---
title: Process Tensors in Open Quantum Systems
url: https://www.emergentmind.com/topics/process-tensors
type: topic
---

# Process Tensors in Open Quantum Systems

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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 [2402.15454].

## 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 [2402.15454].

A complementary microscopic expression starts from joint system-environment unitary propagators \(U_{k+1,k}\) and an initially factorized state \(\rho_S(0)\otimes\rho_E\). In that setting, the process tensor may be written as the unique multi-linear map, or “quantum comb,” satisfying
\[
\rho_S(t_N\mid\{\mathcal A_k\})
=
\Upsilon_{N:0}\star
(\mathcal A_{N-1}\otimes\cdots\otimes\mathcal A_0),
\]
with Choi form
\[
\Upsilon_{N:0}
=
\mathrm{Tr}_E\!\Big[
(U_{N,N-1}\!\otimes\! U_{N,N-1}^*)\cdots
(U_{1,0}\!\otimes\! U_{1,0}^*)
(I\!\otimes\!\rho_E^T)
\Big].
\]
This identifies the process tensor as the compressed multi-time record of all system-environment correlations relevant to future reduced dynamics [2512.16823].

## 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 \(\delta t\), a maximum memory length \(\Delta K_{\max}\) is chosen, and at step \(m\) one forms bath tensors
\[
[b_k]^{\alpha,j}_{\alpha',j'}
=
\delta^j_{j'}\,\delta^\alpha_{\alpha'}\,I_k(\alpha,j),
\]
which are contracted with the existing process-tensor MPO and compressed by SVD, truncating singular values below \(\epsilon_{\mathrm{rel}}\). The resulting compressed MPO represents \(\Upsilon_{m:0}\), and the cost at each step scales roughly as \(O(d^2\chi^3)\), where \(d\) is the system Liouville dimension and \(\chi\) the MPO bond dimension [2402.15454].

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,
\[
\mathcal F^{\mu_0,\dots,\mu_{n-1}}
=
\prod_{i=0}^{n-1}\prod_{j=1}^{i} b^{\mu_j}_{\mu_i}(i-j),
\]
with the two-dimensional triangular tensor network generated by four-leg gates \(\tilde b(k)\). Rotating this network by \(45^\circ\) 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 \(|i-j|\), and one obtains a time-translation-invariant process tensor whose application cost grows with memory depth \(\kappa\), not with total time \(n\) [2603.06840].

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 \(\chi_d\) provides a direct metric for comparing the effective environmental complexity carried by different methods [2406.17719].

## 3. Structural properties and physical interpretation

In Choi form, process tensors satisfy positivity and causality constraints. For a \(k\)-slot process tensor \(T_{k:0}\), one has
\[
T_{k:0}\ge 0,\qquad \mathrm{Tr}[T_{k:0}]=d_S^{2k},
\]
together with the no-future-to-past conditions
\[
\forall j:\quad \mathrm{Tr}_{o_j}[T_{j:0}]
=
I_{i_j}\otimes T_{j-1:0},
\qquad
T_{0:0}=\rho_S.
\]
These identities express the fact that plugging in trace-preserving instruments yields valid states and that interventions at later times cannot alter earlier statistics [2312.04624].

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
\[
T_l:\mathfrak L(\mathcal H_E)\to \mathbb C^{D_l},
\]
which project environment Liouville vectors \(|\beta_l)\) to bond-space amplitudes \(|d_l)\), together with Moore-Penrose pseudoinverses
\[
T_l^+=(T_l^\dagger T_l)^{-1}T_l^\dagger.
\]
The local MPO block may then be written as
\[
\mathcal Q^{(\alpha_l,\alpha'_l)}_{d_l,d_{l-1}}
=
\sum_{\beta_l,\beta_{l-1}}
T_{l,d_l,\beta_l}\,
(\alpha_l,\beta_l|e^{\mathcal L_E\Delta t}|\alpha'_l,\beta_{l-1})\,
T^+_{l-1,\beta_{l-1},d_{l-1}}.
\]
This makes the inner bonds interpretable as a lossy but physically meaningful projection of the environment dynamics [2404.01287].

This interpretation has practical consequences. Observable “closures” constructed with \(T_l^+\) 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 [2404.01287].

## 4. Construction algorithms, compression, and scaling

The basic PT-TEMPO workflow proceeds by discretizing time, keeping only the last \(\Delta K_{\max}\) influence tensors at each step, contracting the new bath tensor with the existing MPO, and compressing by SVD. Convergence is controlled by the triplet \((\delta t,\Delta K_{\max},\epsilon_{\mathrm{rel}})\). This provides a numerically exact construction in the converged limit and a transparent route from influence functionals to compressed MPOs [2402.15454].

For time-translation-invariant process tensors, the original iTEBD construction applies a sequence of four-leg gates \(\tilde b(k)\), performs full SVDs on \((\chi d^2)\times(\chi d^2)\) 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 \(b(k)\) to low effective rank \(\alpha\), then performing partial SVDs on intermediate blocks before the final SVD on a reduced core tensor \(\Theta\). In the reported benchmarks, this reduces end-to-end construction cost from \(\sim d^8\) to \(\sim d^4\), while peak memory drops from \(\sim d^4\) to \(\sim d^2\) or lower with matrix-free SVD, and the final bond dimension \(\chi\) is unchanged [2603.06840].

The MPO form also enables adjoint-style optimal control. Defining the forward “extended” state
\[
\sigma_k^{\mu_k\chi_k}
=
\sum_{\nu_k,\mu_{k-1},\chi_{k-1}}
O[k]^{\mu_k\nu_k}_{\chi_k,\chi_{k-1}}
\,U_k^{\nu_k\mu_{k-1}}
\,\sigma_{k-1}^{\mu_{k-1}\chi_{k-1}},
\]
and a backward costate with terminal condition \(\lambda_T^{\mu_T\chi_T}=-\partial Z/\partial \rho_{S,T}^{\mu_T}\), one obtains local gradient contractions
\[
\frac{\partial Z}{\partial U_q^{\mu_{q-1}\nu_q}}
=
\sum_{\mu_q,\chi_q,\chi_{q-1}}
\lambda_q^{\mu_q\chi_q}
\,O[q]^{\mu_q\nu_q}_{\chi_q,\chi_{q-1}}
\,\sigma_{q-1}^{\mu_{q-1}\chi_{q-1}}.
\]
In this setting, forward and backward propagation scale as
\[
O(T\,S^4\,\chi_d^2),
\]
with \(S\) the system Hilbert-space size and \(\chi_d\) the maximal MPO bond dimension [2406.17719].

## 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 \(R_j\). Within the PT-MPO framework, one inserts superoperators \(V^L,V^R\) 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 \(\tau_4\). 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 [2402.15454].

Process tensors also support numerically exact quantum thermodynamics. In work-counting applications, one introduces a generalized time \(\tau\) carrying both physical time and counting fields, constructs local superoperators \(M_j=e^{\Delta\tau\,\mathcal L_S(\tau_j)}\), and contracts them with a PT-MPO built once from the environment dynamics:
\[
\rho_S(\tau_f)
=
\mathrm{Tr}_E\big[\Upsilon_{N:0}*(M_N\otimes\cdots\otimes M_1)\big].
\]
The work characteristic function \(\Phi(\chi,t_f)\) is then inverted by Fourier transform,
\[
P(W)=\frac{1}{2\pi}\int_{-\chi_{\max}}^{\chi_{\max}} d\chi\, e^{i\chi W}\,\Phi(\chi,t_f).
\]
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 [2512.16823].

In circuit QED, time-translation-invariant process tensors have been used to simulate dispersive qubit readout while treating the full measurement resonator as an \(N\)-level subsystem. With \(\kappa=1000\), \(\Delta t=2\pi/62\,\omega_r^{-1}\), \(\epsilon_{\mathrm{rel}}=10^{-7}\), and \(N=20\), the method reaches times \(t\approx 5/(\eta\omega_r)\) and captures ac-Stark shifts, non-flat \(J(\omega)\), and bath-memory tails missed by Lindblad or simple Purcell formulas [2603.06840].

Projected ensembles of process tensors provide another application domain: quantum chaos diagnostics. For a pure process tensor \(|\Upsilon\rangle\), projection onto a basis \(\{|\vec x\rangle\}\) of multi-time Choi states defines the projected process ensemble
\[
\mathcal E=\{\,p_{\vec x};|\tilde\Upsilon_{R|\vec x}\rangle\,\}_{|\vec x\rangle\in\mathcal H_B}.
\]
Its first moment
\[
\Upsilon_R^{(1)}
=
\sum_{\vec x}p_{\vec x}\,
|\tilde\Upsilon_{R|\vec x}\rangle
\langle \tilde\Upsilon_{R|\vec x}|
\]
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 [2502.13930].

## 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
\[
w_{\mathbf t_n}(\mathbf a_n,\mathbf b_n)
=
\mu\!\bigl(j_{t_1}(a_1)\cdots j_{t_n}(a_n)\bigr)^*
\bigl(j_{t_1}(b_1)\cdots j_{t_n}(b_n)\bigr).
\]
By introducing an ancilla and an extended kernel operator \(K_n\), the Schrödinger-picture process-tensor Choi state is obtained as
\[
\Upsilon_{\mathbf t_n}
=
\mathrm{Tr}_{\mathrm{ancilla}}[L\,K_n],
\]
where \(L\) is the link operator swapping ancilla input and output copies. This establishes an explicit equivalence between AFL correlation kernels and discrete-time process tensors [2109.09256].

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 \(\mathcal W\) satisfying a scale-consistency condition. In a uniform tree, connected two-point correlators decay as a power law,
\[
\langle A'(t_n)A(t_0)\rangle-\langle A'\rangle\langle A\rangle
\sim
(\Delta t+1)^{-\alpha},
\]
with \(\alpha=|\log_2(\lambda_2^2)|>0\), and \(k\)-point correlators can be evaluated with scale-causal-cone algorithms whose cost is \(O((k+1)N d^6)\) for tree height \(N\). In contrast, an MPO with finite bond dimension supports exponentially decaying correlations [2312.04624].

The term “process tensor” also appears in a distinct algebraic usage outside open-quantum-system control. For a real stochastic process \(\{x_k\}\), the \(m\)th-order joint moment tensor
\[
[M^{(m)}]_{i_1\cdots i_m}
=
\mathbb E[x_{i_1}x_{i_2}\cdots x_{i_m}]
\]
is symmetric, and if the process is \(m\)th-order stationary with period \(n\), the tensor is a real symmetric circulant tensor. For even \(m\), such a process-moment tensor is positive semidefinite because
\[
\mathcal M z^m=\mathbb E[(z_1x_1+\cdots+z_nx_n)^m]\ge 0.
\]
This alternative terminology refers to moment tensors of stationary stochastic processes rather than to the multi-time intervention maps central to open quantum dynamics [1312.2752].

Source: https://www.emergentmind.com/topics/process-tensors