Quantifying Spatial Correlations of General Quantum Dynamics

Understanding the role of correlations in quantum systems is both a fundamental challenge as well as of high practical relevance for the control of multi-particle quantum systems. Whereas a lot of research has been devoted to study the various types of correlations that can be present in the states of quantum systems, in this work we introduce a general and rigorous method to quantify the amount of correlations in the dynamics of quantum systems. Using a resource-theoretical approach, we introduce a suitable quantifier and characterize the properties of correlated dynamics. Furthermore, we benchmark our method by applying it to the paradigmatic case of two atoms weakly coupled to the electromagnetic radiation field, and illustrate its potential use to detect and assess spatial noise correlations in quantum computing architectures.

Introduction.-Quantum systems can display a wide variety of dynamical behaviors, in particular depending on how the system is affected by its coupling to the surrounding environment. One interesting feature which has attracted much attention is the presence of memory effects (non-Markovianity) in the time evolution. These typically arise for strong enough coupling between the system and its environment, or when the environment is structured, such that the assumptions of the well-known weak-coupling limit [1][2][3] are no longer valid. Whereas memory effects (or time correlations) can be present in any quantum system exposed to noise, another extremely relevant feature are correlations in the dynamics of different parts of multi-partite quantum systems. Since different parties are commonly (but not always) identified with different places, we will refer to these correlations as spatial correlations in the following.
Developing measures to quantify up to which extent dynamics deviates from Markovian behavior has been the aim of numerous works along the last few years, see e.g. [34][35][36][37][38][39][40][41][42]. However, much less attention has been paid to develop quantifiers of spatial correlations in the dynamics [43,44]. In this work, we aim at filling this gap and propose a theoretical framework to classify and quantify the degree of spatial correlations of multipartite quantum dynamics. Specifically, i) we formulate a general measure for spatial correlations of quantum dynamics without resorting to any specific physical model. To this aim we adopt a resource theory approach, and formulate a fundamental law that any faithful measure must satisfy. ii) Within this framework, we study the properties that a dynamics has to fulfill to be considered as maximally correlated. We find that it is not possible to provide a unique definition in the standard sense of a resource theory. Thus, as an alternative, we motivate one of them and characterize the dynamics complying with it. iii) We apply our measure to the paradigmatic quantumoptical model of two two-level atoms radiating into the electromagnetic vacuum. This case exemplifies the working principle of our measure and quantitatively confirms the expectation that spatial dynamical correlations decay with increasing interatomic distance and for long times. iv) Finally, we illustrate the utility of this formalism in the context quantum computing, where protocols for fault-tolerant quantum information processing rely on certain assumptions on (typically sufficiently small) noise strengths and noise correlations. Specifically, we consider two qubits subject to local thermal baths that suffer some residual interaction which induces correlated noise. Our method reveals the remarkable fact that, under keeping the overall error probability for the two qubits constant, the degree of spatial correlations decays very rapidly as the bath temperature increases. This suggests that, in some situations, noise addition as e.g. by a moderate increase of the environmental temperature, can be beneficial to tailor specific desired noise characteristics. This dynamics is said to be uncorrelated with respect to the subsystems A and B if it can be decomposed as E S = E A ⊗ E B , with CPT maps E A and E B acting on A and B, respectively. Otherwise it is said to be correlated.
In many cases, and with some a priori information about the dynamics, correlated dynamics can be identified by measuring simple correlations between two observables X and Y acting on H A and H B respectively. Indeed, any correlation detected during the time evolution of an initial product state, ρ S = ρ A ⊗ ρ B , witnesses the correlated character of the dynamics. However, since there are correlated maps which do not generate any correlation, further methods are required if we intend to assess completely the degree of dynamical correlation. For that purpose the Choi-Jamio lkowki isomorphism [45,46] turns out to be a useful tool. This is, consider a second d 2 −dimensional bipartite system S = A B , and let |Φ SS be the maximally entangled state between S and S , Here, |j denotes the state vector with 1 at the j-th position and zero elsewhere (canonical basis). The Choi-Jamio lkowki representation of some CPT map E S on S is given by the d 4 −dimensional state where 1 S denotes the identity map acting on S . Construction of the correlation measure.-In order to formulate a faithful measure of spatial correlations for dynamics, we adopt a resource theory approach [47][48][49][50][51][52][53][54]. This is, we may consider correlated dynamics as a resource to perform whatever task that cannot be implemented solely by (composing) uncorrelated evolutions E A ⊗ E B . Then, suppose that the system S undergoes some dynamics given by the map E S , and consider the composition of E S with some uncorrelated map E A ⊗ E B , so that the total dynamics is given by It is clear that any task that we can do with E S by composition with uncorrelated maps can also be achieved with E S by composition with uncorrelated maps. Hence, we assert that the amount of correlation in E S is at least as large as in E S . In other words, the amount of correlations of some dynamics does not increase under composition with posterior uncorrelated dynamics. This is the fundamental law of this resource theory, and any faithful measure of correlations should satisfy it. For the sake of comparison, in the resource theory of entanglement, entanglement is the resource, and the fundamental law is that entanglement cannot increase under application of local operations and classical communication (LOCC) [47].
In this spirit, we introduce a measure of correlations for dynamics via the (normalized) quantum mutual information of the Choi-Jamio lkowski state ρ CJ S , Eq. (2), with S the von Neumann entropy evaluated for the reduced density operators ρ CJ S | AA := Tr BB (ρ CJ S ) and ρ CJ S | BB := Tr AA (ρ CJ S ), and ρ CJ S ; see Fig. 1. The quantitȳ I(E S ) is a faithful measure of how correlated the dynamics given by E S is, as it satisfies the following properties: This follows from the fact that the Choi-Jamio lkowski state of an uncorrelated map is a product state with respect to the bipartition AA |BB [55].
S is a maximally entangled state with respect to the bipartition AA |BB , leading to I(ρ CJ S ) = 2 log d 2 .
iii) The fundamental law is satisfied, where the equality is reached for uncorrelated unitaries This result follows from the monotonicity of the quantum mutual information under local CPT maps, which in turn follows from the monotonicity of quantum relative entropy [56].
Maximally correlated dynamics.-Before computingĪ for some examples it is worth studying which dynamics achieve the maximum valueĪ max = 1 and in what sense they can be called maximally correlated.
From the resource theory point of view, maximally correlated evolutions would be the ones which can generate any arbitrary dynamics by posterior composition with uncorrelated operations -again in analogy to maximally entangled states which are the ones that allow one to prepare any state by applying only LOCC on them. However, such a dynamics, E max S , does not exist: by hypothesis E max S would be able to generate any dynamics, in particular any unitary evolution U S , S . This would imply that both E max S and (E A ⊗E B ) are unitary evolutions as well, so that However, sinceĪ(E S ) is invariant under the composition of uncorrelated unitaries, this result would imply that for any correlated unitary U S ,Ī(U S ) would take the same value [Ī(U max S )], and this is not true as can be easily checked. Therefore this definition is too restrictive.
Alternatively we may adopt the criterion to define maximally correlated maps as those that maximize the degree of correlation Eq.
is a pure state. Therefore E S must be unitary as the Choi-Jamio lkowski state is pure if and only if it represents a unitary map.
Despite the connection with maximally entangled states, the set of maximally correlated operations C := {U S ;Ī(U S ) = 1}, can not be so straightforwardly characterized as it may seem. Note that not all maximally entangled states |Ψ (AA )|(BB ) are valid Choi-Jamio lkowski states. In the supplementary material we provide a detailed proof to the next theorem. Theorem 2. A unitary map U S ∈ C if and only if it fulfills the equation Examples of maximally correlated dynamics are the swap operation exchanging the states of two qubits A and B, U S = U A↔B , and thus also any unitary of the form of (U A ⊗U B )U A↔B . However, not every U S ∈ C falls into this class. For example, the unitary U S = |21 12| + i(|11 21|+|12 11|+|22 22|) belongs to C and it cannot be written as (U A ⊗U B )U A↔B , sinceĪ(U S U A↔B ) = 1/2 = 0. Example: Two-level atoms in the electromagnetic vacuum.-To illustrate the behavior ofĪ(E S ), consider the paradigmatic example of two identical two-level atoms with transition frequency ω interacting with the vacuum of the electromagnetic radiation field. Under a series of standard approximations [55], the dynamics of the reduced density matrix of the atoms ρ S is described by the master equation where σ z j is the Pauli z-matrix for the j-th atom, and σ + j = (σ − j ) † = |e j g| the electronic raising and lowering operators, describing transitions between the exited |e j and ground |g j states. The coefficients a jk depend on the spatial separation r between the atoms. In the limit of r 1/ω they reduce to a jk γ 0 δ jk , whereas for r 1/ω they take the form a jk γ 0 . Here γ 0 is the decay rate of the individual transition between |e and |g [55]. In the first regime the two-level atoms interact effectively with independent environments, while in the second, the transitions are collective and lead to the Dicke model of super-radiance [4].
To quantitatively assess this behavior of uncorrelated/correlated dynamics as a function of r, we compute the measure of correlationsĪ, Eq. (3). The results are shown in Fig. 2. Despite the fact that the value ofĪ depends on time (the dynamical map is E S = e tL ), I decreases as r increases, as expected. Furthermore, the value ofĪ approaches zero for t large enough (see inset plot), except in the limiting case r = 0, because for r = 0 the dynamics becomes uncorrelated in the asymptotic limit, lim t→∞ e tL = E ⊗ E, where E(·) = K 1 (·)K † 1 + K 2 (·)K † 2 with Kraus operators K 1 = 0 0 1 0 and K 2 = 0 0 0 1 ; however for r = 0, lim t→∞ e tL is a correlated map. Thus, we obtain perfect agreement between the rigorous measure of correlationsĪ and the physically expected behavior of two distant atoms undergoing independent noise.
Spatial noise correlations in quantum computing.-Scalable and fault-tolerant quantum computing is predicted to be achievable provided the noise affecting the quan-tum computer is sufficiently weak and not too strongly correlated [57]. However, even if noise correlations decay sufficiently fast with increasing spatial distance between qubits, associated (provable) bounds for the accuracy threshold values can decrease by several orders of magnitude as compared to uncorrelated noise [14]. Thus, it is of essential importance to be able to detect, quantify and possibly reduce without a priori knowledge of the underlying microscopic dynamics the amount of correlated noise. Here, we exemplify the usefulness of the proposed correlation measure for these purposes by applying it to a simple, though paradigmatic model system of two representative qubits from a larger qubit register. We assume that the qubits are exposed to local thermal (bosonic) baths, such as realized e.g. by coupling distant atomic qubits to the surrounding electromagnetic radiation field, and that they interact via a weak ZZ-coupling, which could be caused, e.g., by undesired residual dipolar or van-der-Waals type interactions between the atoms. The "error" dynamics of this system is described by the master equation where ω is the energy difference between the qubit states, J the strength of the residual Hamiltonian coupling, γ 0 is again the decay rate between upper and lower energy level of each individual qubit andn = [exp(ω/T ) − 1] −1 is mean number of bosons with frequency ω in the two local baths of temperature T (assumed to be equal). We assume J and γ 0 to be out of our control and aim at studying the spatial correlations of the errors induced by the interplay of the residual ZZ-coupling and the baths as a function of the bath temperature T and elapsed time t, which in the present context might correspond to the time required to execute one round of quantum error correction [57,58]. Since the overall probability that some error occurs on the two qubits will increase under increasing t and T , we need to fix it for a fair assessment of the correlation of the dynamics. A natural way to do this is by defining the error probability in terms of how close the dynamical map induced by Eq. (8) [excluding the term ω 2 (σ z 1 + σ z 2 ), as this is not considered a source of error] is to the identity map (the case of no errors). Particularly, we can use the fidelity between both Choi-Jamio lkowski states, ρ CJ S for the "error" map and |Φ SS for the identity map, P error = 1 − Φ SS |ρ CJ S |Φ SS . Figure 3 shows the value of amount of dynamical correlations as measured byĪ along a t-T line on which the error probability is constant (P error = 0.1, green line in the inset plot). The numerical data shows that as the temperature increases the correlatedness of errors decreases very rapidly. This unexpected result suggests that by increasing the effective, surrounding temperature one can strongly decrease the non-local character of the noise at the expense of a slightly higher error rates per fixed time t, or constant error rates if the time t for an error correction round can be reduced. Thus, the proposed quantifier might prove essential to meet and certify in a given physical architectures the noise levels and noise correlation characteristics which are required to reach the regime where fault-tolerant scalable quantum computing becomes feasible in practice.
Conclusions.-In this work, we have formulated a general measure for the spatial correlations of quantum dynamics without restriction to any specific model. To that aim we have adopted a resource theory approach and obtained a fundamental law that any faithful quantifier of spatial correlation must satisfy. We have characterized the maximally correlated dynamics, and applied our measure to the paradigmatic example of two atoms radiating in the electromagnetic field, where spatial correlations are naturally related to the separation between atoms. Furthermore, we have illustrated the applicability of the measure in the context of quantum computing, where it can be employed to quantify spatial noise correlations without a priori knowledge of the underlying dynamics, and as a tool for controlling noise characteristics.
Beyond the scope of this work it will be interesting from a fundamental point of view to study how many independent (up to local unitaries) maximally correlated dynamics there are, and how to deal with the case of multi-partite or infinite dimensional systems. Moreover, with the present results, we hope to pave a feasible way to study rigorously the role of spatial correlations in a variety of physical processes, including noise assisted transport, quantum computing and dissipative phase transitions.
We acknowledge interesting discussions with T. Monz where ρ CJ A and ρ CJ B are the Choi-Jamio lkowski states of the maps E A and E B , respectively.
Indeed, if E S = E A ⊗ E B , we have (omitting for the sake of clarity the subindexes in the basis expansion of |Φ SS ): Conversely, if Eq. (S3) holds, then the dynamics has to be uncorrelated because the correspondence between Choi-Jamio lkowski states and dynamical maps is one-toone.
where Ψ (AA )|(BB ) is a maximally entangled state with respect to the bipartition AA |BB . Note that if Ψ (AA )|(BB ) is a maximally entangled state with respect to the bipartition AA |BB , U B↔A Ψ (AA )|(BB ) will be a maximally entangled state state with respect to the bipartition AB|A B = S|S . Since any maximally entangled state with respect to the bipartition S|S can be written asŨ S ⊗Ũ S |Φ SS for some local unitariesŨ S andŨ S , we can write