Foundations and measures of quantum non-Markovianity

An open quantum system S is a quantum system that is coupled to another quantum system E, i.e. its environment. Thus, S can be regarded as a subsystem of the total system S + E consisting of open system plus environment. Denoting the Hilbert spaces of S and E by ${\mathcal {H}}_S$ and ${\mathcal {H}}_E$, respectively, the Hilbert space of the total system is given by the tensor product space:

$$\begin{equation} {\mathcal {H}} = {\mathcal {H}}_S\otimes {\mathcal {H}}_E. \end{equation} \tag{ 1 }$$

The physical states of the total system are described by density matrices ρ, representing positive trace class operators on ${\mathcal {H}}$ with unit trace. This means that ρ ⩾ 0, which implies that ρ is self-adjoint with non-negative eigenvalues, and trρ = 1. The corresponding reduced states of subsystems S and E are given by partial traces over ${\mathcal {H}}_E$ and ${\mathcal {H}}_S$, respectively,

$$\begin{equation} \rho _S = {\rm {tr}}_E \rho , \qquad \rho _E = {\rm {tr}}_S \rho . \end{equation} \tag{ 2 }$$

In the following, we denote the convex set of physical states pertaining to some Hilbert space ${\mathcal {H}}$ by $S({\mathcal {H}})$.

We will assume here that the total system S + E is closed and follows a unitary dynamics described by some unitary time evolution operator

$$\begin{equation} U(t) = \exp [ -{\rm i}Ht ] \qquad (\hbar = 1) \end{equation} \tag{ 3 }$$

with a Hamiltonian of the most general form:

$$\begin{equation} H = H_S \otimes I_E + I_S \otimes H_E + H_I, \end{equation} \tag{ 4 }$$

where H_S and H_E are the self-Hamiltonians of system and environment, respectively, and H_I is any interaction Hamiltonian. Thus, the time dependence of the total system states is given by the von Neumann equation:

$$\begin{equation} \frac{\rm d}{{\rm d}t}\rho (t) = -{\rm i}[H,\rho (t)], \end{equation} \tag{ 5 }$$

with the formal solution

$$\begin{equation} \rho (t) = U(t)\rho (0)U^{\dagger }(t). \end{equation} \tag{ 6 }$$

It turns out that in most cases of practical relevance a full solution of the equations of motion on the microscopic level is impossible. Thus, one of the central goals of the quantum theory of open systems [3] is the development of an effective and analytically or numerically feasible formulation of the dynamical behaviour of a suitably defined reduced set of variables forming the open subsystem S. Given a certain split of the total system into an open system S and environment E, one tries to derive effective equations of motion for the time dependence of the reduced system state ρ_S(t) through an elimination of the environmental variables from the dynamical equations. The main aim is to develop efficient descriptions for a wide class of physical problems and phenomena, such as the dissipation and damping of populations, the relaxation to a thermal equilibrium state, the emergence of non-equilibrium stationary states, the suppression or destruction of quantum coherences, the description of the quantum transport of excitations in complex systems, and the dynamics of quantum correlations and entanglement.

Quantum dynamical maps represent a key concept in the theory of open systems. To introduce this concept, we presuppose (i) that the dynamics of the total system is given by a unitary time evolution (6) and (ii) that the system S and environment E are statistically independent at the initial time, i.e. the initial total system state represents a tensor product state:

$$\begin{equation} \rho (0) = \rho _S(0) \otimes \rho _E(0). \end{equation} \tag{ 7 }$$

While condition (i) can be relaxed to include more general cases, condition (ii) is crucial for the following considerations. We will discuss in section 4 the case of initially correlated system–environment states. On the basis of these assumptions, the open system state at time t ⩾ 0 can be written as follows:

$$\begin{equation} \rho _S(t) = {\rm {tr}}_E \lbrace U(t) \rho _S(0) \otimes \rho _E(0) U^{\dagger }(t) \rbrace . \end{equation} \tag{ 8 }$$

In the following, we consider the initial environmental state ρ_E(0) to be fixed. For each fixed t ⩾ 0, equation (8) then can be seen to represent a linear map

$$\begin{equation} \Phi (t,0): \, S({\mathcal {H}}_S) \longrightarrow S({\mathcal {H}}_S) \end{equation} \tag{ 9 }$$

on the open system's state space $S({\mathcal {H}}_S),$ which maps any initial open system state to the corresponding open system state at time t:

$$\begin{equation} \rho _S(0) \mapsto \rho _S(t) = \Phi (t,0) \rho _S(0). \end{equation} \tag{ 10 }$$

This is the quantum dynamical map corresponding to time t. It is easy to check that it preserves the Hermiticity and the trace of operators, i.e. we have

$$\begin{equation} {\rm {tr}}_S \left\lbrace \Phi (t,0) A \right\rbrace = {\rm {tr}}_S \left\lbrace A \right\rbrace \end{equation} \tag{ 11 }$$

and

$$\begin{equation} \left[ \Phi (t,0) A \right]^{\dagger } = \Phi (t,0)A^{\dagger }. \end{equation} \tag{ 12 }$$

Moreover, Φ(t, 0) is a positive map, i.e. it maps positive operators to positive operators:

$$\begin{equation} A \ge 0 \Longrightarrow \Phi (t,0) A \ge 0. \end{equation} \tag{ 13 }$$

An important further property of dynamical maps is that they are not only positive but also completely positive. Such maps are also known as trace-preserving quantum operations or quantum channels in quantum information and communication theory [11]. We recall that a linear map Φ is completely positive if and only if it admits a Kraus representation [12], i.e. if there are operators Ω_i on the underlying Hilbert space ${\mathcal {H}}_S$, such that

$$\begin{equation} \Phi A = \sum _i \Omega _i A \Omega _i^{\dagger }. \end{equation} \tag{ 14 }$$

Such a map is trace preserving if and only if the normalization condition

$$\begin{equation} \sum _i \Omega _i^{\dagger }\Omega = I_S \end{equation} \tag{ 15 }$$

holds. The original definition of complete positivity, which is equivalent to the existence of a Kraus representation, is the following. Consider for any n = 1, 2, ... the tensor product ${\mathcal {H}}_S\otimes {\mathbb C}^n$, describing the Hilbert space of S combined with an n-level system and the corresponding linear tensor extension of Φ defined by (Φ⊗I_n)(A⊗B) = (ΦA)⊗B. The map Φ⊗I_n thus describes an operation that acts only on the first factor of the composite system and leaves the second factor unchanged. The map Φ is then defined to be n-positive if Φ⊗I_n is a positive map and completely positive if Φ⊗I_n is a positive map for all n. We note that positivity is equivalent to 1-positivity, and for a Hilbert space with a finite dimension $N_S={\rm dim}\,{\mathcal {H}}_S,$ complete positivity is equivalent to N_S-positivity [13].

If we now allow the time parameter t to vary (keeping the initial environmental state ρ_E(0) fixed), we obtain a one-parameter family of dynamical maps:

$$\begin{equation} \left\lbrace \Phi (t,0) \mid t \ge 0, \Phi (0,0) = I \right\rbrace , \end{equation} \tag{ 16 }$$

which contains the complete information on the dynamical evolution of all possible initial system states. Thus, formally speaking, a quantum process of an open system is given by such a one-parameter family of completely positive and trace-preserving (CPT) quantum dynamical maps.

As a simple example, we consider the decay of a two-state system into a bosonic reservoir [3, 14]. The total Hamiltonian of the model is given by equation (4) with the system Hamiltonian

$$\begin{equation} H_S = \omega _{0}\sigma _+\sigma _-, \end{equation} \tag{ 17 }$$

describing a two-state system (qubit) with the ground state |0〉, excited state |1〉 and transition frequency ω₀, where σ₊ = |1〉〈0| and σ₋ = |0〉〈1| are the raising and lowering operators of the qubit. The Hamiltonian of the environment,

$$\begin{equation} H_E = \sum _k\omega _k b_k^\dagger b_k, \end{equation} \tag{ 18 }$$

represents a reservoir of harmonic oscillators with creation and annihilation operators b†_k and b_k satisfying the bosonic commutation relations $[b_k,b_{k^{\prime }}^{\dagger }] = \delta _{kk^{\prime }}$. The interaction Hamiltonian takes the form

$$\begin{equation} H_I = \sum _k \big( g_k \sigma _+ \otimes b_k + g^*_k \sigma _- \otimes b^{\dagger }_k \big). \end{equation} \tag{ 19 }$$

The model thus describes, e.g., the coupling of the qubit to a reservoir of electromagnetic field modes labelled by the index k, with the corresponding frequencies ω_k and coupling constants g_k. Since we are using the rotating wave approximation in the interaction Hamiltonian, the total number of excitations in the system,

$$\begin{equation} N = \sigma _+\sigma _- + \sum _k b_k^{\dagger }b_k, \end{equation} \tag{ 20 }$$

is a conserved quantity. The model, therefore, allows one to derive an analytical expression for the dynamical map (10). Assuming the environment to be in the vacuum state |0〉 initially, one finds

$$\begin{eqnarray} &&\rho _{11}(t) = |G(t)|^2 \rho _{11}(0), \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} &&\rho _{00}(t) = \rho _{00}(0) + (1-|G(t)|^2)\rho _{11}(0), \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} &&\rho _{10}(t) = G(t) \rho _{10}(0), \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} &&\rho _{01}(t) = G^*(t) \rho _{01}(0), \end{eqnarray} \tag{ 24 }$$

where ρ_ij(t) = 〈i|ρ_S(t)|j〉 denote the elements of the interaction picture density matrix ρ_S(t). The function G(t) introduced here is defined to be the solution of the integro-differential equation

$$\begin{equation} \frac{\rm d}{{\rm d}t}G(t) = -\int _0^t {\rm d}t_1 f(t-t_1)G(t_1) \end{equation} \tag{ 25 }$$

corresponding to the initial condition G(0) = 1, where the kernel f(t − t₁) represents a certain two-point correlation function,

$$\begin{eqnarray} f(t-t_1) &=& \langle 0|B(t)B^{\dagger }(t_1)|0\rangle\, {\rm e}^{{\rm i}\omega _0(t-t_1)} \nonumber \\\ms &=& \sum _k |g_k|^2\, {\rm e}^{{\rm i}(\omega _0-\omega _k)(t-t_1)}, \end{eqnarray} \tag{ 26 }$$

of the environmental operators

$$\begin{equation} B(t) = \sum _k g_k b_k {\rm e}^{-{\rm i} \omega _k t}. \end{equation} \tag{ 27 }$$

These results hold for a generic environmental spectral density and the corresponding two-point correlation function. Taking, e.g., a Lorentzian spectral density in resonance with the transition frequency of the qubit, we find an exponential two-point correlation function

$$\begin{equation} f(\tau ) = {\textstyle\frac{1}{2}}\, \gamma _0 \lambda {\rm e}^{-\lambda |\tau |}, \end{equation} \tag{ 28 }$$

where γ₀ describes the strength of the system–environment coupling and λ is the spectral width that is related to the environmental correlation time by τ_E = λ⁻¹. Solving equation (25) with this correlation function, we find

$$\begin{equation} G(t) = {\rm e}^{-\lambda t/2}\left[\cosh \left(\frac{{\rm d}t}{2}\right) +\frac{\lambda }{d}\sinh \left(\frac{{\rm d}t}{2} \right)\right]\!, \end{equation} \tag{ 29 }$$

where $d=\sqrt{\lambda ^2-2\gamma _0\lambda }$.

The simplest example of a quantum process is provided by a semigroup of completely positive dynamical maps, which is often considered as a prototypical example of a quantum Markov process. In this case, one assumes that the set (16) has the additional property

$$\begin{equation} \Phi (t,0) \, \Phi (s,0) = \Phi (t+s,0) \end{equation} \tag{ 30 }$$

for all t, s ⩾ 0 and, hence, has the structure of a semigroup. Under very general mathematical conditions, such a semigroup has an infinitesimal generator ${\mathcal {L,}}$ which allows us to write

$$\begin{equation} \Phi (t,0) = \exp [{\mathcal {L}}t]. \end{equation} \tag{ 31 }$$

Accordingly, the reduced system state ρ_S(t) obeys the master equation

$$\begin{equation} \frac{\rm d}{{\rm d}t} \rho _S(t) = \mathcal {L} \rho _S(t). \end{equation} \tag{ 32 }$$

The complete positivity of the semigroup leads to important statements on the general structure of the generator. The famous Gorini–Kossakowski–Sudarshan–Lindblad theorem [1, 2] states that ${\mathcal {L}}$ is the generator of a semigroup of completely positive quantum dynamical maps if and only if it has the following form:

$$\begin{equation} \mathcal {L}\rho _S \,{=}\,{ -}{\rm i}\left[H_S,\rho _S\right] + \sum _i {\gamma _i \left[ A_i \rho _S A_i^\dagger -\frac{1}{2} \big\lbrace A_i^\dagger A_i,\rho _S\big\rbrace \right]}, \end{equation} \tag{ 33 }$$

where H_S is a system Hamiltonian (which need not coincide with the system Hamiltonian H_S in the microscopic Hamiltonian (4)), A_i are arbitrary system operators, often called the Lindblad operators, describing the various decay modes of the system, and γ_i are the corresponding decay rates. This theorem has many far-reaching consequences and is extremely useful, in particular, in phenomenological approaches, since it guarantees a time evolution that is compatible with general physical principles for any master equation of the above structure. On the other hand, it is in general difficult to justify rigorously the assumption (30) and to derive a quantum master equation of the form (32) starting from a given system–environment model with a microscopic Hamiltonian (4). Such a derivation requires the validity of several approximations, the most important one being the so-called Markov approximation. This approximation presupposes a rapid decay of the two-point correlation functions of those environmental operators that describe the system–environment coupling H_I. More precisely, if τ_E describes the temporal width of these correlations and τ_R is the relaxation or decoherence time of the system, the Markov approximation demands that

$$\begin{equation} \tau _E \ll \tau _R. \end{equation} \tag{ 34 }$$

This means that the environmental correlation time τ_E is small compared to the open system's relaxation or decoherence time τ_R, i.e. that we have a separation of time scales, the environmental variables being the fast and the system variables being the slow variables. The Markov approximation is justified in many cases of physical interest. Typical examples of application are the weak coupling master equation, the quantum optical master equation describing the interaction of radiation with matter [3] and the master equation for a test particle in a quantum gas [15]. However, large couplings or interactions with low-temperature reservoirs can lead to strong correlations resulting in long memory times and in a failure of the Markov approximation. Moreover, the standard Markov condition (34) alone does not guarantee, in general, that the Markovian master equation provides a reasonable description of the dynamics, a situation which can occur, e.g., for finite and/or structured reservoirs [16, 17].

There exists a whole bunch of different theoretical and numerical methods for the treatment of open quantum systems, beyond the assumption of a dynamical semigroup, such as projection operator techniques [18, 19], influence functional and path integral techniques [20], quantum Monte Carlo methods and stochastic wave function techniques [14, 21]. Here, we concentrate on a specific approach that is particularly suited for our purpose and describes the open system dynamics in terms of a time-local master equation.

It is usually expected that the mathematical formulation of quantum processes describing effects of finite memory times in the system must necessarily involve equations of motion, which are non-local in time. In fact, such a description is suggested by the Nakajima–Zwanzig projection operator technique that leads to an integro-differential equation for the reduced density matrix [18, 19]. However, even the presence of strong memory effects does not exclude the description of the dynamics in terms of a quantum master equation that is local in time, as may be seen from the following simple argument. According to equation (10), we have ρ_S(t) = Φ(t, 0)ρ_S(0). Assuming a smooth time dependence, we may differentiate this relation to obtain

$$\begin{equation} \frac{\rm d}{{\rm d}t}\rho _S(t) = \dot{\Phi }(t,0) \rho _S(0), \end{equation} \tag{ 35 }$$

where the dot indicates the time derivative of Φ(t, 0). To obtain a local master equation, we invert relation (10), expressing ρ_S(0) in terms of ρ_S(t), which yields

$$\begin{equation} \frac{\rm d}{{\rm d}t}\rho _S(t) = \dot{\Phi }(t,0) \Phi ^{-1}(t,0) \rho _S(t). \end{equation} \tag{ 36 }$$

Thus, we see that the linear map ${\mathcal {K}}(t)=\dot{\Phi }(t,0) \Phi ^{-1}(t,0)$ represents a time-dependent generator of the dynamics and we obtain a quantum master equation, which is indeed local in time, providing a linear first-order differential equation for the open system state:

$$\begin{equation} \frac{\rm d}{{\rm d}t} \rho _S(t) = \mathcal {K}(t) \rho _S(t). \end{equation} \tag{ 37 }$$

We note that the above argument presupposes that the inverse Φ⁻¹(t, 0) of the map Φ(t, 0) exists. It is possible that the inverse of Φ(t, 0) and, hence, also the time-local generator $\mathcal {K}(t)$ do not exist. Such a situation can indeed occur for very strong system–environment couplings (see below). However, for an analytic time dependence, the inverse of Φ(t, 0) and the generator ${\mathcal {K}}(t)$ do exist apart from isolated singularities of the time axis [22]. It should also be emphasized that Φ⁻¹(t, 0) denotes the inverse of Φ(t, 0) regarded as a linear map acting on the space of operators of the reduced system. The important point is that this does not imply that Φ⁻¹(t, 0) is required to be completely positive. In general, the inverse map is neither completely positive nor even positive. There exists a powerful method for the microscopic derivation of time-local master equations of the form (37), which is known as the time-convolutionless projection operator technique [3, 23–27]. This technique yields a systematic expansion of the generator of the master equation in terms of ordered cumulants and many examples have been treated with this method [28–32].

The generator ${\mathcal {K}}(t)$ of the time-local master equation must of course preserve the Hermiticity and the trace, i.e. we have

$$\begin{equation} \left[ {\mathcal {K}}(t) A \right]^{\dagger } = {\mathcal {K}}(t) A^{\dagger } \end{equation} \tag{ 38 }$$

and

$$\begin{equation} {\rm {tr}}_S \left\lbrace {\mathcal {K}}(t) A \right\rbrace = 0. \end{equation} \tag{ 39 }$$

From these requirements, it follows that the generator must be of the following most general form:

$$\begin{eqnarray} &&\mathcal {K}(t)\rho _S = -{\rm i}\,[H_S(t),\rho _S] \nonumber \\ &&\quad\!\!+\, \sum _i{\gamma _i(t)\left[A_i(t)\rho _S A_i^\dagger (t) -\frac{1}{2}\big\lbrace A_i^\dagger (t)A_i(t),\rho _S\big\rbrace \right]}. \end{eqnarray} \tag{ 40 }$$

We see that the structure of the generator provides a natural generalization of the Lindblad structure, in which the Hamiltonian H_S(t), the Lindblad operators A_i(t) as well as the various decay rates γ_i(t) may depend on time.

We briefly sketch the proof of (40) for a finite-dimensional open system Hilbert space, i.e. ${\rm dim}\,\mathcal {H}_S=N_S$. To this end, we consider a fixed time t and a fixed complete system of operators F_i on $\mathcal {H}_S$, i = 1, 2, ..., N²_S, which are orthonormal with respect to the Hilbert–Schmidt scalar product:

$$\begin{equation} {\rm tr}_S\big\lbrace F_i^{\dagger }F_j\big\rbrace = \delta _{ij}. \end{equation} \tag{ 41 }$$

Without loss of generality, we may further choose

$$\begin{equation} F_{N_S^2}=\frac{1}{\sqrt{N_S}}I_S. \end{equation} \tag{ 42 }$$

According to lemma 2.3 of [1], any linear map ${\mathcal {K}}(t)$ satisfying equations (38) and (39) can then be written as

$$\begin{eqnarray} &&\qquad\mathcal {K}(t)\rho _S = -{\rm i}\,[H_S(t),\rho _S]\nonumber\\ &&\quad \qquad+ \sum _{i,j=1}^{N_S^2-1} c_{ij}(t) \left[ F_i \rho _S F_j^{\dagger } -\frac{1}{2} \big\lbrace F_j^{\dagger } F_i , \rho _S \big\rbrace \right] \end{eqnarray} \tag{ 43 }$$

with a self-adjoint operator H_S(t) and a Hermitian matrix c(t) = (c_ij(t)). Diagonalizing this matrix with the help of a unitary matrix u(t) = (u_ij(t)),

$$\begin{equation} u(t)c(t)u^{\dagger }(t) = {\mbox{diag}}(\gamma _1(t),\gamma _2(t),\ldots ,\gamma _{N_S^2-1}(t)), \end{equation} \tag{ 44 }$$

and introducing the new operators

$$\begin{equation} A_i(t) = \sum _{j=1}^{N_S^2-1} u^{\ast }_{ij}(t) F_j, \end{equation} \tag{ 45 }$$

one obtains the form (40) of the generator.

The structure (40) takes into account the Hermiticity and trace preservation of the dynamics, but does not guarantee its complete positivity. The formulation of necessary and sufficient conditions for the complete positivity of the dynamics of this generator is an important unsolved problem. However, in the case that the rates are positive for all times,

$$\begin{equation} \gamma _i(t) \ge 0, \end{equation} \tag{ 46 }$$

the resulting dynamics is indeed completely positive, since the generator is then in the Lindblad form for each fixed t ⩾ 0.

As an example, let us consider the dynamical map given by equations (21)–(24). In this case, the time-local generator takes the form [14]

$$\begin{eqnarray} &&\mathcal {K}(t)\rho _S = -\frac{{\rm i}}{2}S(t) [\sigma _+\sigma _-,\rho _S]\hspace{40pt} \nonumber \\ &&+\,\gamma (t)\left[ \sigma _-\rho _S\sigma _+ -\frac{1}{2}\left\lbrace \sigma _+\sigma _-,\rho _S\right\rbrace \right], \end{eqnarray} \tag{ 47 }$$

where we have introduced the definitions

$$\begin{equation} \gamma (t) = -2\Re \left(\frac{\dot{G}(t)}{G(t)}\right), \qquad S(t) = -2\Im \left(\frac{\dot{G}(t)}{G(t)}\right)\!. \end{equation} \tag{ 48 }$$

With this generator, equation (37) represents an exact master equation of the model. The quantity S(t) plays the role of a time-dependent frequency shift, and γ(t) can be interpreted as a time-dependent decay rate. Due to the time dependence of these quantities, the process generally does not represent a dynamical semigroup, of course. Note that the generator is finite as long as G(t) ≠ 0; at the zeros of G(t), the inverse of the dynamical map Φ(t, 0) does not exist. An example is provided by the zeros of the function of equation (29), which appear in the strong-coupling regime γ₀ > λ/2.

We can also see explicitly how the standard Markov limit arises in this model. Considering the particular case (29), we observe that in the limit of small α = γ₀/λ, we may approximate $G(t)\approx {\rm e}^{-\gamma _0t/2}$. This approximation can also be obtained directly from equation (25) by replacing the two-point correlation function f(t − t₁) with the delta-function γ₀δ(t − t₁), which is conventionally regarded as the Markovian limit. Equation (48) then yields S(t) = 0 and γ(t) = γ₀, i.e. the generator (47) assumes the form of a Lindblad generator of a quantum dynamical semigroup. The quantity α can also be written as the ratio of the environmental correlations time τ_E = λ⁻¹ and the relaxation time τ_R = γ⁻¹₀ of the system:

$$\begin{equation} \alpha = \frac{\tau _E}{\tau _R}. \end{equation} \tag{ 49 }$$

Thus, we see that the standard Markov condition (34) indeed leads to a Markovian semigroup here. We also mention that the time-convolutionless projection operator technique yields an expansion of the generator (47) in powers of this ratio α [3].

A family of dynamical maps Φ(t, 0) is defined to be divisible if for all t₂ ⩾ t₁ ⩾ 0 there exists a CPT map Φ(t₂, t₁) such that the relation

$$\begin{equation} \Phi (t_2,0) = \Phi (t_2,t_1) \Phi (t_1,0) \end{equation} \tag{ 50 }$$

holds. Note that in this equation, the left-hand side and the second factor on the right-hand side are fixed by the given family of dynamical maps. Hence, equation (50) requires the existence of a certain linear transformation Φ(t₂, t₁), which maps the states at time t₁ to the states at time t₂ and represents a CPT map (which may be a unitary transformation). This definition differs slightly from the usual definition for the divisibility of a CPT map Λ, according to which Λ is said to be divisible if there exist CPT maps Λ₁ and Λ₂, such that Λ = Λ₁Λ₂, where one requires that neither Λ₁ nor Λ₂ is a unitary transformation, for otherwise the statement is trivial [33]. There are many quantum processes that are not divisible. For instance, if Φ(t₁, 0) is not invertible, a linear map Φ(t₂, t₁) that fulfils equation (50) may not exist. Moreover, even if a linear map Φ(t₂, t₁) satisfying equation (50) does exist, this map need not to be completely positive and not even positive.

The simplest example of a divisible quantum process is given by a dynamical semigroup. In fact, for a semigroup, we have $\Phi (t,0)=\exp [\mathcal {L}t],$ and hence, equation (50) is trivially satisfied with the CPT map $\Phi (t_2,t_1)=\exp [\mathcal {L}(t_2-t_1)]$.

Consider now a quantum process given by the time-local master equation (37) with a time-dependent generator (40). The dynamical maps can then be represented in terms of a time-ordered exponential

$$\begin{equation} \Phi (t,0) = {\rm T} \exp \left[\int _{0}^{t}{{\rm d}t^{\prime }\mathcal {K}(t^{\prime })}\right]\!, \quad t \ge 0, \end{equation} \tag{ 51 }$$

where T denotes the chronological time-ordering operator. We can also define the maps

$$\begin{equation} \Phi (t_2,t_1) = {\rm T} \exp \left[\int _{t_1}^{t_2}{{\rm d}t^{\prime }\mathcal {K}(t^{\prime })}\right]\!, \quad t_2 \ge t_1 \ge 0, \end{equation} \tag{ 52 }$$

such that the composition law Φ(t₂, 0) = Φ(t₂, t₁)Φ(t₁, 0) holds by construction. The maps Φ(t₂, t₁) are completely positive, as is required by the divisibility condition (50), if and only if the rates γ_i(t) of the generator (40) are positive functions. Thus, we see that divisibility is equivalent to positive rates in the time-local master equation [7]. To prove this statement suppose first that γ_i(t) ⩾ 0. The generator $\mathcal {K}(t)$ is then in the Lindblad form for each fixed t ⩾ 0, and therefore, equation (52) represents completely positive maps. Conversely, assume that the maps defined by equation (52) are completely positive. It follows from this equation that the generator is given by

$$\begin{equation} {\mathcal {K}}(t) = \left.\frac{\rm d}{{\rm d}\tau }\right|_{\tau =0} \Phi (t+\tau ,t). \end{equation} \tag{ 53 }$$

Since Φ(t + τ, t) is completely positive for all t, τ ⩾ 0 and satisfies Φ(t, t) = I, this generator must be in the Lindblad form for each fixed t, i.e. it must have the form (40) with γ_i(t) ⩾ 0.

The dynamical map Φ(t, 0) given by equations (21)–(24) is completely positive if and only if |G(t)| ⩽ 1. It is easy to verify that Φ(t, 0) can be decomposed as in equation (50), where the map Φ(t₂, t₁) is given by [7]

$$\begin{eqnarray} \rho _{11}(t_2) &=& \left|\frac{G(t_2)}{G(t_1)}\right|^2 \rho _{11}(t_1), \nonumber \\ \rho _{00}(t_2) &=& \rho _{00}(t_1) + \left(1-\left|\frac{G(t_2)}{G(t_1)}\right|^2\right)\rho _{11}(t_1), \nonumber \\ \rho _{10}(t_2) &=& \frac{G(t_2)}{G(t_1)} \rho _{10}(t_1), \nonumber \\ \rho _{01}(t_2) &=& \frac{G^*(t_2)}{G^*(t_1)} \rho _{01}(t_1). \end{eqnarray} \tag{ 54 }$$

It follows from these equations that a necessary and sufficient condition for the complete positivity of Φ(t₂, t₁) is given by

$$\begin{equation} |G(t_2)|\le |G(t_1)|. \end{equation} \tag{ 55 }$$

Thus, we see that the dynamical map of the model is divisible if and only if |G(t)| is a monotonically decreasing function of time. Note that this statement holds true also for the case that G(t) vanishes at some finite time. The rate γ(t) in equation (48) can be written as

$$\begin{equation} \gamma (t) = -\frac{2}{|G(t)|}\frac{\rm d}{{\rm d}t}|G(t)|. \end{equation} \tag{ 56 }$$

This shows that any increase of |G(t)| leads to a negative decay rate in the corresponding generator (47) and illustrates the equivalence of the non-divisibility of the dynamical map and the occurrence of a temporarily negative rate in the time-local master equation demonstrated above.

The trace norm of a trace class operator A is defined by

$$\begin{equation} ||A|| = {\rm {tr}} |A| = {\rm {tr}} \sqrt{A^{\dagger }A}. \end{equation} \tag{ 57 }$$

If A is self-adjoint with eigenvalues a_i, this formula reduces to

$$\begin{equation} ||A|| = \sum _i |a_i|. \end{equation} \tag{ 58 }$$

Hence, the trace norm of a self-adjoint operator is equal to the sum of the moduli of its eigenvalues (counting their multiplicities). The trace norm leads to a natural and useful measure for the distance between two quantum states represented by positive operators ρ¹ and ρ² with unit trace, which is known as the trace distance:

$$\begin{equation} D(\rho ^1,\rho ^2) = {\textstyle\frac{1}{2}}||\rho ^1-\rho ^2|| = {\textstyle\frac{1}{2}}\, {\rm {tr}}|\rho ^1-\rho ^2|. \end{equation} \tag{ 59 }$$

The trace distance is well defined and finite for all pairs of quantum states and provides a metric on the space $S({\mathcal {H}})$ of physical states. We list some of the most important mathematical properties of this metric which will be needed later on (most of the proofs may be found in [10, 11]).

• (1)  
  The trace distance between any pair of states satisfies

  $$\begin{equation} 0 \le D(\rho ^1,\rho ^2) \le 1. \end{equation} \tag{ 60 }$$

  Of course, we have D(ρ¹, ρ²) = 0 if and only if ρ¹ = ρ², while the upper bound is reached, i.e. D(ρ¹, ρ²) = 1 if and only if ρ¹ and ρ² are orthogonal, which means that the supports of ρ¹ and ρ² are orthogonal. (The support is defined to be the space spanned by the eigenstates with nonzero eigenvalue.) Moreover, D(ρ¹, ρ²) is obviously symmetric in the input arguments and satisfies the triangular inequality:

  $$\begin{equation} D(\rho ^1,\rho ^2) \le D(\rho ^1,\rho ^3) + D(\rho ^3,\rho ^2). \end{equation} \tag{ 61 }$$
• (2)  
  If ρ¹ = |ψ¹〉〈ψ¹| and ρ² = |ψ²〉〈ψ²| are pure states, the following explicit formula for the trace distance can easily be derived:

  $$\begin{equation} D(\rho ^1,\rho ^2) = \sqrt{1-|\langle \psi ^1 | \psi ^2 \rangle |^2}. \end{equation} \tag{ 62 }$$

  If the underlying Hilbert space is two dimensional (qubit), spanned by the basis states |1〉 and |0〉, the trace distance between two states with the matrix elements ρ¹_ij and ρ²_ij is found to be

  $$\begin{equation} D(\rho ^1,\rho ^2) = \sqrt{a^2+|b|^2}, \end{equation} \tag{ 63 }$$

  where a = ρ¹₁₁ − ρ²₁₁ is the difference of the populations and b = ρ¹₁₀ − ρ²₁₀ is the difference of the coherences of the two states.
• (3)  
  The trace distance is sub-additive with respect to tensor products of states which means that

  $$\begin{equation} D(\rho ^1\otimes \sigma ^1,\rho ^2\otimes \sigma ^2) \le D(\rho ^1,\rho ^2) + D(\sigma ^1,\sigma ^2). \end{equation} \tag{ 64 }$$

  In addition, we have

  $$\begin{equation} D(\rho ^1\otimes \sigma ,\rho ^2\otimes \sigma ) = D(\rho ^1,\rho ^2). \end{equation} \tag{ 65 }$$
• (4)  
  The trace distance is invariant under unitary transformations U:

  $$\begin{equation} D(U\rho ^1U^{\dagger },U\rho ^2U^{\dagger }) = D(\rho ^1,\rho ^2). \end{equation} \tag{ 66 }$$

  More generally, all trace-preserving and completely positive maps, i.e. all trace-preserving quantum operations Λ are contractions of the trace distance:

  $$\begin{equation} D(\Lambda \rho ^1,\Lambda \rho ^2) \le D(\rho ^1,\rho ^2). \end{equation} \tag{ 67 }$$

  Note that the condition of the trace preservation is important here and that this inequality also holds for the larger class of trace-preserving positive maps [34].
• (5)  
  The trace distance can be represented as the maximum of a certain functional:

  $$\begin{equation} D(\rho ^1,\rho ^2) = \max _{\Pi } {\rm {tr}}\lbrace \Pi (\rho ^1-\rho ^2)\rbrace . \end{equation} \tag{ 68 }$$

  The maximum is taken over all projection operators Π. Alternatively, one can take the maximum over all positive operators A with A ⩽ I. Note that this formula is symmetric, i.e. we can also write D(ρ¹, ρ²) = max _Πtr{Π(ρ² − ρ¹)} where, however, the maximum is then assumed for a different projection Π.

The trace distance between two quantum states ρ¹ and ρ² has a direct physical interpretation, which is based on the representation (68). Consider two parties, Alice and Bob. Alice prepares a quantum system in one of two states ρ¹ or ρ² with the probability $\frac{1}{2}$ each, and then sends the system to Bob. It is Bob's task to find out by a single measurement on the system whether the system state was ρ¹ or ρ². It turns out that Bob cannot always distinguish the states with certainty, but there is an optimal strategy that allows him to achieve the maximal possible success probability given by

$$\begin{equation} P_{\max } = {\textstyle\frac{1}{2}} [1 + D(\rho ^1,\rho ^2) ]. \end{equation} \tag{ 69 }$$

Thus, we see that the trace distance represents the bias in favour of the correct state identification that Bob can achieve through an optimal strategy. The trace distance D(ρ¹, ρ²) can therefore be interpreted as a measure for the distinguishability of the quantum states ρ¹ and ρ² [8–10].

According to equation (68), Bob's optimal strategy consists in measuring the projection Π for which the maximum in this relation is assumed, and in associating the outcome Π = 1 with the state ρ¹ and the outcome Π = 0 with the state ρ². Under the condition that the system state was ρ¹, he then has correctly identified this state with the probability tr{Πρ¹}, while under the condition that the system state was ρ², his answer is correct with the probability tr{(I − Π)ρ²}. Since both possibilities occur with a probability of $\frac{1}{2}$, we obtain the success probability

$$\begin{eqnarray} P_{\max } &=& {\textstyle\frac{1}{2}}{\rm {tr}}\lbrace \Pi \rho ^1\rbrace + {\textstyle\frac{1}{2}}\,{\rm {tr}}\lbrace (I-\Pi )\rho ^2\rbrace \nonumber \\\ms &=& {\textstyle\frac{1}{2}} [1 + {\rm {tr}}\lbrace \Pi (\rho ^1-\rho ^2)\rbrace ] \nonumber \\\ms &=& {\textstyle\frac{1}{2}} [1 + D(\rho ^1,\rho ^2) ], \end{eqnarray} \tag{ 70 }$$

which proves equation (69). We further see that a state identification with certainty, P_max = 1, can be achieved if and only if D(ρ¹, ρ²) = 1, i.e. if and only if ρ¹ and ρ² are orthogonal, in which case Bob's optimal strategy is to measure the projection Π onto the support of ρ¹ (or of ρ²).

As we have seen in section 3.1, the trace distance D(ρ¹, ρ²) between two quantum states ρ¹ and ρ² can be interpreted as the distinguishability of these states. An important conclusion from this interpretation is that according to the contraction property of equation (67), no CPT quantum operation can increase the distinguishability between quantum states. Consider two initial states ρ¹_S(0) and ρ²_S(0) of an open system and the corresponding time-evolved states

$$\begin{equation} \rho _S^{1,2}(t) = \Phi (t,0)\rho _S^{1,2}(0). \end{equation} \tag{ 71 }$$

Since the dynamical maps Φ(t, 0) are CPT, the trace distance between the time-evolved states can never be larger than the trace distance between the initial states:

$$\begin{equation} D\big(\rho _S^1(t),\rho _S^2(t)\big) \le D\big(\rho _S^1(0),\rho _S^2(0)\big). \end{equation} \tag{ 72 }$$

Thus, no quantum process describable by a family of CPT dynamical maps can ever increase the distinguishability of a pair of states over its initial value. Of course, this general feature does not imply that D(ρ¹_S(t), ρ_S²(t)) is a monotonically decreasing function of time.

We can interpret the dynamical change of the distinguishability of the states of an open system in terms of a flow of information between the system and its environment. When a quantum process reduces the distinguishability of states, information is flowing from the system to the environment. Correspondingly, an increase of the distinguishability signifies that information flows from the environment back to the system. The invariance under unitary transformations (66) indicates that information is preserved under the dynamics of closed systems. On the other hand, the contraction property of equation (67) guarantees that the maximal amount of information the system can recover from the environment is the amount of information earlier flowed out of the system.

Our definition for quantum non-Markovianity is based on the idea that for Markovian processes any two quantum states become less and less distinguishable under the dynamics, leading to a perpetual loss of information into the environment. Quantum memory effects thus arise if there is a temporal flow of information from the environment to the system. The information flowing back from the environment allows the earlier open system states to have an effect on the later dynamics of the system, which implies the emergence of memory effects [5].

In view of these considerations, we thus define a quantum process described in terms of a family of quantum dynamical maps Φ(t, 0) to be non-Markovian if and only if there is a pair of initial states ρ^{1, 2}_S(0), such that the trace distance between the corresponding states ρ^{1, 2}_S(t) increases at a certain time t > 0:

$$\begin{equation} \sigma \big(t,\rho _S^{1,2}(0)\big) \equiv \frac{\rm d}{{\rm d}t}D\big(\rho _S^1(t),\rho _S^2(t)\big) > 0, \end{equation} \tag{ 73 }$$

where σ(t, ρ^{1, 2}_S(0)) denotes the rate of change of the trace distance at time t corresponding to the initial pair of states.

This definition for quantum non-Markovianity has many important consequences for the general classification of quantum processes. In particular, it implies that all divisible families of dynamical maps are Markovian, including of course the class of quantum dynamical semigroups. To prove this statement suppose that Φ(t, 0) is divisible. For any pair of initial states ρ^{1, 2}_S(0), we then have

$$\begin{equation} \rho _S^{1,2}(t+\tau )=\Phi (t+\tau ,t)\rho _S^{1,2}(t), \quad t,\tau \ge 0. \end{equation} \tag{ 74 }$$

Since Φ(t + τ, t) is a CPT map, we can apply the contraction property (67) to obtain

$$\begin{equation} D\big(\rho _S^1(t+\tau ),\rho _S^2(t+\tau )\big)\le D\big(\rho _S^1(t),\rho _S^2(t)\big). \end{equation} \tag{ 75 }$$

This shows that for all divisible dynamical maps the trace distance decreases monotonically and that, therefore, these processes are Markovian.

Hence, we see that non-Markovian quantum processes must necessarily be described by non-divisible dynamical maps and by time-local master equations whose generator (40) involves at least one temporarily negative rate γ_i(t). However, it is important to note that the converse of this statement is not true, i.e. there exist non-divisible dynamical processes that are Markovian in the sense of the above definition. For such processes, the effect of the contribution of the decay channels with a negative rate is overcompensated by the channels with a positive decay rate, resulting in a total information flow directed from the open system to the environment. Further implications are discussed in [35], and specific examples for this class of processes are constructed in [36, 37].

We have interpreted the change of the distinguishability of quantum states as arising from an exchange of information between the open system and its environment. If the distinguishability decreases, e.g., we say that information is flowing from the open system into the environment. The more precise meaning of this statement is that the available relative information on the considered pair of states, that is, the information which enables one to distinguish these states, is lost from the open system, i.e. when only measurements on the open system's degrees of freedom can be performed. This does not imply that the information lost is now contained completely in the reduced states of the environment, but instead, this information can also be carried by the system–environment correlations. And vice versa, if we say that information flows from the environment back to the system, this means that the distinguishability of the open system states increases because information is regained by the open system, information which was carried by the environmental degrees of freedom or by the correlations between the degrees of freedom of system and environment (see also section 4).

For a non-Markovian process described by a family of dynamical maps Φ(t, 0), information must flow from the environment to the system for some interval of time, and thus, we must have σ > 0 for this time interval. A measure of non-Markovianity should measure the total increase of the distinguishability over the whole time evolution, i.e. the total amount of information flowing from the environment back to the system. This suggests defining a measure $\mathcal {N}(\Phi )$ for the non-Markovianity of a quantum process through [5]

$$\begin{equation} \mathcal {N}(\Phi ) = \max _{\rho _S^{1,2}(0)} \int _{\sigma >0}{{\rm d}t\,\sigma \big(t,\rho _S^{1,2}(0)\big)}. \end{equation} \tag{ 76 }$$

The time integration is extended over all time intervals (a_i, b_i) in which σ is positive and the maximum is taken over all pairs of initial states. Due to equation (73), the measure can be written as

$$\begin{eqnarray} &&\mathcal {N}(\Phi ) = \max _{\rho ^{1,2}(0)}\sum _i{\left[D\big(\rho _S^1(b_i),\rho _S^2(b_i)\big) -D\big(\rho _S^1(a_i),\rho _S^2(a_i)\big)\right]}. \nonumber\\ \end{eqnarray} \tag{ 77 }$$

To calculate this quantity, one first determines for any pair of initial states the total growth of the trace distance over each time interval (a_i, b_i) and sums up the contribution of all intervals. $\mathcal {N}(\Phi )$ is then obtained by determining the maximum over all pairs of initial states.

To illustrate our definition of non-Markovianity and the measure (76), we again refer to the example of the dynamical map given by equations (21)–(24). According to equation (63), the time evolution of the trace distance corresponding to any pair of the initial states ρ¹_S(0) and ρ²_S(0) is given by

$$\begin{equation} D\big(\rho _S^1(t),\rho _S^2(t)\big) = |G(t)|\sqrt{|G(t)|^2a^2+|b|^2}, \end{equation} \tag{ 78 }$$

where a = ρ¹₁₁(0) − ρ²₁₁(0) and b = ρ¹₁₀(0) − ρ²₁₀(0). The time derivative of this expression yields

$$\begin{equation} \sigma \big(t,\rho _S^{1,2}(0)\big) = \frac{2|G(t)|^2a^2 + |b|^2}{\sqrt{|G(t)|^2a^2 + |b|^2}} \frac{\rm d}{{\rm d}t}|G(t)|. \end{equation} \tag{ 79 }$$

We conclude from this equation that the trace distance increases at time t if and only if the function |G(t)| increases at this point of time. It follows that the process is non-Markovian, $\mathcal {N}(\Phi )>0$, if and only if the dynamical map is non-divisible, which in turn is equivalent to a temporarily negative rate γ(t) in the time-local master equation with generator (47).

Consider, e.g., the case of an exponential correlation function, which leads to expression (29) for the function G(t). For small couplings, γ₀ < λ/2, this function decreases monotonically. The dynamical map is then divisible, the rate γ(t) is always positive and the process is Markovian. However, in the strong-coupling regime, γ₀ > λ/2, the function |G(t)| starts to oscillate, showing a non-monotonic behaviour. Consequently, the dynamical map is then no longer divisible and the process is non-Markovian. Note that in the strong-coupling regime the rate γ(t) diverges at the zeros of G(t). However, the master equation can still be used to describe the evolution between successive zeros, and therefore, the connection between non-Markovianity, divisibility and a negative rate in the master equation remains valid. There is thus a threshold γ₀ = λ/2 for the system–reservoir coupling below which $\mathcal {N}(\Phi )=0$. One finds, as expected, that for γ₀ > λ/2 the non-Markovianity increases monotonically with increasing system–environment coupling [7]. Moreover, it is easy to show with the help of equation (78) that the maximum in equation (76) is attained for a = 0 and |b| = 1 [38]. This means that the optimal pairs of initial states correspond to antipodal points on the equator of the Bloch sphere representing the qubit.

The non-Markovianity measure (76) has been employed recently for the theoretical characterization and quantification of memory effects in various physical systems. An application to the information flow in the energy-transfer dynamics of photosynthetic complexes has been developed in [39] and to ultracold atomic gases in [40]. Further applications to memory effects in the dynamics of qubits coupled to spin chains [41] and to complex quantum systems with regular and chaotic dynamics [42] have been reported. A series of examples and details of the relation between the classical and the quantum concepts of non-Markovianity are discussed in [43].

The non-Markovianity measure (76) represents a novel experimentally accessible quantity. Quite recently, two experiments determining this quantity have been carried out employing photons moving in optical fibres or birefringent materials [44, 45]. In these experiments, the open system is provided by the polarization degrees of freedom of the photons, while the environment is given by their frequency (mode) degrees of freedom. It has been demonstrated that a complete determination of the measure is experimentally possible, including the realization of the maximization over initial states in the definition (76). It was shown further that a careful preparation of the initial environmental state allows one to control the information flow between the system and its environment and to observe the transition from the Markovian to the non-Markovian regime through quantum-state tomography carried out on the open system. This means that an experimental quantification and control of memory effects in open quantum systems is indeed feasible, which could be useful in the development of quantum memory and communication devices.

While in the general case of correlated initial states a quantum dynamical map can only be defined for a restricted set of states [47], we can of course always introduce a linear map

$$\begin{equation} \Lambda (t,0): \, S({\mathcal {H}}_S\otimes {\mathcal {H}}_E) \longrightarrow S({\mathcal {H}}_S) \end{equation} \tag{ 80 }$$

on the total system's state space $S({\mathcal {H}}_S\otimes {\mathcal {H}}_E)$ by means of

$$\begin{equation} \rho (0) \mapsto \rho _S(t) = \Lambda (t,0) \rho (0) = {\rm {tr}}_E \lbrace U(t) \rho (0) U^{\dagger }(t) \rbrace . \ \end{equation} \tag{ 81 }$$

This map takes any initial state ρ(0) of the total system to the corresponding reduced open system state ρ_S(t) at time t. Since unitary transformations and partial traces are CPT maps, the composite maps Λ(t, 0) are again CPT maps. Considering a pair of total initial states ρ^{1, 2}(0) and the corresponding open system states at time t,

$$\begin{equation} \rho ^{1,2}_S(t) = \Lambda (t,0) \rho ^{1,2}(0), \end{equation} \tag{ 82 }$$

we then have the following by use of the contraction property (67) for CPT maps:

$$\begin{equation} D\big(\rho ^1_S(t),\rho ^2_S(t)\big) \le D(\rho ^1(0),\rho ^2(0)), \end{equation} \tag{ 83 }$$

which means that the distinguishability of the open system states can never be larger than the distinguishability of the total initial states. Subtracting the initial trace distance of the open system states, we obtain

$$\begin{eqnarray} &&\fl D\big(\rho ^1_S(t),\rho ^2_S(t)\big) - D\big(\rho ^1_S(0),\rho ^2_S(0)\big) \le I\big(\rho ^1(0),\rho ^2(0)\big)\nonumber\\ &&\equiv D\big(\rho ^1(0),\rho ^2(0)\big) - D\big(\rho ^1_S(0),\rho ^2_S(0)\big). \end{eqnarray} \tag{ 84 }$$

The increase of the trace distance between the open system states is thus bounded from above by the quantity I(ρ¹(0), ρ²(0)) on the right-hand side of this equation. This quantity represents the distinguishability of the total initial states minus the distinguishability of the initial open system states. It can be interpreted as the information on the total system states, which is outside the open system, i.e. which is inaccessible through local measurements on the open system. Thus, we see that the increase of the distinguishability of the open system states and, hence, the flow of information to the open system are bounded by the information, which is outside the open system at the initial time.

We now show how the upper bound of equation (84) is connected to the correlations in the initial states. To this end, we use twice the triangular inequality (61) to obtain

$$\begin{eqnarray} &&\fl D\big(\rho ^1(0),\rho ^2(0)\big) \le D\big(\rho ^1(0),\rho _S^1(0)\otimes \rho _E^1(0)\big)\nonumber\\ &&+\, D\big(\rho ^2(0), \rho _S^1(0)\otimes \rho _E^1(0)\big)\nonumber\\ &&\le D\big(\rho ^1(0),\rho _S^1(0)\otimes \rho _E^1(0)\big) + D\big(\rho ^2(0), \rho _S^2(0)\otimes \rho _E^2(0)\big)\nonumber\\ &&+\, D\big(\rho _S^1(0)\otimes \rho _E^1(0),\rho _S^2(0)\otimes \rho _E^2(0)\big). \end{eqnarray} \tag{ 85 }$$

With the help of the subadditivity (64) of the trace distance, we find

$$\begin{eqnarray} &&D\big(\rho _S^1(0)\otimes \rho _E^1(0),\rho _S^2(0)\otimes \rho _E^2(0)\big)\quad\nonumber\\ &&\quad \le D\big(\rho _S^1(0),\rho _S^2(0)\big) + D\big(\rho _E^1(0),\rho _E^2(0)\big). \end{eqnarray} \tag{ 86 }$$

Combining this with equations (85) and (84), we finally obtain

$$\begin{eqnarray} &&\fl D\big(\rho ^1_S(t),\rho ^2_S(t)\big) - D\big(\rho ^1_S(0),\rho ^2_S(0)\big)\nonumber\\ &&\le D\big(\rho ^1(0),\rho _S^1(0)\otimes \rho _E^1(0)\big) + D\big(\rho ^2(0),\rho _S^2(0)\otimes \rho _E^2(0)\big) \nonumber \\ &&+\, D\big(\rho _E^1(0),\rho _E^2(0)\big). \end{eqnarray} \tag{ 87 }$$

For any total system state ρ, the trace distance D(ρ, ρ_S⊗ρ_E) between ρ and the product of its marginals ρ_S⊗ρ_E represents a measure for the correlations in the state ρ, which quantifies how well ρ can be distinguished from the corresponding product state ρ_S⊗ρ_E. Thus, equation (87) demonstrates that a dynamical increase of the trace distance of the open system states over the initial value,

$$\begin{eqnarray} &&D\big(\rho ^1_S(t),\rho ^2_S(t)\big) - D\big(\rho ^1_S(0),\rho ^2_S(0)\big)>0, \end{eqnarray} \tag{ 88 }$$

implies that the corresponding initial environmental states ρ^{1, 2}_E(0) are different or that at least one of the total initial states ρ^{1, 2}(0) is correlated.

As an example, we consider the special case of an initial pair of states given by a correlated state ρ¹(0) and the uncorrelated product of its marginals ρ²(0) = ρ¹_S(0)⊗ρ_E¹(0). Hence, we have ρ¹_S(0) = ρ_S²(0) and ρ¹_E(0) = ρ_E²(0), and equation (87) reduces to

$$\begin{equation} D\big(\rho ^1_S(t),\rho ^2_S(t)\big) \le D\big(\rho ^1(0),\rho _S^1(0)\otimes \rho _E^1(0)\big). \end{equation} \tag{ 89 }$$

Thus, the increase of the trace distance between ρ¹_S(t) and ρ²_S(t) (which is zero initially) is bounded from above by the amount of correlations in the initial state ρ¹(0).

We further remark that in the case of two uncorrelated states with the same environmental state, i.e. ρ^{1, 2}(0) = ρ^{1, 2}_S(0)⊗ρ_E(0), the right-hand side of the inequality (87) vanishes, and we are led again to the contraction property (72) for CPT dynamical maps on the reduced state space. Thus, equation (87) represents a generalization of the contraction property of dynamical maps on the reduced state space.

Equation (87) leads to an experimentally realizable scheme for the local detection of correlations in an unknown total system's initial state ρ¹(0) as follows [46]. First, one generates a second reference state ρ²(0) by applying a local trace-preserving quantum operation ${\mathcal {E}}$:

$$\begin{equation} \rho ^2(0) = ({\mathcal {E}}\otimes I) \rho ^1(0). \end{equation} \tag{ 90 }$$

The operation ${\mathcal {E}}$ acts only on the variables of the open system and can be realized, e.g., by the measurement of an observable of the open system, or by a unitary transformation induced, e.g., through an external control field. It is easy to check that ρ¹(0) and ρ²(0) lead to the same reduced environmental state, i.e. we have ρ¹_E(0) = ρ²_E(0), and hence, equation (87) yields

$$\begin{eqnarray} &&\fl D\big(\rho ^1_S(t),\rho ^2_S(t)\big) - D\big(\rho ^1_S(0),\rho ^2_S(0)\big)\nonumber\\ &&{\le}\, D\big(\rho ^1(0),\rho _S^1(0)\,{\otimes}\, \rho _E^1(0)\big) \,{+}\, D\big(\rho ^2(0), \rho _S^2(0)\otimes \rho _E^2(0)\big).\nonumber\\ \end{eqnarray} \tag{ 91 }$$

This inequality shows that any dynamical increase of the trace distance between the open system states over the initial value implies the presence of correlations in the initial state ρ¹(0). In fact, if one finds that the left-hand side of the inequality is greater than zero, then either ρ¹(0) or ρ²(0) must be correlated. If ρ¹(0) was uncorrelated, then also ρ²(0) must be uncorrelated since it is obtained from ρ¹(0) through application of a local operation. Thus, any increase of the trace distance of the open system states over the initial value witnesses correlations in ρ¹(0).

We note that this strategy for the local detection of initial correlations requires only local control and measurements of the open quantum system. It neither demands knowledge about the structure of the environment or of the system–environment interaction, nor a full knowledge of the initial system–environment state ρ¹(0). Moreover, there is no principal restriction on the operation ${\mathcal {E}}$ used to generate the second state ρ²(0), which makes the scheme very flexible in practice. In fact, experimental realizations of the scheme have been reported recently [48, 49]. Further examples and applications to the study of correlations in thermal equilibrium states are discussed in [50]. Moreover, by taking ${\mathcal {E}}$ to be a pure dephasing operation, the scheme enables the local detection of nonclassical correlations, i.e. of total initial states with nonzero quantum discord [51]. This fact has been shown very recently in [52], where also a statistical approach to initial correlations on the basis of random matrix theory has been developed.