Two people read the same book, but one turns the page first and then looks at the text, while the other does the opposite. It would seem the result is the same. But in the quantum world, the order of operations changes everything.
This work examines models that output sequences of symbols, like a ticket machine. Inside such a machine, there is a hidden state that gets updated, and from it a visible symbol is extracted. Usually, the state is updated first, then the symbol is output. The authors reversed the order: output first, then update. It turns out that the two variants produce different symbol chains, and this difference can be noticed even if you wait a very long time or carefully choose the initial state.
But there is an exception. If both models are derived from the same classical model, then the order does not matter: they produce identical chains. This means the difference arises only from quantum effects, when the hidden state can be in multiple variants simultaneously. Such sensitivity to order helps reveal quantum memory that cannot be reduced to classical memory.
as written by the authors arXiv:2602.19120
We introduce a class of causal hidden quantum Markov models (cHQMMs) that reverse the usual order of hidden updates and emissions compared to conventional HQMMs. Using a simple qubit model with a rotating hidden state and sharp measurements, we show that these two architectures-emission then transition versus transition then emission-generally produce different quantum processes. They can be distinguished by measurements at arbitrarily late times, no matter how the hidden system is initialized, and even when the two models start from different initial states. This means that the two orders of operations lead to genuinely different observable behaviors that cannot be reconciled by waiting longer or by choosing special initial conditions. At the same time, we prove that the two architectures become equivalent when they arise from entangled liftings of classical hidden Markov models, sharing the same classical statistics. This identifies a clear dividing line between classical and genuinely quantum hidden memory. Our findings highlight causal HQMMs as a useful tool for studying and distinguishing quantum memory effects in sequential processes.
Even if two quantum devices are started from different initial states, the difference in the order of operations can still be detected after arbitrarily long times.