Process Tensors in Open Quantum Systems

• Process tensors are the most general objects for describing open quantum system dynamics under discrete interventions, integrating all non-Markovian correlations into a single framework.
• They are defined operationally through multilinear maps and Choi-state representations, and implemented numerically as matrix product operators for efficient simulation.
• This framework supports applications in spectroscopy, quantum thermodynamics, and chaos diagnostics by unifying diverse simulation methods and control strategies.

to=arxiv_search.search ,最新高清无码专区 code 尚度json {"query":"process tensor open quantum systems matrix product operator non-Markovian", "max_results": 10} to=arxiv_search.search 大发彩票官网 code to=arxiv_search ՞նչ code 微信公众号天天中彩票json {"query":"process tensor non-Markovian open quantum systems", "max_results": 5} Process tensors are the most general objects that describe the dynamics of an open quantum system under arbitrary external interventions at a discrete set of times. In the operational formulation, a process tensor is a multilinear map from a sequence of completely-positive trace-nonincreasing maps to a final reduced system state; in the Choi representation, it is a positive multi-time operator on the tensor product of input and output system spaces at successive times. In this form, all non-Markovian correlations between system and environment are collected into a single object that can be contracted with control operations, measurements, or propagators to recover final states and multi-time observables (Wit et al., 2024).

1. Operational definition and Choi-state representation

At discrete times $t_0,t_1,\dots,t_{n-1}$, one may intervene on the system by applying maps $\mathcal A_k$. The process tensor $\mathcal T_{n:0}$ is the multilinear map

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$

where $\rho_n$ is the reduced system state at $t_n$. Equivalently, one represents $\mathcal T_{n:0}$ by its Choi state $\Upsilon_{n:0}$, defined on the tensor product of system input and output spaces at each time step, so that

$$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$

This formulation makes explicit that sequential control and observation are not appended to the dynamics after the fact; they are part of the definition of the object itself (Wit et al., 2024).

A complementary microscopic expression starts from joint system-environment unitary propagators $U_{k+1,k}$ and an initially factorized state $\mathcal A_k$0. In that setting, the process tensor may be written as the unique multi-linear map, or “quantum comb,” satisfying

$\mathcal A_k$1

with Choi form

$\mathcal A_k$2

This identifies the process tensor as the compressed multi-time record of all system-environment correlations relevant to future reduced dynamics (Shubrook et al., 18 Dec 2025).

2. Tensor-network and matrix-product-operator realizations

For numerical work, process tensors are commonly represented as matrix product operators. In PT-TEMPO, time is discretized into uniform steps $\mathcal A_k$3, a maximum memory length $\mathcal A_k$4 is chosen, and at step $\mathcal A_k$5 one forms bath tensors

$\mathcal A_k$6

which are contracted with the existing process-tensor MPO and compressed by SVD, truncating singular values below $\mathcal A_k$7. The resulting compressed MPO represents $\mathcal A_k$8, and the cost at each step scales roughly as $\mathcal A_k$9, where $\mathcal T_{n:0}$0 is the system Liouville dimension and $\mathcal T_{n:0}$1 the MPO bond dimension (Wit et al., 2024).

For Gaussian environments with a coupling operator diagonal in some basis, the process tensor can be identified with the discrete Feynman-Vernon influence functional. In that case,

$\mathcal T_{n:0}$2

with the two-dimensional triangular tensor network generated by four-leg gates $\mathcal T_{n:0}$3. Rotating this network by $\mathcal T_{n:0}$4 turns the time-translation direction into a spatial axis, allowing contraction into an infinite translationally invariant MPS/MPO for the process tensor. When the bath correlation function depends only on time differences, the influence coefficients depend only on $\mathcal T_{n:0}$5, and one obtains a time-translation-invariant process tensor whose application cost grows with memory depth $\mathcal T_{n:0}$6, not with total time $\mathcal T_{n:0}$7 (Cochin et al., 6 Mar 2026).

The MPO language also serves as a unifying representation for several non-Markovian simulation methods. Hierarchical equations of motion, stochastic Liouville-von Neumann methods, auxiliary-mode mappings, reaction-coordinate approaches, pseudomode approaches, and path-integral TEMPO can all be recast as process-tensor MPOs. In that formulation, the MPO bond dimension $\mathcal T_{n:0}$8 provides a direct metric for comparing the effective environmental complexity carried by different methods (Ortega-Taberner et al., 2024).

3. Structural properties and physical interpretation

In Choi form, process tensors satisfy positivity and causality constraints. For a $\mathcal T_{n:0}$9-slot process tensor $$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$0, one has

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$1

together with the no-future-to-past conditions

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$2

These identities express the fact that plugging in trace-preserving instruments yields valid states and that interventions at later times cannot alter earlier statistics (Dowling et al., 2023).

A more refined interpretation emerges from the inner bonds of a compressed PT-MPO. Those bonds do not merely quantify complexity through their dimensions; they represent the subspace of the full environment Liouville space that hosts environment excitations most relevant to subsequent open-system dynamics. Formally, one introduces lossy linear maps

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$3

which project environment Liouville vectors $$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$4 to bond-space amplitudes $$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$5, together with Moore-Penrose pseudoinverses

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$6

The local MPO block may then be written as

$$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$7

This makes the inner bonds interpretable as a lossy but physically meaningful projection of the environment dynamics (Cygorek et al., 2024).

This interpretation has practical consequences. Observable “closures” constructed with $$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$8 permit extraction of environment observables, mixed system-environment observables, photon number, current, and energy partitioning directly from the compressed process tensor, rather than only reduced system observables (Cygorek et al., 2024).

4. Construction algorithms, compression, and scaling

The basic PT-TEMPO workflow proceeds by discretizing time, keeping only the last $$\mathcal T_{n:0}:(\mathcal A_0,\mathcal A_1,\dots,\mathcal A_{n-1})\longmapsto \rho_n,$$9 influence tensors at each step, contracting the new bath tensor with the existing MPO, and compressing by SVD. Convergence is controlled by the triplet $\rho_n$0. This provides a numerically exact construction in the converged limit and a transparent route from influence functionals to compressed MPOs (Wit et al., 2024).

For time-translation-invariant process tensors, the original iTEBD construction applies a sequence of four-leg gates $\rho_n$1, performs full SVDs on $\rho_n$2 objects, truncates to a bond dimension set by tolerance, restores canonical form, and swaps sites. A modified iTEBD algorithm introduces intermediate compression steps by first precompressing $\rho_n$3 to low effective rank $\rho_n$4, then performing partial SVDs on intermediate blocks before the final SVD on a reduced core tensor $\rho_n$5. In the reported benchmarks, this reduces end-to-end construction cost from $\rho_n$6 to $\rho_n$7, while peak memory drops from $\rho_n$8 to $\rho_n$9 or lower with matrix-free SVD, and the final bond dimension $t_n$0 is unchanged (Cochin et al., 6 Mar 2026).

The MPO form also enables adjoint-style optimal control. Defining the forward “extended” state

$t_n$1

and a backward costate with terminal condition $t_n$2, one obtains local gradient contractions

$t_n$3

In this setting, forward and backward propagation scale as

$t_n$4

with $t_n$5 the system Hilbert-space size and $t_n$6 the maximal MPO bond dimension (Ortega-Taberner et al., 2024).

5. Spectroscopy, control, thermodynamics, and many-body diagnostics

One major application is non-Markovian spectroscopy. In two-dimensional electronic spectroscopy, the signal is a sum of four-time correlation functions $t_n$7. Within the PT-MPO framework, one inserts superoperators $t_n$8 at the appropriate time-step legs, contracts through the network, and obtains exact non-Markovian multi-time correlators in one tensor-network sweep per final time $t_n$9. In the regimes studied, the method reproduces peak positions, broadenings, and cross-peaks that Markovian master equations miss when the bath is structured, coupling is intermediate, or temperature is low (Wit et al., 2024).

Process tensors also support numerically exact quantum thermodynamics. In work-counting applications, one introduces a generalized time $\mathcal T_{n:0}$0 carrying both physical time and counting fields, constructs local superoperators $\mathcal T_{n:0}$1, and contracts them with a PT-MPO built once from the environment dynamics: $\mathcal T_{n:0}$2 The work characteristic function $\mathcal T_{n:0}$3 is then inverted by Fourier transform,

$\mathcal T_{n:0}$4

Applied to a Landauer erasure protocol beyond weak-coupling, Markovian, and slow-driving limits, this yields full work distributions with quantum features that are invisible in the first two moments (Shubrook et al., 18 Dec 2025).

In circuit QED, time-translation-invariant process tensors have been used to simulate dispersive qubit readout while treating the full measurement resonator as an $\mathcal T_{n:0}$5-level subsystem. With $\mathcal T_{n:0}$6, $\mathcal T_{n:0}$7, $\mathcal T_{n:0}$8, and $\mathcal T_{n:0}$9, the method reaches times $\Upsilon_{n:0}$0 and captures ac-Stark shifts, non-flat $\Upsilon_{n:0}$1, and bath-memory tails missed by Lindblad or simple Purcell formulas (Cochin et al., 6 Mar 2026).

Projected ensembles of process tensors provide another application domain: quantum chaos diagnostics. For a pure process tensor $\Upsilon_{n:0}$2, projection onto a basis $\Upsilon_{n:0}$3 of multi-time Choi states defines the projected process ensemble

$\Upsilon_{n:0}$4

Its first moment

$\Upsilon_{n:0}$5

recovers the Renyi-2 quantum dynamical entropy and spatiotemporal entanglement, while higher moments reveal entanglement structures that distinguish chaotic, integrable, and many-body localized regimes more sharply than first-moment diagnostics alone (O'Donovan et al., 19 Feb 2025).

6. Relation to quantum stochastic processes, alternative geometries, and terminology

The process-tensor formalism is closely related to the operator-algebraic theory of quantum stochastic processes. In the Heisenberg-picture AFL framework, one works with non-commutative correlation kernels

$\Upsilon_{n:0}$6

By introducing an ancilla and an extended kernel operator $\Upsilon_{n:0}$7, the Schrödinger-picture process-tensor Choi state is obtained as

$\Upsilon_{n:0}$8

where $\Upsilon_{n:0}$9 is the link operator swapping ancilla input and output copies. This establishes an explicit equivalence between AFL correlation kernels and discrete-time process tensors (Nurdin et al., 2021).

Beyond MPO chains, alternative tensor-network geometries have been proposed. Process trees replace the linear time geometry by a binary tree in scale space built from fine-graining bricks $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$0 satisfying a scale-consistency condition. In a uniform tree, connected two-point correlators decay as a power law,

$$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$1

with $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$2, and $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$3-point correlators can be evaluated with scale-causal-cone algorithms whose cost is $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$4 for tree height $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$5. In contrast, an MPO with finite bond dimension supports exponentially decaying correlations (Dowling et al., 2023).

The term “process tensor” also appears in a distinct algebraic usage outside open-quantum-system control. For a real stochastic process $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$6, the $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$7th-order joint moment tensor

$$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$8

is symmetric, and if the process is $$\rho_n = \mathrm{Tr}_{\mathrm{in},\mathrm{out}} \Bigl[ \Upsilon_{n:0}\, (\mathcal I\!\otimes\!\mathcal A_{n-1}) \otimes\cdots\otimes (\mathcal I\!\otimes\!\mathcal A_0) \Bigr].$$9th-order stationary with period $U_{k+1,k}$0, the tensor is a real symmetric circulant tensor. For even $U_{k+1,k}$1, such a process-moment tensor is positive semidefinite because

$U_{k+1,k}$2

This alternative terminology refers to moment tensors of stationary stochastic processes rather than to the multi-time intervention maps central to open quantum dynamics (Chen et al., 2013).