Structure of quantum stochastic processes with finite Markov order

Non-Markovian quantum processes exhibit different memory effects when measured in different ways; an unambiguous characterization of memory length requires accounting for the sequence of instruments applied to probe the system dynamics. This instrument-specific notion of quantum Markov order displays stark differences to its classical counterpart. Here, we explore the structure of quantum stochastic processes with finite memory length in detail. We begin by examining a generalized collision model with memory, before framing this instance within the general theory. We detail the constraints that are placed on the underlying system-environment dynamics for a process to exhibit finite Markov order with respect to natural classes of probing instruments, including deterministic (unitary) operations and sequences of generalized quantum measurements with informationally complete repreparations. Lastly, we show how processes with vanishing quantum conditional mutual information form a special case of the theory. Throughout, we provide a number of representative, pedagogical examples to display the salient features of memory effects in quantum processes.

Figure 1

Generalized collision model with memory. The top row depicts a standard memoryless collision model. With time running left to right, the system $S$ (green) interacts unitarily at each time step once with each of a number of uncorrelated, fresh ancillary states $A_j$ that constitute the environment (orange); the collision is represented by the gray boundary. Following the dynamics at $t=1$, the $A_1$ ancilla has been used and so stores information about the initial state of the system, indicated by the purple color and dashed outline (see $A_1$ at $t=2)$. However, each successive portion of evolution proceeds through an interaction with a fresh ancilla that has not yet interacted with the system. Thus, any memory of the system's history cannot influence the future evolution, leading to Markovian dynamics. The bottom row shows a generalized collision model, where the system is allowed to interact with multiple ancillas between time steps. Here, once the system interacts with $A_0$ and $A_1$ after $t=1$, again, these ancillary states can store information about the initial system state. The next step of dynamics following $t=2$ involves $A_1$ again; thus, the future dynamics are conditioned on the history. In this way, the ancillas serve to propagate memory effects through the process.

Figure 2

Finite-length memory with respect to trash-and-prepare protocol. The underlying system-environment dynamics for the generalized collision model described in the main text, interspersed with the trash-and-prepare protocol applied to the system. Any possible influence stemming from the history persists to impact the future for at most $\ell =3$ time steps before being trashed. For instance, the red paths signify the degrees of freedom that can be affected by the initial preparation, whereas the black ones cannot be. The final state is a function of only the most recent $\ell$ preparations, $\{{\sigma }_1^S,{\sigma }_2^S,{\sigma }_3^S\}$, and entirely independent of the initial system state, ${\rho }_0^S$. Any other instrument sequence on the system, e.g., a measurement rather than a trash-and-prepare instrument, would open up a pathway for the initial state ${\rho }_0^S$ to influence the future state ${\rho }_4^S$.

Figure 3

Operational description of quantum stochastic processes. Any quantum stochastic process can be modeled as arising from a system interacting unitarily with a suitably sized inaccessible environment. To characterize such evolution, one must interrogate the system. Hence, we allow for sequences of probing operations, ${\mathcal{O}}_j$ (green), to be applied to the system throughout the dynamics; these are the most general transformations allowable, taking input states to output states $(\mathtt{i},\mathtt{o}$ respectively). The final state of the system can be described uniquely through a multilinear mapping on only the space of operations applied. This is precisely the process tensor, ${\mathrm{{\rm Y}}}_{n:1}$, represented as everything within the yellow, dashed boundary.

Figure 4

Instrument-specific quantum Markov order. An instrument sequence ${\mathcal{J}}_M$, comprising (temporally correlated) CP operations $\left\{O_M^{\left(x_M\right)}\right\}$ (green) across a sequence of time steps of length $\ell$, is applied to a process ${\mathrm{{\rm Y}}}_{FMH}$ (yellow). The process is said to have Markov order $\ell$ with respect to this instrument sequence if, for each possible realization of the instrument, $x_M$, the history (red, ${\mathrm{{\rm Y}}}_H^{\left(x_M\right)})$ and future (blue, ${\mathrm{{\rm Y}}}_F^{\left(x_M\right)})$ parts of the process are rendered conditionally independent. Here, for illustrative purposes, $M=\{k-1^{\mathtt{o}},\cdots ,k-{\ell }^{\mathtt{o}}\}$; i.e., any possible measurement performed at time step ${k-\ell }^{\mathtt{i}}$ is considered part of the history, and its outcomes are conditionally independent of any future statistics. Equation (11) calculates the mutual information between the conditional history and future processes, which vanishes for processes with finite Markov order.

Figure 5

Finite-memory with respect to a unitary instrument sequence. The $SE$ dilation of a single time step for a process whose historic influence on the future is blocked only by the sequence of unitary operations on the system $\{{\mathcal{V}}_{k-\ell },\cdots ,{\mathcal{V}}_{k-1}\}$. Everything inside the yellow, dashed boundary, including the unitary operation ${\mathcal{V}}_j^{†}$, is part of the inaccessible process; we only have the choice of what operation is applied in the green outlined box. The “cutting protocol” described in the main text is depicted here in purple: The ancillary counter, $C$, registers the number of successive successful Bell basis measurements on the $SA$ system, which is reprepared as a maximally entangled pair, ${\psi }^{AS}$, at each time step. When the counter reaches $\ell$, the current environment state is discarded and a fresh one, ${\tau }_{j+1}^E$, is prepared to govern the future evolution. If the counter has not reached $\ell$, the environment is left to mediate correlations from the history to the future.

Figure 6

Finite memory with respect to an informationally complete sequence. Initially, a tripartite state ${\rho }_{Y{2}^{\mathtt{i}}{1}^{\mathtt{i}}}$ is constructed as per Eq. (17), with subsystems $1^{\mathtt{i}},2^{\mathtt{i}}$ of it fed out at consecutive time steps as described in the text. The states fed back into the process on spaces $1^{\mathtt{o}},2^{\mathtt{o}}$ are fed forward as inputs to the CPTP map ${\mathrm{\Lambda }}_{3^{\mathtt{i}}Y{2}^{\mathtt{o}}{1}^{\mathtt{o}}}$ defined in Eq. (18). Upon applying any combination of the correct IC causal break sequence $\{{\sigma }_{1^{\mathtt{o}}}^{\left(x\right)},{\mathrm{\Pi }}_{2^{\mathtt{i}}}^{\left(y\right)},{\sigma }_{2^{\mathtt{o}}}^{\left(z\right)}\}$, one of $d^6$ final output states ${\rho }_{3^{\mathtt{i}}}^{\left(xyz\right)}$ are output by the process in the future, each of which is conditionally independent of each historic ${\rho }_{1^{\mathtt{i}}}^{\left(y\right)}$. If any other operations are applied, correlations will arise between the history and future in general.

Figure 7

Process with nonvanishing quantum conditional mutual information. The environment is a four-dimensional ancilla. Its state is a coherent superposition of the basis states $\left\{\right|0\rangle ,\cdots ,|3\rangle \}$. The system-environment evolution is a control unitary, which implements one of the the four Pauli rotations $\mathcal{V}:=\{\mathcal{I},\mathcal{X},\mathcal{Y},\mathcal{Z}\}$ on the system depending on the state of the ancilla (see top panel). The history-blocking instrument sequence consists of feeding in one half of a Bell pair and at the next measuring the system and the other half in the Bell basis at the next time step. For each outcome of this instrument, one can infer which of the four Pauli rotations was applied, and the history and future processes are conditionally independent. For illustrative purposes, the bottom panel depicts the conditional processes that arise from successful implementation of the operation $\frac{1}{4}{\mathrm{\Psi }}^{+}$, which occurs with probability ${\left|\alpha \right|}^2$.

Figure 8

Generalized collision models with memory. In addition to introducing memory into collision models with repeated system-ancilla collisions (as in Sec. 2), memory can be built into collision models by allowing for (i) ancilla-ancilla interactions (top row) and (ii) an initially correlated environment (bottom row). (Note that the legend is as per Figs. 1 and 2). The top left panel depicts a schematic of the dynamics where ancilla-ancilla interactions (yellow boundary) are interleaved between the system-ancilla collisions (gray boundary). At the second time step, $A_2$ already has knowledge of the state of the system at $t=1$ mediated via the $A_1{A}_2$ interaction. As such, the future dynamics is conditioned on the initial system state. The top right panel displays the corresponding circuit diagram. Here, for arbitrary operations on the system ${\mathcal{O}}_j$, it is clear that the ancilla-ancilla interactions provide a possible path of influence from the history to the future state; hence, such a process generically displays infinite Markov order with respect to any instrument sequence (as shown by the red path). For instance, even with the application of a sequence of trash-and-prepare operations, the final output state can still be influenced by the initial preparation. The bottom left panel depicts a schematic of the dynamics where the ancillas constituting the environment begin in a correlated state (represented by the orange line connecting them). Here, as soon as the system interacts with a part of the correlated ancillary state, all other ancillary systems can store information about the initial state of the system, thereby influencing the future dynamics. Finally, the bottom-right panel displays the corresponding circuit diagram for this case. Again, allowing for arbitrary operations on the system level, the initial correlations in the environment provide a mechanism for the history to influence the future over an infinite length of time.

Figure 9

Choi-Jamiołkowski isomorphism for the process tensor. The process tensor map, ${\mathcal{T}}_{n:1}$, for any quantum stochastic process can be represented as a many-body state, ${\mathrm{{\rm Y}}}_{n:1}$, through the extended Choi-Jamiołkowski isomorphism depicted above and as described in Eq. (B1). For each time step $j\in \{1,\cdots ,n\}$, half of an ancillary system that has been prepared as an un-normalized maximally entangled state, ${\mathrm{\Psi }}^{A_j{B}_j}$, is swapped into the process, represented by the blue crosses. The resulting $\left(2n-1\right)$-body quantum state ${\mathrm{{\rm Y}}}_{n:1}$ contains equivalent information to the map ${\mathcal{T}}_{n:1}$.

Figure 10

Instrument dependence of classical Markov order with fuzzy measurements. Here we depict two classical processes to highlight the instrument dependence of Markov order when sharp measurements are not assumed (a legend is provided in the rightmost panel). On the left, the process of Example t7 is shown, defined by the transition probabilities depicted at each time step. Here, if one is able to record observations sharply, i.e., measure the random variable $X=\{a,b,c\}$, the process is clearly Markovian; however, if one cannot measure at that resolution and, e.g., the measurement apparatus only records fuzzy statistics of the random variable $Y=\{a,b\cup c\}$, as depicted by the blue, dashed box, the process would be classified as non-Markovian. In the middle panel, the process of Example t8 is shown. Here, three bits are initially prepared as described in the text, and each bit is fed out of the process at successive time steps. The preparation is such that if the second bit is sharply measured to be in the state 0, bits 1 and 3 are perfectly correlated; if the second bit is in state 1, bits 1 and 3 are perfectly anticorrelated; while on average, i.e., with respect to the fuzzy measurement coarse graining over everything in the blue, dashed box, bits 1 and 3 are completely uncorrelated.

Figure 11

Process with nonvanishing quantum conditional mutual information but finite Markov order with respect to (fuzzy) orthogonal projectors. The tripartite state ${\rho }_{3^{\mathtt{i}}{2}^{\mathtt{i}}{1}^{\mathtt{i}}}$ as defined in Eq. (E4) is depicted on the left. Here, if an experimenter cannot distinguish between measurement outcomes in the $\left\{\right|0\rangle ,|1\rangle \}$ subspace of ${\mathcal{H}}_{2^{\mathtt{i}}}$, represented by the blue, dashed boxes, then the conditional state ${\rho }_{3^{\mathtt{i}}{1}^{\mathtt{i}}}^{\left(x\right)}$ for each outcome is product. If, on the other hand, the experimenter can resolve sharp measurements in the $\left\{\right|0\rangle ,|1\rangle \}$ subspace and implement, e.g., the operations $\left\{O_{2^{\mathtt{i}}}^{\left(x\right)}\right\}=\left\{{\mathrm{\Pi }}_{2^{\mathtt{i}}}^{\left(x\right)}\right\}$, then for each outcome realized, the conditional state ${\rho }_{3^{\mathtt{i}}{1}^{\mathtt{i}}}^{\left(x\right)}$ is a (correlated) Werner state, defined in Eq. (E1). The process is such that this state is initially prepared, with subsystems fed out to the experimenter over a sequence of time steps. Whatever is fed back into the process is discarded by the process itself; hence, the process tensor is written as per Eq. (E5). As described above, the fuzzy (orthogonal) measurement at time step $2^{\mathtt{i}}$ blocks the influence of history on the future, although a sharp measurement resolving all three outcomes does not. Lastly, the quantum CMI for this process does not vanish.