We develop a product-level trace-Dobrushin theory for finite-dimensional quantum channel products and apply it to deterministic and stationary random inhomogeneous matrix product states in left-canonical CPTP gauge. For a product of channels, the centered trace-Dobrushin coefficient quantifies the residual dependence on the input state, and its decay is the criterion for trace-norm forgetting. In the deterministic setting, this decay is equivalent to asymptotic replacement by a moving replacement channel. For two-sided products, pullback forgetting produces a unique boundary state, which determines the canonical replacement family. For stationary random CPTP cocycles, submultiplicativity of the product coefficient yields a trace-Dobrushin Lyapunov exponent. We prove that the almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the \(\varrho\)-mixing profile of the channel environment tends to zero, we obtain annealed super-polynomial estimates, while independence gives annealed exponential estimates. Finally, we transfer these estimates to inhomogeneous matrix product states whose auxiliary transfer maps are CPTP. These channel estimates transfer to deterministic and stationary random inhomogeneous MPS, giving infinite-volume limits of trace-closed finite-volume states, quantitative boundary stability, and correlation bounds governed by the same auxiliary product coefficients.
We develop a product-level trace-Dobrushin theory for finite-dimensional quantum channel products and apply it to deterministic and stationary random inhomogeneous matrix product states in left-canonical CPTP gauge. For a product of channels, the centered trace-Dobrushin coefficient quantifies the residual dependence on the input state, and its decay is the criterion for trace-norm forgetting. In the deterministic setting, this decay is equivalent to asymptotic replacement by a moving replacement channel. For two-sided products, pullback forgetting produces a unique boundary state, which determines the canonical replacement family. For stationary random CPTP cocycles, submultiplicativity of the product coefficient yields a trace-Dobrushin Lyapunov exponent. We prove that the almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the (\varrho)-mixing profile of the channel environment tends to zero, we obtain annealed super-polynomial estimates, while independence gives annealed exponential estimates. Finally, we transfer these estimates to inhomogeneous matrix product states whose auxiliary transfer maps are CPTP. These channel estimates transfer to deterministic and stationary random inhomogeneous MPS, giving infinite-volume limits of trace-closed finite-volume states, quantitative boundary stability, and correlation bounds governed by the same auxiliary product coefficients.
We prove annealed functional central limit theorems for finite pattern counts in the measurement record of discrete-time quantum trajectories, with the instrument applied at each step determined by an invertible, probability-preserving base dynamical system. When the base is ergodic, under summable strong-mixing coefficients of the instrument process and a summable uniform annealed trace-norm forgetting rate for the associated non-selective channel cocycle, we establish a joint functional CLT for bounded vector-valued functions of finite outcome blocks under the annealed law determined by the dynamically stationary state. We then extend this limit to every measurable random initial state, yielding a universal functional CLT with unchanged stationary centering and asymptotic covariance. We also provide practical sufficient criteria ensuring the existence and uniqueness of the dynamically stationary state and the required annealed trace-norm forgetting. We illustrate the results through a broad family of examples, including disordered walk-type models generated by finite group actions, measurement followed by preparation, and instruments with reset components. The results apply to general disordered quantum instruments and are not restricted to the perfect-measurement regime; they complement the law of large numbers established by Ekblad, Moreno-Nadales, and Pathirana (2026) for the same disordered setting and provide a disordered counterpart of the homogeneous CLT of Attal, Guillotin-Plantard, and Sabot (2014).
We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectations of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: (ρ)-mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in (ρ) (resp. (β)) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.
We prove that every injective Matrix Product State is the unique ground state of a simple hopping theory. We start by studying the low energy spectrum of parent Hamiltonians of injective Matrix Product States in a particular long range and system size limit under the validity of an asymptotic regime with low particle density. We show that in this limit a natural first quantization arises. This allows us to compute a specific type of low energy spectrum. This spectrum depends solely on the properties of a quantum channel, i.e. transfer matrix of the ground state, and not on any other details of the ground-state. We also review normal quantum channels for which the expression is more simplified. The construction possibly has some interesting uses for the study of quantum and classical Markov processes which we briefly expose. As an application, we revisit the notion of (many-body)-localization with our framework. Our calculations revealed that translational invariant Matrix Product States can be interpreted as a stationary sea of particles. As a next step rather than starting from some local Hamiltonian with random potentials, we consider fluctuations of the local tensors of a continuous one-parameter family of Matrix Product States. Localization in 1-dimension, is then understood from a simple study of spectral and mixing properties of finite dimensional quantum channels.
We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II error exponent equals the regularized Umegaki channel relative entropy minimized over the alternatives, and both defining limits exist. Inputs may be entangled across channel uses and with a reference system. The main technical result constructs exactly trace-preserving approximations with exponentially small diamond error, dominated in completely positive order by free channels at every rate above the common exponent. The proof combines uniform auxiliary map approximation, iterative reduction of the domination rate, and tensor amplification. It yields asymptotic equipartition with trace-preserving smoothing at every subexponentially vanishing error, an exponential strong converse, and stability under vanishing diamond-norm perturbations. Under additional permutation invariance and closure under insertion of a fixed faithful input state and discarding of its output, we also prove an explicit finite-block completion bound. This quantitative construction uses weighted discarding, local operator corrections, and comparison with auxiliary extensions. Applications include state-preserving, coherence-restricted, covariant, entanglement-breaking, and positive-partial-transpose channels. For isometric targets against entanglement-breaking or positive-partial-transpose alternatives, we obtain exact finite-block testing and smoothing formulas.
We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.
We prove that every injective Matrix Product State is the unique ground state of a simple hopping theory. We start by studying the low energy spectrum of parent Hamiltonians of injective Matrix Product States in a particular long range and system size limit under the validity of an asymptotic regime with low particle density. We show that in this limit a natural first quantization arises. This allows us to compute a specific type of low energy spectrum. This spectrum depends solely on the properties of a quantum channel, i.e. transfer matrix of the ground state, and not on any other details of the ground-state. We also review normal quantum channels for which the expression is more simplified. The construction possibly has some interesting uses for the study of quantum and classical Markov processes which we briefly expose. As an application, we revisit the notion of (many-body)-localization with our framework. Our calculations revealed that translational invariant Matrix Product States can be interpreted as a stationary sea of particles. As a next step rather than starting from some local Hamiltonian with random potentials, we consider fluctuations of the local tensors of a continuous one-parameter family of Matrix Product States. Localization in 1-dimension, is then understood from a simple study of spectral and mixing properties of finite dimensional quantum channels.