Quantum Stochastic Processes and Quantum non-Markovian Phenomena

The field of classical stochastic processes forms a major branch of mathematics. Stochastic processes are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences. When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers. Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities. This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum-information theory. We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain. Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes. We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.

Popular Summary

Predicting the spread of a disease or fluctuations at the stock market, modeling chemical reactions, and computing photon arrival probabilities; all of these are amongst the many problems firmly footed in the field of classical stochastic processes, a vast branch of mathematics with numerous practical applications in social, biological, and physical sciences.

Today we are in the midst of a second quantum revolution with more and more sophistication in engineered quantum devices that aims to process information exploiting the rules of quantum mechanics. A key obstacle in the way of building reliable quantum-information processors is noise, which too constitutes a stochastic process, but one that has to be modeled quantum mechanically. It is thus crucial to employ a fully quantum theory of stochastic processes to—amongst others—better characterize and mitigate quantum noise. However, the structure of quantum mechanics has proven cumbersome to a straightforward generalization of the classical theory to the quantum realm and scientists have struggled to develop a complete theory of quantum stochastic processes.

Recent developments in quantum-information theory have made it possible to overcome the old challenges and to provide a comprehensive framework that can account for genuine quantum stochastic phenomena. The current theory stands on a firm mathematical and conceptual ground and contains the classical theory as a limiting case. This tutorial lays out the fundamentals of the basic structures of quantum stochastic processes, maps out their derivation starting from fully classical considerations, and elucidates why these structures indeed provide a comprehensive description of all conceivable quantum stochastic processes.

While taming quantum noise is an important application in its own right, quantum stochastic processes promise far more theoretical and practical applications in the future, including understanding the nature of complex noise. Consequently, this topic will see tremendous growth in the future and will end up at least as large as its classical counterpart, and possibly as impactful.

Figure 1

Classical die processes. (a) Fair toss; (b) perturbed toss; and (c) the perturbation strength depends on the history.

Figure 2

Random, Markovian, and non-Markovian processes. Top panel shows the circuits for random and Markovian die. In these cases, there are no extra lines of communication between the tosses (represented by boxes). Only the system carries the information forward for a Markov process. The bottom panel shows the non-Markovian die. Here, information is sent between tosses (represented by boxes) in addition to what the system carries, which is the memory of the past states of the system (die). This memory is then denoted by the thick line. The memory has to carry the information about the state of the die in the past four tosses to determine the intensity of the next perturbation.

Figure 3

Memory in non-Markovian processes. For processes with memory, besides the state of the system at present, we need additional information about the past to correctly predict future statistics. If only the probability of the next outcome is of interest, then a map ${\mathrm{\Gamma }}^{(m)}$ of the form of Eq. (24) is sufficient, if all future probabilities are to be computed via the concatenation of a single map, then $\mathrm{\Xi }$, given in Eq. (26), is required. Together, the system and memory undergo Markovian dynamics.

Figure 4

Markov chains. Given a time series of coin flips we can deduce any of the above hidden Markov models. At memory length 2 we have a deterministic process and therefore longer memory will not yield any more information. In other words, the Markov order of the process in (c) is 2.

Figure 5

Continuous process. A stochastic process is a joint probability distribution over all times. From a physical perspective, we can think of it as the probability of observing a trajectory $s_k$. This is highly desirable when talking about the motion of a Brownian particle. However, this interpretation requires some caution as there are cases where trajectories may not be smooth or even continuous.

Figure 6

Hierarchy of multitime processes. A stochastic process is the joint probability distribution over all times. Of course, in practice one looks only at finite-time statistics. However, the set of all $k$-time probability distributions $\{{\mathbb{P}}_{{\mathbf{T}}_k}\}$ contain, as marginals, all $j$-time probability distributions $\{{\mathbb{P}}_{{\mathbf{T}}_j}\}$ for $j<k$. Moreover, the set of two and three time distributions plays a significant role in the theory of stochastic processes.

Figure 7

Stroboscopic divisible non-Markovian process. At each time $t_j$, each of the possible outcomes $0$ and $1$ occurs with probability $1/2$ (for example, they could be drawn from urns with uniform distributions). At the final time $t_4$, the observed outcome is equal to the sum (modulo 2) of the previous three outcomes. While the stochastic map between any two points in time is completely random—and thus the process is divisible—the overall joint probability distribution shows multitime memory effects (as laid out in the text).

Figure 8

Nonconvex decomposition. All states in the $x$-$y$ plane of the Bloch sphere, including the pure states, can be described by the basis state ${\hat{\varrho }}_1$, ${\hat{\varrho }}_2$, and ${\hat{\varrho }}_4$ in Eq. (69). However, only the states in shaded region will be convex mixtures of these basis states. Of course, no pure state can be expressed as a convex mixture.

Figure 9

Quantum process tomography. Any quantum channel $\mathcal{E}$ can be reconstructed by preparing a basis of input states and measuring the corresponding output states ${\rho }_j^{\prime }=\mathcal{E}[{\hat{\varrho }}_j]$ with an informationally complete POVM. The corresponding duals $\{{\hat{D}}_j\}$, $\{{\hat{\mathrm{\Delta }}}_k\}$ and the outcome probabilities then allow for the reconstruction of $\mathcal{E}$ according to Eq. (86).

Figure 10

Choi-Jamiołkowski isomorphism. A map $\mathcal{E}:\mathcal{B}({\mathcal{H}}^{\mathtt{i}})\to \mathcal{B}({\mathcal{H}}^{\mathtt{o}})$ can be mapped to a matrix ${\mathrm{{\rm Y}}}_{\mathcal{E}}\in \mathcal{B}({\mathcal{H}}^{\mathtt{o}}\otimes {\mathcal{H}}^{\mathtt{i}})$ by letting it act on one half on a maximally entangled state. Note that, for ease of notation, here, we let $\mathcal{E}$ act on $\mathcal{B}({\mathcal{H}}_{{\mathtt{i}}^{\prime }})$, such that ${\mathrm{{\rm Y}}}_{\mathcal{E}}\in \mathcal{B}({\mathcal{H}}^{\mathtt{o}}\otimes {\mathcal{H}}^{\mathtt{i}})$. As ${\mathcal{H}}^{{\mathtt{i}}^{\prime }}\cong {\mathcal{H}}^{\mathtt{i}}$, this is merely a relabeling and not of conceptual relevance.

Figure 11

Stinespring dilation. Any CPTP map on the system $S$ can be represented in terms of a unitary on a larger space $SA$ and final discarding of the additional degrees of freedom. The corresponding unitary can be computed by means of Eq. (105). Here, for simplicity, we drop the explicit distinction between input and output spaces we normally employ.

Figure 12

A simple quantum process that violates the assumptions of the KET. Successive measurements of the spin of a spin-$\frac{1}{2}$ particle do not allow one to predict the statistics if the intermediate measurement is not conducted. Here, measuring in the $x$ basis is invasive, and thus summing over the respective outcomes is not the same as not having done the measurement at all.

Figure 13

Perturbed coin with interventions. Between measurements, the coin—which initially shows heads—is perturbed and stays on its side with probability $p$ and flips with probability $1-p$, leading to a stochastic matrix $\mathrm{\Gamma }$ between measurements. Using their instrument, upon measuring an outcome, the experimenter flips the coin. Here, this is shown for the sequence of outcomes $th$. For most values of the probability $p$, this process—despite being fully classical—does not satisfy the requirement of the KET.

Figure 14

Spatial measurements. Alice, Bob, Charlie, and David perform measurements on a seven-partite quantum state $\rho$. Both Bob and Charlie have access to two parts of said state, respectively, but while Bob can perform correlated measurements on said systems, Charlie can only access them independently. The probabilities corresponding to the respective outcomes are computed via the Born rule [see Eq. (139)].

Figure 15

Complete positivity and trace preservation for superchannels. A superchannel is said to be CP if it maps CP maps to CP maps (even when acting on only a part of them), and we call it CPTP if it maps CPTP maps to CPTP maps. Here, ${\mathcal{T}}_{(t:0)}$ is CP (CPTP) if for all CP (CPTP) maps $\mathcal{A}$ and all possible ancilla sizes, the resulting map ${\mathcal{A}}^{\prime }$ is also CP (CPTP). Note that for the TP part, it is already sufficient that ${\mathcal{T}}_{(t:0)}$ maps all CPTP maps on the system to a unit-trace object.

Figure 16

General quantum stochastic process. The system of interest is coupled to an unknown environment and probed at times ${\mathbf{T}}_k=\{t_0,t_1,\dots ,t_k\}$ with corresponding CP maps ${\mathcal{A}}_{{\mathbf{x}}_{{\mathbf{T}}_k}}=\{{\mathcal{A}}_{x_0},{\mathcal{A}}_{x_1},\dots ,{\mathcal{A}}_{x_k}\}$. In between measurements, the system and the environment together undergo closed, i.e., unitary dynamics. The corresponding multitime joint probabilities can be computed by means of the process tensor corresponding to the process at hand (depicted by the gray dotted outline).

Figure 17

Choi state of a process tensor. At each time, half of an unnormalized maximally entangled state is fed into the process. For better bookkeeping, all spaces are labeled by their respective time. The resulting many-body state ${\mathrm{{\rm Y}}}_{{\mathbf{T}}_k}$ contains all spatiotemporal correlations of the corresponding process as spatial correlations.

Figure 18

Trace conditions on process tensors. Displayed is the pertinent part of Fig. 17. As $\mathrm{tr}\circ \mathcal{U}=\mathrm{tr}$ for all CPTP maps $\mathcal{U}$, tracing out the final degree of freedom of ${\mathrm{{\rm Y}}}_{{\mathbf{T}}_k}$, denoted by $k^{\mathtt{i}}$, amounts to a partial trace of ${\mathrm{\Phi }}_{k-1^{\mathtt{o}}}^{+}$. This, in turn, yields a tensor product between ${\mathbb{1}}_{{k-1}^{\mathtt{i}}}$ and a process tensor on one step less. As in Fig. 17, the swap operation is represented by a vertical line with crosses at its ends.

Figure 19

Tester element. In the most general case, an experimenter can correlate the system of interest with an ancilla (here, initially in state $|\mathrm{\Psi }⟩$), use said ancilla again at the next time, etc., and make a final measurement with outcome $\mathbf{x}$ in the end. As the unitaries ${\mathcal{V}}_j$ can also act trivially on parts of the ancilla, this scenario includes all conceivable measurements an experimenter can perform. Summing over the outcomes ${\mathbf{x}}_{{\mathbf{T}}_k}$ amounts to tracing out the ancillas, thus yielding a proper comb (compare with Fig. 16). Note that the inputs (outputs) of the resulting tester elements correspond to the outputs (inputs) of the process tensor, and the system of interest corresponds to the top line, not the bottom line.

Figure 20

Action of a process tensor on a tester element. “Contracting” a process tensor (depicted in blue) with a temporally correlated measurement, i.e., a tester element (depicted in green), yields the probability for the occurrence of said tester element.

Figure 21

Dilation of the Choi state of a process tensor. Up to normalization, Eqs. (192) and (193) together yield a quantum circuit for the implementation of the Choi state of a two-step process tensor that consists only of pure states and isometries (which could be further dilated to unitaries).

Figure 22

Process tensor corresponding to Fig. 21. Rearranging the wires of the circuit of Fig. 21 and removing the maximally entangled states (i.e., undoing the CJI), yields the representation of a process tensor we already encountered in the previous section. As above, note that ${\mathcal{H}}_0^{\mathtt{o}}\cong {\mathcal{H}}_{0^{\prime }}^{\mathtt{o}}$ and ${\mathcal{H}}_1^{\mathtt{o}}\cong {\mathcal{H}}_{1^{\mathrm{\prime }}}^{\mathtt{o}}$, such that this is indeed the correct dilation of ${\mathrm{{\rm Y}}}_{0^{\mathtt{i}}{0}^{\mathtt{o}}{1}^{\mathtt{i}}{1}^{\mathtt{o}}{2}^{\mathtt{i}}}$.

Figure 23

Consistency condition for quantum stochastic processes. Letting a process tensor act on an identity (here at time $t_2$) yields the correct process tensor on the remaining times.

Figure 24

Hierarchy of multitime quantum processes. A quantum stochastic process is the process tensor over all times. Of course, in practice one looks only at finite-time statistics. However, the generalized extension theorem tells us that the set of all $k$-time process tensors $\{{\mathrm{{\rm Y}}}_{{\mathbf{T}}_k}\}$ contain, as marginals, all $j$-time probability distributions $\{{\mathrm{{\rm Y}}}_{{\mathbf{T}}_j}\}$ for $j<k$. Moreover, the set of two and three time processes plays a significant roles in the theory of quantum stochastic processes. Here, we display only a small part of the multifaceted structure of non-Markovian quantum processes. For a much more comprehensive stratification, see Refs. [7, 10].

Figure 25

“Trajectories” of a quantum stochastic process. An open quantum process is fully described once all joint probabilities for sequences of outcomes are known for all possible instruments an experimenter can employ to probe the process. Like in the classical case, each sequence of outcomes can be considered a trajectory, but unlike in the classical case, there is no ontology attached to such trajectories. Additionally, each sequence of outcomes in the quantum case corresponds to a sequence of measurement operators, not just labels. If both the process and the allowed (noninvasive) measurements are diagonal in the same fixed basis, then the above figure coincides with Fig. 5, where trajectories of classical stochastic processes are considered. Importantly, while in classical physics, only probabilistic mixtures of different trajectories are possible, quantum mechanics allows for the coherent superposition of “trajectories” [279].

Figure 26

(Quantum) causal network. Performing different interventions allows for the causal relations between different events (denoted by $X_j$) to be probed. For example, in the figure the event $B_1$ directly influences the events $C_3$ and $A_2$, while $A_3$ influences only $B_4$. As not all pertinent degrees of freedom are necessarily in the control of the experimenter, such scenarios can equivalently be understood as an open system dynamics. Any such scenario can be described by a process tensor [248], and the GET applies, even though active interventions must be performed to discern causal relations. For example, the events $D_3,D_4,B_5$ could be successive (e.g., at times $t_3,t_4$ and $t_5$) spin measurements in $z$, $x$, and $z$ direction, respectively. Summing over the results of the spin measurement in $x$ direction at $t_4$ would not yield the correct probability distribution for two measurements in $z$ direction at $t_3$ and $t_5$ only, but consistency still holds on the level of process tensors (see also Sec. 4 5d).

Figure 27

Determining whether a quantum process is Markovian. Generalized testers (multitime instruments) ${\mathbf{A}}_{{\mathbf{x}}_{-}}$ and ${\mathbf{A}}_{{\mathbf{x}}_{+}}$ are applied to the system during a quantum process, where the subscripts represent the outcomes. The testers are chosen to implement a causal break at a timestep $t_k$, which ensures that the only way the future outcomes depend on the past if the process is non-Markovian. Thus by checking if the future depends on the past for a basis of instruments we can certify the process to be Markovian or not.

Figure 28

A CP-divisible but non-Markovian process. A qubit system is prepared in states $|x\pm ⟩$ and evolves along with an environment. The uninterrupted dynamics of the system is pure dephasing, which is certified as Markovian by two-point witnesses. However, when an instrument $X$ is applied at time $t$, the system dynamics reverse and the system returns to its original state, which is only possible in the presence of non-Markovian memory.

Figure 29

Choi state for the shallow pocket model. Each wire of the Choi state of the shallow pocket model [defined in Eq. (217)] corresponds to a different time (where $t_1^{-}$ and $t_1^{+}$ are the input and output wire at time $t_1$, respectively). The intermediate system-environment unitaries $U_t$ are given by Eq. (216). Note that, in contrast to previously depicted Choi states, here, the environmental degree of freedom $E$ is not traced out yet.

Figure 30

Three-step process. Graphic provided as reference for the considerations of Sec. 6c1.

Figure 31

MPO representation of a multitime process $\mathrm{{\rm Y}}$. Under the assumptions that all the unitaries in the process are the same—which, for example, holds true when the times are equidistantly spaced and the generating Hamiltonian is time independent, then the resulting process tensor can be written as a product of an initial matrix $\breve{\rho }$ and matrices $\breve{\mathsf{U}}$, together with a final partial trace (bent orange line on the right). The degrees of freedom that they share (orange lines) are the bond dimension of the MPO, the remaining open indices correspond to the spaces the resulting MPO lives on. The smallest bond dimension of all possible MPO representations of a given $\mathrm{{\rm Y}}$ can be taken as a measure of non-Markovianity of $\mathrm{{\rm Y}}$.

Figure 32

Quantum Markov order. If a process ${\mathrm{{\rm Y}}}_{FMH}$ has finite Markov order with respect to an instrument and tester ${\mathcal{J}}_M=\{{\mathbf{A}}_{{\mathbf{x}}_M}\}$ on the memory block, then the application of each of the elements of ${\mathcal{J}}_M$ leaves the process in a tensor product between future $F$ and history $H$.

Figure 33

(Quantum) Markov order network. A process with finite quantum Markov order with parts of $M$ kept by $H$ and $F$. The top panel shows the first process, in which parts of the common cause state $|e^{+}⟩$ is sent to $L^{\mathtt{i}}$ and $|e^{-}⟩$ is sent to $R^{\mathtt{i}}$. The process in the bottom panel has the recipients flipped. The process tensor is depicted in gray, and entanglement between parties are color coded in green and maroon. The overall process is a probabilistic mixture of both scenarios. Still the process has finite Markov order because it is possible to differentiate between the scenarios by making a parity measurement on $M$.