Two kinds of randomness
A quantum-information preprint proposes a way to separate randomness that remains irreducible from randomness that only appears because a process’s memory has not yet been accounted for. Its central question is how much apparent randomness in a stationary, fully quantum process — one treated as stable over time — can be explained by temporal memory.
The framework calls the long-run component the quantum entropy rate: the continuing growth of a process’s uncertainty. The paper’s theorem says this rate is finite and equals the limiting conditional process entropy for the stationary processes it considers. That conditional quantity cannot increase as the observation window grows; it decreases or stays level and is bounded below by zero, so it converges.
Quantum excess entropy captures the other component. It is the accumulated amount by which finite-window entropy sits above the long-run rate — the apparent randomness that temporal correlations can eventually explain. When the relevant limits are finite, excess entropy also equals the quantum mutual information between asymptotically long past and future halves, measuring how much information those two sides share.
A memory constraint
To make temporal memory calculable, the framework represents stationary quantum processes as normalized process-tensor Choi states. In plain language, this packages a process unfolding through time into a quantum state, allowing information-theoretic tools normally used for spatial quantum states to be applied to correlations across time.
For a time-homogeneous recurrent quantum circuit — one that repeats the same memory-and-interaction pattern — the paper derives a resource constraint. If the memory settles into a stationary state, excess entropy cannot exceed twice the entropy of that memory. A circuit with very little stationary memory entropy therefore cannot reproduce a process carrying a large finite store of past–future correlation. The bound is not guaranteed to be tight in every circuit; equality requires additional unitarity and purification conditions.
A small circuit, three regimes
The illustrative calculation uses a recurrent circuit with a single qubit serving as memory. The memory starts in a maximally mixed stationary state, the circuit includes Bell-state interactions, and the calculation follows a six-step window. The illustration compares three settings of a depolarization parameter: a noiseless case, an intermediate case and full depolarization.
In the noiseless setting, with lambda equal to 0, the estimated entropy rate was approximately 0 while estimated excess entropy was approximately 2, and the memory bound was saturated. In the intermediate setting, with lambda equal to 0.02, the corresponding estimates were about 0.13 and 1.79. Under full depolarization, with lambda equal to 1, excess entropy was 0 and the estimated entropy rate was about 1.55; the text describes that regime as lacking memory-mediated temporal correlations.
What the estimates can and cannot show
Those figures are model estimates from a finite window, not exact values for an infinite process. Across the simulated regimes, linear extrapolation approximately approached the proposed entropy rate and brought estimated excess entropy into line with past–future quantum mutual information. That provides a consistency check within the illustrative setup, but it does not establish the exact asymptotic quantities.
The formal framework also has a defined boundary: it assumes that quantum excess entropy is finite. Processes with divergent excess entropy are left for future work. The memory inequality applies to the stated time-homogeneous recurrent-circuit setting and is not necessarily saturated by every circuit.
The result is therefore a mathematical way to distinguish continuing process-entropy growth from temporal structure that can be explained through memory. It offers a bound linking that memory-related structure to the entropy of a recurrent circuit’s stationary memory, while leaving broader process families and divergent cases unresolved.
The supplied document is an arXiv version-one quantum-information preprint, with the finite-window circuit serving as an illustrative model for the proposed quantities.
Paper data and sources
Original title: How much randomness in a quantum process can be explained using memory?
Authors: Derek D. C. Chang, Graeme D. Berk, Mile Gu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text