The quantum symmetric simple exclusion process (QSSEP) is a recent extension of the symmetric simple exclusion process, designed to model quantum coherent fluctuating effects in noisy diffusive systems. It models stochastic nearest-neighbor fermionic hopping on a lattice, possibly driven out-of-equilibrium by boundary processes. We present a direct formulation in the continuum, and establish how this formulation captures the scaling limit of the discrete version. In the continuum, QSSEP emerges as a non-commutative process, driven by free increments, conditioned on the algebra of functions on the ambiant space to encode spatial correlations. We actually develop a more general framework dealing with conditioned orbits with free increments which may find applications beyond the present context. We view this construction as a preliminary step toward formulating a quantum extension of the macroscopic fluctuation theory.
Non-equilibrium phenomena, whether classical or quantum, are ubiquitous in Nature, but their understanding remains far less profound than those at equilibrium. Over the past decade, significant conceptual progress has been made for classical non-equilibrium systems, driven by exact analysis of simple model systems, such as the symmetric simple exclusion process (SSEP) or its asymmetric counterpart (ASEP). See [15,19] for a review. These progresses culminated in the formulation of the macroscopic fluctuation theory (MFT), an effective theory describing transports and their fluctuations in diffusive classical systems [10].
However, the questions whether -and how-the macroscopic fluctuation theory might be extended to the quantum realm remains open [3]. A quantum analogue of MFT should not only describe diffusive transport and its fluctuations, but also capture quantum coherent phenomena and their fluctuations, including quantum interferences, correlations, and entanglement dynamics in out-of-equilibrium diffusive systems. Recent progress has been achieved through two complementary approaches. The first approach, based on the analysis of random quantum circuits, has led to an emerging membrane picture for entanglement production in quantum chaotic systems [20,21,27,30]. For a review, see [16]. A parallel approach involves analyzing quantum exclusion processes, which extend classical exclusion processes into the quantum domain. See [2] for a review. A paradigmatic example is the quantum symmetric simple exclusion process (QSSEP) [6], a stochastic evolution of fermions hopping along a one-dimensional chain. While its averaged dynamics reproduces the classical SSEP, studying QSSEP has revealed remarkable features: (i) transport fluctuations in such noisy quantum systems are typically classical, with sub-leading fluctuations of quantum origin [1,8], and (ii) off-diagonal quantum correlations persist beyond decoherence, revealing a rich structure deeply connected to free probability theory [17].
Yet, these studies rely on discrete models, followed by continuous scaling limits. Our goal is here to formulate QSSEP directly in the continuum, bypassing the discrete-to-continuum transition.
The QSSEP dynamics is fully characterized by the stochastic evolution of the N × N matrix of two-point functions, (G s ) ij := Tr(ρ s c † i c j ), where c † i , c i are the fermionic annihilation-creation operators and ρ s the state (density matrix) of the system at time s, with i, j = 1, • • • , N labeling the sites of the chain. Specific aspects of QSSEP ensure that its matrix of two-point functions satisfies a stochastic differential equation (SDE) -this is not true for generic quantum exclusion processes but is a special feature of QSSEP-, of the following form, G s+ds = e idhs G s e -idhs + boundary terms, (1.1) where the stochastic Hamiltonian increments dh s are some instances of structured matrix valued Brownian increments. The QSSEP exists in three variants, each associated with distinct algebraic structures: (i) the “periodic” case, defined on a circle, whose Hamiltonian increments is naturally associated to the loop algebra of su(N ) and without boundary terms, (ii) the “closed” case, defined on an interval, with Hamiltonian increments associated to su(N ) and without boundary terms, and finally, (iii) the “open” case, also defined on an interval, with Hamiltonian increments identical to those of the closed case but with supplementary boundary processes. In the large N limit, these boundary terms fix the particle densities (denoted n a , n b ) at the two boundaries, and drive the system out-of-equilibrium for n a ̸ = n b . See Appendix A for a more precise definition and notation.
Analyzing the scaling limit (N → ∞, with a diffusive rescaling of time and space, t = s/N 2 ∈ R + and x = i/N ∈ [0, 1]) reveals that the relevant quantities are the cyclic moments of G of the form
, p ≥ 0, in which the successive indices form a cycle. They scale as O(N -p ) and depend on the index positions x k = i k /N . To capture this dependence, we introduce test diagonal matrices ∆k ’s and consider dressed moments,
Expanding these moments linearly in ∆k ’s yields the cyclic moments back. They can naturally be viewed as moments of random matrices conditioned on the sub-algebra of diagonal matrices.
From this discussion, we identify two essential properties that QSSEP in the continuum should satisfy:
(a) The limiting process should be defined in terms of conditioned measures. In the large size limit, the algebra of diagonal matrices becomes the algebra L ∞ [0, 1] of bounded functions on [0, 1], since to any function ∆ ∈ L ∞ [0, 1] we may associate the diagonal N × N matrix ∆ with entries ∆ii = ∆(i/N ). Thus, the scaling limit of QSSEP should be defined in terms of processes on some filtered algebra (A t ) t∈R + , conditioned on L ∞ [0, 1].
(b) The limiting process should be driven by free independent increments. In the large size limi
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