What is the operational complexity of quantum space series through causal memory?

Complexity is measured by memory, not by entanglement alone

The paper asks what makes one quantum process more complex than another, and answers it operationally: structure is the memory a system must retain to causally generate a correlated sequence of quantum outputs. This mirrors classical computational mechanics, where structural complexity is the minimum memory of a faithful, causal model, but here both the generated process and the model memory are treated quantum mechanically.

The object of study is a quantum space series: a one-dimensional chain of quantum systems emitted one site at a time by a time-invariant evolution of a persistent memory, where outputs are retained and measured but never fed back to interact with the generator. Because outputs are not returned, the sequence carries spatial quantum correlations between sites rather than the temporal correlations that invasive interventions would create. The paper proves these series satisfy the Kolmogorov consistency conditions, so their marginals are well defined even though not measuring differs from an unselective measurement in general quantum processes.

A memory-minimal model needs no discarded environment

A general model here keeps a memory system, interacts it with an ancilla each timestep, emits an output, and discards junk into an environment. The central result is that for stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, a model minimising both statistical and topological memory costs can be chosen to generate the outputs isometrically, with no discarded environment.

That matters because it removes the usual worry that apparent memory savings are hidden in correlations with thrown-away junk. Causality is imposed across sharp measurement histories, so the requirement does not privilege a preferred measurement basis. The minimal model is then uniquely specified by its canonical matrix product state representation, which fixes both the minimum memory dimension and the minimum memory entropy.

Memory dimension and memory entropy are distinct costs

The paper keeps two complementary complexity measures separate. The topological complexity counts the size of the memory, expressed in bits, while the statistical complexity measures how much information is stored in the memory's steady-state distribution. A model can be minimal in one sense and not the other, so the two numbers should not be substituted for each other.

The canonical matrix product state determines both quantities at once. The minimum memory dimension comes from the representation's structure, and the minimum memory entropy comes from its Schmidt spectrum across a past-future partition. This is why the paper can claim that causal memory minimisation, not an externally imposed tensor-network ansatz, is what produces the matrix product state form.

Past-future entanglement entropy equals irreducible memory

The paper's sharpest identification is that the quantum statistical complexity is precisely the entanglement entropy across the past-future bipartition. Under this framework, entanglement is not merely a non-classical resource attached to the process; it is the irreducible memory required for faithful causal generation of the quantum space series itself.

This extends earlier connections between quantum model memory and entanglement, which were established for quantum models of classical stochastic processes, to the unmeasured quantum sequence. The relevant correlations are those across the past-future cut of the chain, so the quantity is defined by the process's own structure rather than by a chosen observation record.

Finite causal memory forces a vanishing entropy rate

Within the finite-dimensional, primitive causal class, the paper establishes three structural consequences: the fundamental entropy rate vanishes, causal asymmetry is absent, and quantum Markov order is infinite. These properties look pathological from classical computational mechanics, where positive entropy rate and finite Markov order are routine.

The paper explains the contrast by the richer informational structure available quantum mechanically. A memoryful classical process with positive entropy rate, embedded as a quantum space series diagonal in a fixed product basis, admits no finite-dimensional primitive causal model, so it falls outside the class studied here even if it has a finite classical causal model. The distinction turns on requiring causality for all sharp measurement choices, not just one.

Quantum complexity sits below the measured classical complexity

The examples make the gap concrete. For the quantum perturbed coin and the quantum Nemo process, block entropies converge to the excess quantum entropy while myopic fundamental entropy rates converge to zero; at 1000 timesteps the deviations are below 10^-12 for the block entropy and below 10^-15 for the entropy rate, indicating the values have essentially reached their asymptotic quantities.

For the modelled classical Nemo process, the classical statistical complexity is about (1.58, 1.15, 1.56) across the three parameter sets, while the full quantum space series and the quantum model of the classical process have quantum statistical complexity about (1.25, 0.781, 1.32). The quantum statistical complexity of the space series is therefore smaller than the classical statistical complexity of the classical process. Excess entropy also differs: about (2.50, 1.56, 2.64) for the quantum space series versus about (0.585, 0.499, 0.866) for the classical process, showing that measurement in the computational basis loses some past-future correlations present in the unmeasured series. The classical process retains non-zero asymptotic randomness, about (0.318, 0.619, 0.624), while the space series has zero entropy rate at all parameter values.

The conclusions hold for the unmeasured quantum sequence

The vanishing entropy rate, absence of causal asymmetry, and infinite Markov order are statements about the unmeasured quantum sequence within the finite-dimensional, primitive causal class. Classical processes obtained by measuring its outputs may still show extensive randomness and different memory requirements, as the Nemo comparison illustrates.

The framework also assumes stationarity and memoryfulness, and it deliberately excludes feedback from interventions on emitted outputs into the generating dynamics. Extending it to genuinely spatiotemporal quantum processes, and to observers restricted to local probes, fixed measurement bases, finite-depth collective operations, or non-adaptive protocols, is left as future work. Those observer-dependent notions of uncertainty, memory, and complexity are where the intrinsic picture is expected to meet semiclassical descriptions built from measurement statistics.

Compare unmeasured quantum memory against measured classical memory

The paper's practical priority is a comparison rule: when judging how complex a quantum process is, do not read complexity off a measured classical record. Measure the irreducible memory of the unmeasured quantum space series through the past-future entanglement entropy of its canonical matrix product state, and keep the memory dimension and memory entropy as separate costs.

The Nemo numbers show why this matters. The same underlying process can look more complex once a computational-basis measurement is imposed, because that measurement discards past-future correlations and introduces non-zero asymptotic randomness that the unmeasured series does not have. The decisive threshold is whether a finite-dimensional, primitive causal model exists for the unmeasured series at all; if it does, the entropy rate must vanish, and any positive-entropy-rate classical process embedded diagonally in a fixed product basis falls outside that class.

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