Extracting Bayesian networks from multiple copies of a quantum system

Despite their theoretical importance, dynamic Bayesian networks associated with quantum processes are currently not accessible experimentally. We here describe a general scheme to determine the multi-time path probability of a Bayesian network based on local measurements on independent copies of a composite quantum system combined with postselection. We further show that this protocol corresponds to a non-projective measurement. It thus allows the investigation of the multi-time properties of a given local observable while fully preserving all its quantum features.

An interesting property of dynamic Bayesian networks is that they allow to specify the local dynamics of a composite quantum system conditioned on its global state.The Bayesian network formalism thus preserves all the quantum features of the system, especially quantum correlations [18] and quantum coherence [19].As a result, it permits to go beyond the standard two-projectivemeasurement scheme [20][21][22], which, owing to its projective nature, destroys off-diagonal density matrix elements.This characteristic has recently been exploited to derive fully quantum fluctuation theorems that not only account for the quantum nonequilibrium dynamics of a driven system [23], as in the two-projective-measurement approach, but also fully capture both quantum correlations and quantum coherence at arbitrary times [13][14][15].However, while a number of methods to implement the two-projective-measurement approach (and its variants) have been both theoretically developed [24][25][26][27] and experimentally demonstrated [28][29][30][31][32][33][34], to date, no such protocol exists for dynamic Bayesian networks.
In this paper, we introduce a general experimental scheme to extract dynamic Bayesian networks using identical copies of a quantum system.The use of multiple copies has been popularized in quantum information theory to achieve, for example, entanglement detection [35][36][37][38][39][40][41][42][43] or quantum state estimation [44][45][46][47][48][49], and has re-cently been considered in quantum thermodynamics to reduce quantum backaction [50,51].In the following, we first employ independent copies of a quantum system combined with postselection [52] to reconstruct the path probability of a dynamic Bayesian network.The latter quantity determines the multi-time properties of a given local observable without requiring full state tomography, which is in general extremely costly to realize [53].We moreover introduce a positive-operator-valued measure (POVM) [52] such that the path probability directly results from global measurements of correlated copies in a broadcast state [54].We further show that a no-go theorem for the characterization of work fluctuations in coherent quantum systems discussed in Ref. [50] does not apply to such a POVM.A well-defined nonequilibrium quantum work distribution may consequently be obtained for driven systems with initial coherence.We finally illustrate our findings by concretely evaluating the two-point path probabilities for a coherent qubit and for a quantum correlated pair of qubits.
Dynamic Bayesian networks.We consider an isolated quantum system initially prepared in a (global) state with spectral decomposition, ρ = s P s |s s|.The system may be multipartite or single-partite.During its unitary evolution, ρ t = U t ρU † t , the populations P s remain constant and the basis elements rotate from |s to |s t = U t |s .Let us now introduce arbitrary (local) basis sets {|x 0 }, {|x 1 }, . . ., {|x N } at (N + 1) specific points in time, t = t 0 , t 1 , . . ., t N (Fig. 1).These bases are not necessarily compatible with the bases {|s t } in the presence of quantum correlations or quantum coherence.
The central quantity of a dynamic Bayesian network is the joint probability distribution [1][2][3][4] 1) may determined by performing local measurements Mx on independent copies of a quantum system and postselecting the outcomes, Eq. ( 3).(c) Alternatively, one may obtain the same statistics by performing a global measurement on correlated copies prepared in a broadcast state, Eq. ( 4). (s, It is a proper probability distribution, in the sense that it is non-negative and all its marginals are non-negative.It also contains the complete information about the multi-time properties of the (local) variable x, while fully preserving the quantum features of the system, in contrast to the two-projective measurement scheme [20][21][22].It is the key quantity involved in the study of the nonequilibrium properties of small composite systems [5][6][7][8][9][10][11][12][13][14][15][16].We next describe an experimental protocol to determine Eq. ( 1) based on multiple identical copies of the quantum system and postselection.Experimental scheme.We begin, for simplicity, by treating the case of the two-point distribution P (x 0 , x 1 ) at time t 0 = 0 and a later time t 1 .To this end, we consider two independent copies ρ ⊗ ρ of the system.We assume, as done in the two-projective-measurement scheme, that the eigenbasis of the system has been determined.The protocol consists of two stages: In a first step, each copy is measured in the (global) eigenstate |s by applying the projector Π s ⊗ Π s with Π s = |s s|.This results in the state (Π s ⊗ Π s ) (ρ ⊗ ρ) = P 2 s Π s ⊗ Π s .In a second step, half of the copies are projected at t = t 0 in the (local) state |x 0 , while the second half is projected at t = t 1 in the (local) state |x 1 , for a given (global) state |s .The corresponding measurement operator reads The joint probability distribution P (x 0 , x 1 ) of the dynamic Bayesian network then follows by summing Eq. ( 2) over all (global) trajectories s.We emphasize that this protocol only relies on local measurements of each copy.Moreover, since two-projective-measurement experiments already determine distributions by repeating measurements on many identically prepared systems [28][29][30][31][32][33][34], the above scheme may be realized without much additional experimental effort.The generalization to an arbitrary sequence of times, t 0 , t 1 , • • • , t N , is straightforward.It involves (N + 1) in-dependent copies, ρ ind = ⊗ n ρ, and the measurement operator In this case, the multipoint joint probability distribution ( 1) is given by The path probability ( 1) is thus obtained from the conditional expectation value of M x on postselected states.Generalized measurement operators.The most general measurements in quantum theory are so-called positiveoperator-valued measures (POVMs) [52].Such quantum measurements may always be realized as ordinary projective measurements on an enlarged system [52].In order to derive the POVM corresponding to the measurement of the path probability (1), we note that Eq. (3) may be written as the expectation value where [54].Like the case of (N + 1) independent copies, this state has the property that if we take the partial trace over all except one of the subsystems, we always recover the original state ρ.Thus, locally, each copy is in state ρ, although, globally, they are in a quantumcorrelated state.The multipoint joint probability distribution (1) of a dynamic Bayesian network may therefore be evaluated either using independent copies and postselection or directly as the outcomes of the operator M x on a broadcast state of correlated copies.We now introduce a completely positive tracepreserving map, labeling the eigenstates of the system, such that the broadcast state can be constructed from (N + 1) independent copies as ρ bro = E(⊗ n ρ) [55].Using the cyclic property of the trace, we obtain with the positive semidefinite operators , they form a POVM [52].The set of operators J x define the general quantum measurement of the path probability (1) of a dynamic Bayesian network on (N + 1) independent copies.
Connection with a no-go theorem for quantum work.Reference [50] has recently examined general measurement schemes to evaluate the statistics of nonequilibrium work performed on coherent systems.In this instance, the observable x is the energy of the system and the work distribution is given as the expectation P (w) = Tr[(⊗ n ρ)W (w)], with the general work POVM The main conclusion of Ref. [50] is that no POVM exists such that (i) the average work corresponds to the difference of average energy for closed quantum systems (first law) and (ii) the work statistics agree with the two-projectivemeasurement method for states with no coherence in the energy basis (classical state limit), even if multiple copies are accessible.In other words, it does not seem possible to simultaneously obey the first law of thermodynamics and respect the classical-state limit in coherent systems.However, this result is based on the assumption that the measurement operator does not depend on the state ρ, that is, no information about the initial state is available.By contrast, we have here shown that the Bayesian-network approach allows the determination of the joint probability distribution (1), and, in turn, of the nonequilibrium work distribution for coherent (as well as correlated multipartite) systems, by relaxing this restriction and assuming that the eigenbasis of the system has been determined.In a sense, the hypothesis of state independence, which was based on a universality argument, thus seems too strong.As a matter of fact, even the evaluation of the classical work statistics along single trajectories in stochastic thermodynamics presupposes knowledge of the driven potential [17].In addition, there exists many quantum protocols that require information about the eigenbasis of the system, from the two-projectivemeasurement scheme [20][21][22] to optimal cloning [60,61] and quantum parameter estimation [62,63].
Compared with other methods to specify quantum work distributions, such as the two-projectivemeasurement scheme [20][21][22], the work-operator formalism [56,57] or the quasiprobability approach [58,59], the dynamic-Bayesian-network framework [13,14,16] appears to be currently the only one leading to quantum work distributions that (i) are measurable, that is, are described by a POVM, (ii) satisfy nonequilibrium fluctuation theorems and (iii) apply to coherent systems (Table I) [50].It hence comes across as a powerful tool to study nonequilibrium quantum processes of composite systems.
Examples.We finally illustrate our results by computing the two-point path probability (1) for two paradigmatic thermodynamic examples for work extraction [64] and heat exchange [65]: a driven coherent qubit and a FIG. 2. Path probability P (+, +), Eq. ( 6), for a driven coherent qubit as a function of temperature T for various values of the parameter a.The Bayesian network results (a = 0) are generally smaller than that of the two-projective measurement scheme (a = 0), except for very low temperatures.They further plateau at a constant value for large temperatures.correlated pair of qubits at two different temperatures.
A) Driven coherent qubit.We consider the minimal example of a qubit with Hamiltonian H t = g t σ z , whose gap is adiabatically varied from g t0 to g t1 .The system is assumed to be initially in state ρ = ρ th +ασ x , with parameter α and thermal distribution ρ th = exp(−βg 0 σ z )/Z; here Z = Tr[exp(−βg 0 σ z )] denotes the partition function at inverse temperature β = 1/T .The qubit exhibits coherences in the energy basis when α = 0.As a consequence, the eigenbasis |s ± differs from the energy basis   [20][21][22], work operators [56,57], quasiprobabilities [58,59] and Bayesian networks [13,14,16].Only the latter one yields work densities that are measurable, obey fluctuation relations and apply to coherent systems.

Fluctuation Coherent Measurable theorems processes Projective measurements
Operators of work Quasiprobabilities Bayesian networks abilities of the dynamic Bayesian network are and a similar expression for P (−, −) with b replaced with −b.Both formulas fully account for the presence of coherence (a = 0) in contrast to the two-projectivemeasurement approach, in which the first measurement would completely destroy coherences, effectively setting a = 0.As a consequence, this would invariably lead to and, analogously, P (−, −) TPM = exp(+βg 0 )/Z. Figure 2 shows that quantum coherence significantly affects the joint distribution P (+, +), except for very low temperatures: P (+, +) is in general smaller than P (+, +) TPM and plateaus at a constant value at high temperatures.B) Correlated pair of qubits.We next consider a pair of qubits AB in the initial global state with Taking the expectation value over two independent copies ρ AB ⊗ ρ AB , we eventually find where we have defined ξ = 2a 2 +1 and γ = exp(2∆β)ξ+1.Equation ( 9) entirely captures quantum correlations (a = FIG.3. Path probability P (+−, −+), Eq. ( 9), for a pair of correlated qubits AB as a function of the temperature TB for various values of the parameter a (and constant TA = 0.4).The Bayesian network results (a = 0) strongly differ from that of the two-projective measurement scheme (a = 0), except for very low temperatures, and exhibit nonmonotonic behavior.
0) between the two qubits at t 0 and t 1 , contrary to the two-projective-measurement result to which it reduces for a = 0. Figure 3 displays the behavior of P (+−, −+) as a function of T B for fixed T A .We observe that quantum correlations have a nontrivial (nonmonotonic) influence on the path probability (1).These effects vanish again in the limit of low temperatures.
Conclusions.We have introduced a general experimental scheme to extract a dynamic Bayesian network from multiple copies of a multipartite quantum system.We have specifically shown how to determine the multipoint path probability (1) from local measurements of independent copies combined with postselection.This joint probability density characterizes the multi-time properties of a given local observable, fully including quantum coherence and quantum correlations, without requiring state tomography.We have further argued that this protocol may be regarded as a global generalized measurement and derived the corresponding POVM.In view of its versatility, the present method can be implemented on many experimental platforms, including nuclear magnetic resonance [28], trapped ions [29], cold atoms [31] and superconducting qubits [32].We thus expect it to find broad applications in diverse fields, ranging from quantum many-body physics and quantum information theory to nonequilibrium quantum thermodynamics.

FIG. 1 .
FIG. 1.(a) Diagrammatic representation of the unitary evolution of a (possibly composite) quantum system with (global) states |st = Ut|s , and the set of all possible (local) paths |x0 → |x1 → |x2 which can be associated with this evolution.(b) The path probability P (x0, x1, • • • , xN ), Eq. (1) may determined by performing local measurements Mx on independent copies of a quantum system and postselecting the outcomes, Eq. (3).(c) Alternatively, one may obtain the same statistics by performing a global measurement on correlated copies prepared in a broadcast state, Eq. (4).

TABLE I .
Comparison of different approaches to characterize work fluctuations in driven quantum systems: two projective measurements