, for arbitrary spectral characteristics of the control field fluctuations. We further identify families of states exhibiting time-invariant entanglement for arbitrary orientations of the external field. 0031-9007=15=115(1)=010404(5)

To set the stage, let us consider a collection of N noninteracting atomic two-level systems with identical energy splitting ℏω controlled, e.g., by a homogeneous magnetic field. Integration over the unavoidable fluctuations of the latter’s strength will induce a probability distribution pðωÞ of the characteristic energy splitting, and the N-atom quantum state at time t therefore needs to be described by the statistical operator Z ρðtÞ ¼ pðωÞUω ðtÞ⊗N ρð0ÞU†ω ðtÞ⊗N dω; ð1Þ provided the field fluctuations occur on time scales which are longer than the time t over which the N-atom state is propagated by the unitary Uω ðtÞ⊗N. In order to assess the open-system time evolution of the quantum correlations inscribed into the N-atom system, it is convenient to derive an explicit expression for ρðtÞ in terms of the spectral distribution pðωÞ characterizing the fluctuations. The single-atom propagator U ω ðtÞ ¼ e−iHω t=ℏ is generated by the time-independent single-atom Hamiltonian Hω ¼ ðℏω=2Þn · σ, with σ ¼ ðσ x ; σ y ; σ z Þ being the vector of the Pauli matrices and n the orientation of the field. Hω describes atomic dipoles interacting with electromagnetic fields as, e.g., the electronic qubits in trapped-ion quantum registers [14,17]. Introducing pairs of orthogonal projectors Λ ¼ ðI2  n · σÞ=2, we can rewrite the time evolution operator for a collection of N atoms as U ω ðtÞ⊗N ¼ ðe−iωt=2 Λþ þ eiωt=2 Λ− Þ⊗N ¼

N X

eiωtðj−N=2Þ Θj ;

ð2Þ

j¼0

where we have defined the operators

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Θj ¼

X 1 V ½Λ⊗j ⊗ Λ⊗N−j V †s ; þ j!ðN − jÞ! s∈Σ s −

ð3Þ

N

© 2015 American Physical Society

PHYSICAL REVIEW LETTERS

PRL 115, 010404 (2015)

week ending 3 JULY 2015

where ΣN denotes the symmetric group and V s ¼ P i1 …iN jisð1Þ …isðNÞ ihi1 …iN j represents the permutations s ∈ ΣN in the operator space of N qubits. The ensembleaveraged state after time t, ρðtÞ ¼

N X

Mjk ðtÞΘj ρð0ÞΘk ;

ð4Þ

j;k¼0

is then fully characterized by the Toeplitz matrix MðtÞ, whose elements M jk ðtÞ ¼ φ½ðjR− kÞt are generated by the characteristic function φðtÞ ¼ pðωÞeiωt dω of the probability distribution pðωÞ. Bochner’s theorem [18] ensures that MðtÞ is a Hermitian semipositive definite matrix for all t. Diagonalization leads to the canonical Kraus form [19] ρðtÞ ¼ ϵt;0 ½ρð0Þ ¼

N X

Ai ðtÞρð0ÞA†i ðtÞ;

ð5Þ

i¼0

P pffiffiffiffiffiffiffiffiffi where the Kraus operators Ai ðtÞ ¼ Nj¼0 λi ðtÞ λij ðtÞΘj contain the eigenvalues λi ðtÞ and the components of the eigenvectors λi ðtÞ of MðtÞ. Note, from the structure of Ai ðtÞ, that the above defined Kraus operators mediate an effective interaction between the individual qubits—with its origin in the spatial homogeneity of the external field. These environment-induced interactions are able to create discord-type quantum correlations [17] and, as we will show in this Letter, given the appropriate control of n, can uphold multipartite entanglement at all times for arbitrary intensity fluctuations. Using the fact that the operators Λ are orthogonal projectors in C2 , we can immediately show that both of the operators fΘj gj and fAi ðtÞgi satisfy the condition P † P † i Ai ðtÞAi ðtÞ ¼ i Θi Θi ¼ I2N , which ensures that the map ϵt;0 defined in (5)—from now on called the “collective dephasing” map—is not only completely positive but also trace preserving for all t [19,20]. For absolutely continuous distribution functions [24], the characteristic function vanishes asymptotically; i.e., limt→∞ φðtÞ ¼ 0. We then have that limt→∞ MðtÞ ¼ INþ1 , and thus the Kraus operators reduce to limt→∞ Ai ðtÞ ¼ Θi . The asymptotic P N-qubit state is thus given by ρs ¼ limt→∞ ρðtÞ ¼ i Θi ρð0ÞΘi . Because the operators Θi depend exclusively on the magnetic field direction n, the latter completely determines the properties of the asymptotic state. To gain some intuition on the time evolution of the entanglement properties, e.g., of an N-ion quantum register under the action of the collective dephasing map, we first consider two-qubit states with maximally mixed reduced density matrices (also called Bell-diagonal states). Such states allow for a simple geometric representation since they are fully characterized by the matrix βij ¼ trðρ · σ i ⊗ σ j Þ [21]. There always exist unitary operations UA and UB such that UA ρU †B has a diagonal β matrix, β ¼ diagðd1 ; d2 ; d3 Þ, while ρ and UA ρU†B have the same

FIG. 1 (color online). Dynamics of a bipartite system with the initial state ρ0 ¼ I4 =20 þ 4jΦ− ihΦ− j=5. (a) The evolution of the state under Lorentzian- (green) and box-distributed (red) noise, C0;1 ðωÞ and B0;1 ðωÞ (see the text for a definition), respectively, pffiffiffi with a magnetic field direction n ¼ ð2; 1; 1ÞT = 6, is depicted inside the tetrahedron of Bell-diagonal states. The inner octahedron marks the set of separable states. (b) Depending on the orientation of the magnetic field, the concurrence remains constant T [for n ¼ ð1; pffiffi0; ffi 0Þ , orange], decays to a finite value [forT np¼ ffiffiffi T ð2; 1; 1Þ = 6, blue], or decays to zero [for n ¼ ð0; 1; 1Þ = 2, red]. The decay is monotonic for Lorentzian (dashed lines), or nonmonotonic for box-distributed (solid lines) noise.

separability properties [27]. This allows us to associate with each density matrix a point d ¼ ðd1 ; d2 ; d3 ÞT ∈ R3 and, because of positivity, any such point must lie inside a tetrahedron [Fig. 1(a)] of vertices ð−1; −1; −1ÞT , ð−1; 1; 1ÞT , ð1; −1; 1ÞT , and ð1; 1; −1ÞT , which represent the four Bell states [21]. Inside this tetrahedron we distinguish an inner octahedron, which contains the separable states, from the four remaining corners, which consist of the entangled states [21] and are labeled by the Bell state they contain (e.g., jΨ− i corner). In this setting Wootters’s concurrence [22] is simply the distance P from the faces of the octahedron: CðdÞ ¼ 1=2 maxf0; i jdi j − 1g. Equidistant points, parallel to the surfaces of the octahedron, form the “isoconcurrence” planes. In the tetrahedron, the collective dephasing evolution is always constrained onto a plane defined by TrβðtÞ ¼ Trβð0Þ [20]. In the jΨ− i corner, these planes coincide with isoconcurrence planes, which implies that entanglement is preserved for all of these states for arbitrary directions of the magnetic field. This leads to a finite-measure set of states with time-invariant concurrence, despite the fact

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PRL 115, 010404 (2015)

PHYSICAL REVIEW LETTERS

that those states do evolve in time, ρðtÞ ≠ ρð0Þ [29]. For entangled states outside the jΨ− i corner, we can use Eq. (4) to predict concurrence as P3 the final 2 Cd;f ðnÞ ¼ 1=2 maxf0; i¼1 ð1 − 2ni Þdi − 1g, where n ¼ ðn1 ; n2 ; n3 ÞT and d ¼ ðd1 ; d2 ; d3 ÞT characterizes the initial state [20]. Thus, by solving for n, we can always find a field direction such that the entanglement is preserved at all times. This can be seen from the longtime limit in Fig. 1(b), whereas the transient time evolution depends on pðωÞ, as we will discuss later. Collective interactions become particularly relevant in multipartite settings, where decoherence and dissipation can be strongly enhanced [11,33–35]. To analyze the effect of collective dephasing on multipartite entanglement, analytic expressions à la Wootters [22] are not available. Intricate hierarchies of multipartite entanglement [36] can, however, be characterized efficiently by resorting to separability criteria based on inequalities [37,38]. An N-partite state ρ is called k separable if it can be written as a mixture of states of the form ρ ¼ ρA1 ⊗ … ⊗ ρAk , where A1 …Ak label a division of the N parties into k subgroups. For instance, the matrix elements in an arbitrary basisPof any k-separable N-qubit P density matrix ρ satisfy 0≤i