From the Heisenberg to the Schrödinger Picture:
Quantum Stochastic Processes and Process Tensors^††thanks: To appear in Proceedings of the 60th IEEE Conference on Decision and Control (CDC), Dec. 13-15, 2021

Modern probability theory as formulated by Kolmogorov [1] underpins the theory of stochastic processes, stochastic systems and stochastic control [2, 3]. Similarly, beginning with the seminal work of von Neumann on the axiomatization of quantum mechanics [4], quantum probability theory has emerged as a non-commutative generalization of probability theory [5]. It provides a natural setting for a theory of quantum stochastic processes as a non-commutative generalization of the classical theory of Kolmogorov. A major departure of the quantum setting from the classical one is that the random outcomes of sequential measurements on quantum stochastic processes do not in general satisfy the Kolmogorov consistency conditions and hence cannot be described as classical stochastic processes with well-defined sample paths.

A general quantum probabilistic theory of quantum stochastic processes was introduced in the seminal work of Accardi, Frigerio and Lewis (AFL) [6]. The formulation is given in the Heisenberg picture, generalizing the Kolmogorovian theory of classical stochastic processes. This is in the sense that observables as quantum random variables evolve with time while the state of the system is kept fixed, just as how random variables evolve in time in a classical stochastic process while the probability measure on the underlying classical probability space remains fixed. Quantum stochastic processes as operator-valued processes are defined independently of single or sequential measurements that may be performed on the process at any time. However, measurements and their probabilistic outcomes are accounted for by the correlation kernels of quantum stochastic processes, playing a similar role to the family of finite-dimensional distributions for classical stochastic processes.

Recent efforts in trying to understand and characterize temporal quantum noise in engineered quantum systems have led to the consideration of alternative formalisms for defining, describing and witnessing non-Markovian quantum processes, see, e.g., [7]. Unlike [6], these formalisms are developed in the Schrödinger picture, with an emphasis on the transformations of states (described by a density operator) of the system of interest. The process tensor formalism was introduced in [8] to overcome the limitations of conventional descriptions of non-Markovian dynamics based on the reduced dynamics of the system in the Schrödinger picture (as linear transformations of the system’s states). This includes addressing initial system-environment correlation and multi-time interventions.

However, there have been so far no studies to reconcile the process tensor formalism to the well-established AFL theory and its subsequent developments. Since both formalisms are concerned with related objects and the same physics but set in different pictures (Heisenberg vs Schrödinger), one would expect a close relationship between the two. In this paper, we connect the process tensor to AFL theory through the correlation kernels of quantum stochastic processes. In particular, it is shown how process tensors can be recovered from extended correlation kernels incorporating ancillas. Along the way, we also give a tutorial style overview on quantum stochastic processes, multi-time correlations and sequential measurements on quantum systems. In particular, we highlight subtle points surrounding multi-time correlations and sequential measurements, emphasizing the latter’s departure from Kolmogorovian classical stochastic processes.

A quantum probability space is a pair $(\mathscr{X},\mu)$ where $\mathscr{X}$ is a von Neumann algebra of bounded operators on some Hilbert space $\mathfrak{h}$ (containing the identity operator $I_{\mathscr{X}}$) and and $\mu$ is a unital normal state on $\mathscr{X}$ (unital meaning $\mu(I_{\mathscr{X}})=1$). Recall that a von Neumann algebra is a *-algebra of operators equipped with addition, multiplication (as composition of operators) and involution ^∗ (defined as the adjoint of the operator) and is closed with respect to the normal topology of sub-algebras of $\mathrm{B}(\mathfrak{h})$. For normal states $\mu$, there exists a density operator $\rho$ on $\mathfrak{h}$ such that $\mu(X)={\rm tr}(\rho X)$ for all $X\in\mathscr{X}$; see [9] and the references therein. We will also write $\mu(\cdot)$ using the quantum expectation notation $\langle\cdot\rangle$.

A classical probability space $(\Omega,\mathcal{F},P)$ can be viewed as a Banach algebra $L^{\infty}(\Omega,\mathcal{F},P)$ of essentially bounded random variables on $(\Omega,\mathcal{F},P)$. A quantum probability space $(\mathscr{C},\mu)$, with $\mathscr{C}$ commutative¹¹1Meaning that the elements of $\mathcal{C}$ are commuting with one another. and $\mu$ a unital normal state, is *-isomorphic to a classical probability space $(L^{\infty}(\Omega,\mathcal{F},\nu),\mathbb{E})$, where the expectation operator $\mathbb{E}(X)=\int_{\Omega}X(\omega)P(d\omega)$ for some measure $P$ that is absolutely continues with respect to $\nu$. This *-isormorphism is a bijective map $\iota:(\mathscr{C},\mu)\rightarrow(L^{\infty}(\Omega,\mathcal{F},\nu),\mathbb{E})$ with the properties $\iota(ab)=\iota(a)\iota(b)$ and $\iota(b^{*})=\overline{\iota(b)}$ for any $a,b\in(\mathscr{C},\mu)$. The *-isomorphism be tween a classical probability space and commutative von Neumann algebra is known as the Spectral Theorem, see, e.g., [9, Theorem 3.3].

The physical interpretation of the quantum probability space $(\mathscr{X},\mu)$ is as follows. The underlying Hilbert space of the operators in $\mathscr{X}$ is the Hilbert space of an associated quantum mechanical system. Observables of the system are the self-adjoint operators in $\mathscr{X}$, which are also quantum random variables (i.e., quantum analogues of real-valued random variables). The quantum expectation of an observable $X$ is given by $\mu(X)$. Events $E\in\mathscr{X}$ are projection operators $(E=E^{*}=E^{2})$ and the probability of an event $E$ is given by $\mu(E)$. Only commuting events $E$ can have a joint probability distribution that satisfy the Kolmogorov consistency conditions, non-commuting events cannot be assigned a joint probability distribution. This can be seen as a direct consequence of the Spectral Theorem: since non-commuting events form the elements of a non-commutative algebra it cannot be mapped to a classical probability space.

Let $T\subseteq\mathbb{R}$. A quantum stochastic process over a von Neumann algebra $\mathscr{B}\subseteq\mathrm{B}(\mathfrak{h})$ is a triplet $(\mathscr{A},\{j_{t}\}_{t\in T},\mu)$, with $\mathscr{A}$ another von Neumann algebra of operators, possibly over another Hilbert space $\mathfrak{k}$, $\mu$ a normal state on $\mathscr{A}$, and $j_{t}:\mathscr{B}\rightarrow\mathscr{A}$ $\forall t\in T$ is a *-homomorphism from $\mathscr{B}$ to $\mathscr{A}$, $j_{t}(XY)=j_{t}(X)j_{t}(Y)$ and $j_{t}(X^{*})=j_{t}(X)^{*}$ for any $X,Y\in\mathscr{B}$. Note that $(\mathscr{A},\mu)$ is a quantum probability space.

Since a collection of non-commuting random variables will not have a joint probability distribution, for quantum stochastic processes one considers the more general notion of correlation kernels. For any positive integer $n$, given time tuple $\mathbf{t}_{n}\in T^{n}$, and vectors $\mathbf{a}_{n}=(a_{1},\ldots,a_{n})^{\top}\in\mathscr{B}^{n}$ and $\mathbf{b}_{n}=(b_{1},\ldots,b_{n})^{\top}\in\mathscr{B}^{n}$, correlation kernels $w_{\mathbf{t}_{n}}$ are complex functions on $\mathscr{B}^{n}\times\mathscr{B}^{n}$ of the form

where $j_{\mathbf{t}_{n}}(\mathbf{a}_{n})=j_{t_{n}}(a_{n})j_{t_{n-1}}(a_{n-1})\ldots j_{t_{1}}(a_{1})$. For properties of correlation kernels, see [6, Proposition 1.2]. If $\mathbf{a}_{n}=\mathbf{b}_{n}=\mathbf{E}_{n}$, where the elements $E_{j}$, $j=1,\ldots,n$, of $\mathbf{E}_{n}$ are mutually commuting events (projection operators) in $\mathscr{B}$ then $w_{\mathbf{t}_{n}}(j_{\mathbf{t}_{n}}(\mathbf{E}_{n})^{*}j_{\mathbf{t}_{n}}(\mathbf{E}_{n}))$ gives the joint probability distribution of these events. Otherwise, it gives the probability for the events to occur in that specific time order. For the remainder of the paper, for concreteness we take $\mathfrak{h}$ to be embeddable as a subspace of $\mathfrak{k}$, $\mathscr{A}\subseteq\mathrm{B}(\mathfrak{k})$ and $\mathscr{B}$ to be embeddable as a sub-algebra of $\mathscr{A}$. We consider $j_{t}$ of the form $j_{t}(\cdot)=U_{t}^{*}(\cdot)U_{t}$, with $U_{t}$ a unitary operator on $\mathfrak{h}$ and define $j_{t}^{\star}$ via $j_{t}^{\star}(\cdot)=U_{t}(\cdot)U_{t}^{*}$. Note that by definition, $j_{t}^{\star}$ is the essentially the inverse of $j_{t}$, in the sense that $j_{t}\circ j_{t}^{\star}=j_{t}^{\star}\circ j_{t}=I$, where $I$ is an identity map. We also define $j_{t_{1},t_{2}}(\cdot)=j_{t_{2}}\circ j_{t_{1}}^{\star}(\cdot)=U_{t_{2}}^{*}U_{t_{1}}(\cdot)U_{t_{1}}^{*}U_{t_{2}}$ and in a similar fashion define $j_{t_{1},t_{2}}^{\star}=j_{t_{1}}\circ j_{t_{2}}^{\star}$.

In many cases, we can consider experiments where several measurements are made at specific times over a time interval.

We insert the requirement of nontriviality to ensure that we have a hierarchy - an order $n$ experiment is not a special case of a higher order experiment. Incompatible observations being made at the same time in the same trial are precluded. Of course, the family itself can include incompatible measurements. However, we may make incompatible measurements within the same experiment so long as they are made at different times. The notion of completeness just means that we do as many measurements as possible, restricted to $n$ distinct times within the interval of interest. The aim is to be exhaustive in what is measured.

In an order 1 experiment, each trial involves measuring the system at a single time. In an order 2 family the experimenter measures observables at different times in the same trial and thereby obtains the two-point statistical correlations between different quantities at different times - something not available from the data collected in an order 1 experiment. An order $n$ experiment contains information not available from lower order experiments.

In both classical and quantum theories, order 1 measurements lead to the notion of an “instantaneous state”. For instance, in quantum mechanics we obtain empirically from order one experiments the expectations $\langle X(t)\rangle$: this should take the form $\mathrm{tr}(\rho_{t}X)$ and by varying the observable $X$ measured we determine a density matrix $\rho_{t}$. As such, a complete family of order 1 experiments over the time interval reveals the set of “instantaneous states”, but this is still only partial information. We now focus on what we can obtain empirically from a family of experiments of order $n$.

In quantum theory, the most general multi-time correlation that can be estimated from experiment are those of the form [6]

where the times are ordered as $0\leq t_{1}<t_{2}<\cdots<t_{n}\leq T,$ and the operators are all in the Heisenberg picture at the indicated times. We refer to these as pyramidal time-ordered correlations. The essential feature is that we have increasing times as we work in from the outermost operators into the centre. For instance, suppose an order $n$ experiment involves measuring a $k$th observable at time $t_{k}$ and that this results in an answer $\omega_{k}$. Let $Q_{k}(\omega)$ give the corresponding projection in the Heisenberg picture. (For yes/no experiments we have $Q_{k}(\text{yes})=P_{k}(t_{k})$ and $Q_{k}(\text{no})=I-P_{k}(t_{k})$.) From the experiment we may estimate the empirical probabilities

where $\omega_{k}$ is the answer to the $k$ measurement at time $t_{k}$. From the fact that $\sum_{\omega}Q_{k}(\omega)=I$, we find

This may be rephrased as follows.

However, the projections at different times are not assumed to commute with each other. As a result, the finite dimensional distributions need not satisfy Kolmogorov’s consistency conditions in any of the arguments, other than the very last one. This is a key feature of quantum theory and is the basis for results such as Bell’s Theorem. As this can be misunderstood and lead to erroneous results or conclusions, such as when statistical inference methods based on the existence of joint distributions are applied to the outcomes of non-commuting sequential measurements, it will be revisited in more detail in the next section.

It follows from Section 3 that the correlation kernels $w_{\mathbf{t}_{n}}$ as defined in (1) are intimately related to sequential measurements on quantum stochastic processes. In this section we explicitly illustrate the subtleties of these measurements, which can result in a sequence of random outcomes that fail the Kolmogorov consistency conditions and are therefore not classical stochastic processes.

For simplicity of discussion, consider a discrete-valued observable $X_{j}\in\mathscr{B}$ (i.e., $X$ has a most a countable number of eigenvalues) with all eigenvalues distinct. If a measurement of $X_{j}$ is made at time $t_{j}$, the random outcome $m_{j}$ of the measurement will correspond to the application of a projection operator $P_{m_{j}}\in\mathscr{B}$ corresponding to the eigenvalue $\lambda_{m_{j}}$ of $X_{j}$ that is observed. The probability of sequentially observing the outcomes $m_{1},m_{2},\ldots,m_{n}$ at times $t_{1}<t_{2}<\ldots<t_{n}$ in this order under the evolution of the quantum stochastic process is given by:

Caution is now due. In the quantum context, marginalization over any of the variables $m_{j}$ for any $j<n$ does not in general hold except over the last one at time $t_{n}$ (a violation of the Kolmogorov consistency conditions). That is, in general

where a hat ($\,\widehat{\cdot}\,$) above a variable indicates that the variable is dropped from the list of arguments. In the following, to emphasize this we give some simple but explicit examples. We mention that the general relationship (3) is the basis for violation of the Leggett-Garg inequalities [10] in sequential measurements in quantum mechanics, which is essentially a statement about the failure of the Kolmogorov consistency conditions [11, §7 and Eq. (8.5)]. For further discussions on these issues, we refer to [12, 13]. Measurements that satisfy $[j_{t_{k}}(P_{m_{k}}),j_{t_{l}}(P_{m_{l}})]=0$ for all $k,l$ are referred to as quantum non-demolition (QND) measurements. QND measurements produce a classical stochastic process with a well-defined joint probability distribution for any collection of sampled points from the process.

We first motivate and introduce the notion of a discrete-time process tensor. We start by recalling the definition of quantum operations and quantum instruments, see, e.g., [14].

The set of all such quantum operations is denoted by $\mathcal{O}(\mathfrak{h})$. A special quantum operation is the “do nothing” or identity operation $\mathrm{Id}$, defined by $\mathrm{Id}(X)=X$ for all $X\in\mathrm{B}(\mathfrak{h})$.

Consider a system (labelled by a subscript $s$) with a Hilbert space $\mathfrak{h}_{s}$ interacting with an environment (labelled by a subscript $e$) with a Hilbert space $\mathfrak{h}_{e}$, and let $\mathfrak{h}_{se}=\mathfrak{h}_{s}\otimes\mathfrak{h}_{e}$. The state of the system and environment is initially in the (not necessarily factored) state $\rho_{se}$ and their joint state undergoes a joint unitary evolution between times $t_{j}$ and $t_{j+1}$ given by the map $\mathcal{U}_{t_{j},t_{j+1}}^{se}(\cdot)=U_{t_{j},t_{j+1}}^{se}(\cdot)U_{t_{j},t_{j+1}}^{se*}$, with $U^{se}_{t_{j},t_{j+1}}$ unitary and $U^{se}_{t_{j},t_{j}}=I$. At the discrete-times $0\leq t_{1}<t_{2}<\ldots<t_{n}$ they can undergo measurements performed directly on the system, or after interacting the system (but not the environment) with some freshly prepared ancillas (i.e., ancillas that have been used are not reused on subsequent measurements) followed by measurements of compatible observables on the system and/or ancillas. This is described by a quantum instrument $\mathcal{I}_{t_{j}}=(\mathfrak{h}_{s},\Omega_{j},\mathcal{F}_{j},\mathcal{M}_{j})$ at time $t_{j}$. Given the events $A_{1},\ldots,A_{n}$ with $A_{j}\in\mathcal{F}_{j}$, the unnormalised system-environment density operator at the time $t_{n}$ is given by:

Define the map $\mathcal{T}_{\mathbf{t}_{n}}$ via

then we can write $\sigma_{t_{n}}=\mathcal{T}_{\mathbf{t}_{n}}(\mathcal{M}_{1}(A_{1}),\ldots,\mathcal{M}_{n}(A_{n})).$ The probability of observing the events $A_{1},\ldots,A_{n}$ at the times $t_{1},t_{2},\ldots,t_{n}$ is given by

and the system density operator $\rho_{t_{n}}$ at time $t_{n}$ is then simply the normalized version of $\mathrm{tr}_{\mathfrak{h}_{e}}(\sigma_{t_{n}})$ given by

The map $\mathcal{T}_{\mathbf{t}_{n}}$ defined by (5) can be viewed as a real multilinear map from an ordered sequence of quantum operations $(O_{1},O_{2},\ldots,O_{n})$, corresponding to $(\mathcal{M}_{1}(A_{1}),\mathcal{M}_{2}(A_{2}),\ldots,\mathcal{M}_{n}(A_{n}))$, to an unnormalised density operator. As such, the right hand side of (5) can be viewed as a real linear map on the tensor product of quantum operations $\mathcal{O}(\mathfrak{h}_{s})^{\otimes n}$, with the sequence $(O_{1},\ldots,O_{n})$ being mapped to the algebraic tensor product $O_{1}\otimes O_{2}\otimes\cdots\otimes O_{n}$. General elements of $\mathcal{O}(\mathfrak{h}_{s})^{\otimes n}$ are linear combinations of such tensor product maps and limits thereof. They correspond to “correlated” measurements that involve the use of the same ancillas at different time points or the presence of correlated states between distinct ancillas at different times. It has been shown that such maps, in the special case of finite discrete-valued measurements, have the properties [8, 13]

1. (i)

   $\mathrm{tr}(\mathcal{T}_{\mathbf{t}_{n}}(O))\leq 1$ $\forall O\in\mathcal{O}({\mathfrak{h}_{s})}^{\otimes n}$.

2. (ii)

   Complete positivity as a map from $\mathcal{O}(\mathfrak{h}_{s})^{\otimes n}$ to $\mathrm{S}(\mathfrak{h}_{s})$.

3. (iii)

   Containment, for any $\mathbf{s}_{m}\subset\mathbf{t}_{n}$ $(m<n)$ it holds that $\mathcal{T}_{\mathbf{s}_{m}}(O_{s_{1}}\otimes\cdots\otimes O_{s_{m}})=\mathcal{T}_{\mathbf{t}_{n}}(O^{\prime}_{t_{1}}\otimes\cdots\otimes O^{\prime}_{t_{n}})$, where $O^{\prime}_{t_{j}}=O_{t_{j}}$ if $t_{j}\in\mathbf{s}_{m}$, otherwise $O_{t_{j}}=\mathrm{Id}$.

We are now ready to define the process tensor [8, 13] but stated in a more general form that allows for continuous-valued measurements.

When the system Hilbert space $\mathfrak{h}_{s}$ is finite dimensional, process tensors can be represented as generalized Choi many-body states and can be cast into a matrix-product-operator form. In this case, these representations make manipulation of process tensors convenient [8].

We now show the relation of the process tensor to a quantum stochastic process of AFL through the correlation kernels (1).

Let $\mathcal{B}_{s}$ and $\mathcal{B}_{e}$ be von Neumann algebras over the system and environment Hilbert space $\mathfrak{h}_{s}$ and $\mathfrak{h}_{e}$, respectively. The composite space for the system is environment is the quantum probability space $(\mathcal{B}_{se},\mu_{se})$, where $\mathcal{B}_{se}=\mathcal{B}_{s}\otimes\mathcal{B}_{e}$ and $\mu_{se}$ is a normal state on $\mathcal{B}_{se}$. Note that the state $\mu_{se}$ is not necessarily of the factored form $\mu_{s}\otimes\mu_{e}$ for some states $\mu_{s}$ and $\mu_{e}$ on the system and environment, respectively, since the system and environment can be initially entangled or correlated. Take the time to be $T=[0,\infty)$. We define a quantum stochastic process $\mathcal{Q}_{se}$ over $\mathcal{B}_{s}$ as $\mathcal{Q}_{se}=(\mathcal{B}_{se},\{j^{se}_{t}\}_{t\in T},\mu_{se})$.

We now attach to $\mathcal{Q}_{se}$ an ancillary system with quantum probability space $(\mathcal{B}_{a},\mu_{a})$, where $\mathcal{B}_{a}$ is a von Neumann algebra over the ancilla Hilbert space $\mathfrak{h}_{a}$. We then define another quantum stochastic process over $\mathcal{B}_{as}=\mathcal{B}_{a}\otimes\mathcal{B}_{s}$ as $\mathcal{Q}_{ase}=(\mathcal{B}_{ase},\{j^{ase}_{t}\}_{t\in T},\mu_{ase})$, where $\mathcal{B}_{ase}=\mathcal{B}_{as}\otimes\mathcal{B}_{e}$, $\mu_{ase}=\mu_{a}\otimes\mu_{se}$, and $j^{ase}_{t}$ acts as

That is, $j^{ase}_{t}$ acts non-trivially only on a factor in $\mathcal{B}_{se}$.

For any time tuple $\mathbf{t}_{n}$ with $0\leq t_{1}<t_{2}<\ldots<t_{n}$, the correlation kernel $w^{ase}_{\mathbf{t}_{n}}$ for $\mathcal{Q}_{ase}$ is given by

where the components of $\mathbf{a}_{n}$ and $\mathbf{b}_{n}$ are operators on $\mathcal{B}_{as}$.

By the polarizing identity,

it suffices to consider correlation kernels with $\mathbf{b}_{n}=\mathbf{a}_{n}$. Choose $\mathfrak{h}_{a}$, ancilla operators $V_{1,r_{1}},\ldots,V_{n,r_{n}}$ for $r_{j}=1,\ldots,\chi_{j}$ (with $\chi_{j}$ a nonnegative integer) and $j=1,\ldots,n$, and state $\mu_{a}$ such that $\mu_{a}(V_{1,r^{\prime}_{1}}^{*}\cdots V_{n,r^{\prime}_{n}}^{*}V_{n,r_{n}}\cdots V_{1,r_{1}})=\mu_{a}(V_{1,r_{1}}^{*}\cdots V_{n,r_{n}}^{*}V_{n,r_{n}}\cdots V_{1,r_{1}})\prod_{j=1}^{n}\delta_{r_{j}r^{\prime}_{j}}$ for all $r_{j},r^{\prime}_{j}$, where $\delta_{jk}$ is the Kronecker delta. Let $a_{j}\in\mathcal{B}_{as}$ of the form $a_{j}=\sum_{r_{j}=1}^{\chi_{j}}V_{j,r_{j}}\otimes W_{j,r_{j}}$, with $V_{j,r_{j}}\in\mathcal{B}_{a}$ and $W_{j,r_{j}}\in\mathcal{B}_{s}$. For this choice of $a_{j}$ and using the fact $\mu_{ase}(\cdot)=\mathrm{tr}(\rho_{a}\otimes\rho_{se}(\cdot))$ for some density operators $\rho_{a}$ on $\mathfrak{h}_{a}$ and $\rho_{se}$ on $\mathfrak{h}_{s}\otimes\mathfrak{h}_{e}$, we have that,

where $\alpha_{r_{1},\ldots,r_{n}}=\mu_{a}(V_{1,r_{1}}^{*}\cdots V_{n,r_{n}}^{*}V_{n,r_{n}}\cdots V_{1,r_{1}})\geq 0$ and $\mathcal{W}_{j,r_{j}}:\mathcal{B}_{s}\rightarrow\mathcal{B}_{s}$ is map defined by $\mathcal{W}_{j,r_{j}}(\cdot)=W_{j,r_{j}}(\cdot)W_{j,r_{j}}^{*}$. Note that in the development above we have identified $\mathcal{W}_{j,r_{j}}$ with its ampliation $\mathcal{W}_{j,r_{j}}\otimes I$ on $\mathcal{B}_{se}$.

Define the linear operator $\mathcal{T}^{s}_{\mathbf{t}_{n}}$ via,