This paper develops a theoretical framework for nonlinear stochastic Schrödinger equations (SSEs) driven by quantum Bernoulli noise (QBN), providing mathematical foundations for modeling open quantum systems in quantum information processing. We establish the well-posedness of [Formula: see text]-strong solutions for linear SSEs and construct solutions for the nonlinear equations via Girsanov transformations. Our approach utilizes operator lifting techniques on the Hilbert space [Formula: see text] and validates the framework through numerical simulations of qubit-reservoir systems. The results demonstrate higher state fidelity compared to Lindblad models under strong coupling, with applications to quantum error correction and real-time control in near-term quantum devices.
This paper develops a theoretical framework for nonlinear stochastic Schrödinger equations (SSEs) driven by quantum Bernoulli noise (QBN), providing mathematical foundations for modeling open quantum systems in quantum information processing. We establish the well-posedness of [Formula: see text]-strong solutions for linear SSEs and construct solutions for the nonlinear equations via Girsanov transformations. Our approach utilizes operator lifting techniques on the Hilbert space [Formula: see text] and validates the framework through numerical simulations of qubit-reservoir systems. The results demonstrate higher state fidelity compared to Lindblad models under strong coupling, with applications to quantum error correction and real-time control in near-term quantum devices.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation. In this paper, we study linear stochastic Schrödinger equations (LSSEs) associated with QBN in the space of square integrable complex-valued Bernoulli functionals. We first rigorously prove a formula concerning the number operator N on Bernoulli functionals. And then, by using this formula as well as Mora and Rebolledo’s results on a general LSSE [C. M. Mora and R. Rebolledo, Infinite. Dimens. Anal. Quantum Probab. Relat. Top. 10, 237–259 (2007)], we obtain an easily checking condition for a LSSE associated with QBN to have a unique Nr-strong solution of mean square norm conservation for given r≥0. Finally, as an application of this condition, we examine a special class of LSSEs associated with QBN and some further results are proven.
Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K⊗h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K⊗h, which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K⊗h. In particular, we obtain the spectral decomposition of the tensor operator IK⊗N, where IK means the identity operator on K and N is the number operator in h, and give a representation of IK⊗N in terms of operators IK⊗∂k*∂k, k≥0, where ∂k and ∂k* are the annihilation and creation operators on h, respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.
Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of Chen and Wang [J. Math. Phys. 58(5), 053510 (2017)].
The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.
Quantum Bernoulli noises (QBNs) refer to the annihilation and creation operators acting on the space [Formula: see text] of square integrable Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. In this paper, we consider the Markov evolution of an open quantum system interacting with QBNs. Let [Formula: see text] be the system space of an open quantum system interacting with QBNs. Then [Formula: see text] just describes the coupled quantum system. In the framework of [Formula: see text], we first construct a quantum Markov semigroup that respects the interactions between the system and QBNs, and then we prove that under some mild conditions the semigroup has faithful invariant states. To support our main results mentioned above, we prove several technical propositions and theorems about operators defined in [Formula: see text]. Some other results are also obtained.
We present a universal framework for simulating N-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs dXt=A(t)Xtdt+∑j=1JBj(t)XtdWtj to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.