# Prior Art: Local Observables vs Full-State Fidelity in Quantum Dynamics

Date: 2026-09-10

Thread objective: prior-art search for whether, in classical simulation of quantum many-body dynamics, local observables to fixed accuracy are known to be cheaper than representing the full state to fixed fidelity.

## Executive Map

**Evidence class: Established.** The global-state/MPS entanglement barrier is real and old: after generic quenches in 1D, entanglement grows linearly in time, and an MPS with bond dimension `chi` has entanglement at most `log chi`, so global fidelity requires `chi` exponential in time. Key sources: Calabrese and Cardy 2005; Vidal 2003/2004; Schuch, Wolf, Verstraete, Cirac 2008; Schuch, Wolf, Vollbrecht, Cirac 2008; Prosen and Znidaric 2007.

**Evidence class: Established.** Osborne 2006 proves an operator-norm/MPO approximation to the full 1D propagator using resources polynomial in system size `n` and inverse error, but exponential in `|t|`; consequently global simulation is efficient for `|t| = O(log n)`. The explicit theorem is stronger/different than a local-observable-only theorem: it approximates the propagator, not just one local expectation value. URL: https://arxiv.org/abs/quant-ph/0508031

**Evidence class: Established.** Entropy scaling alone does not decide MPS simulability unless one specifies which entropy and which approximation norm. Schuch et al. 2008 prove: logarithmic Renyi entropy with alpha < 1 implies efficient MPS approximability; superlogarithmic Renyi entropy with alpha > 1 or linear von Neumann entropy rules it out; bounded von Neumann entropy alone does not imply MPS approximability. URL: https://arxiv.org/abs/0705.0292

**Evidence class: Established plus Serious speculation.** There is a substantial counter-literature in which local observables, hydrodynamic quantities, or transport coefficients are obtained at bounded or modest cost while the true state/operator is no longer faithfully represented: DMT, DAOE, TDVP-after-entanglement-barrier, purification/MPDO methods, and operator-space methods. These are not general polynomial-time algorithms for arbitrary local observables under arbitrary local Hamiltonians; they use physical assumptions such as ETH/thermalization, high temperature, diffusion/hydrodynamics, dissipation/noise, integrability, or extrapolation.

**Evidence class: Established.** General local observable prediction cannot be classically easy in the worst case unless BQP collapses to classical polynomial time. The acceptance probability of a quantum circuit is a single-qubit local measurement, and local Hamiltonian dynamics can implement universal quantum computation. The precise hardness is worst-case and asymptotic; it does not forbid finite-size/finite-time numerical separations.

## 1. Osborne 2006: What Is Actually Proved

**Paper.** Tobias J. Osborne, "The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently," Phys. Rev. Lett. 97, 157202 (2006), arXiv:quant-ph/0508031. URL: https://arxiv.org/abs/quant-ph/0508031 DOI: https://doi.org/10.1103/PhysRevLett.97.157202

**Evidence class: Established.** Osborne proves an efficient approximation to the time-evolution operator for arbitrary 1D local spin systems, not merely a heuristic tDMRG observation. Abstract quote: "an arbitrarily good approximation to the propagator `e^{itH}` for a 1D lattice of `n` quantum spins with hamiltonian `H` may be obtained with polynomial computational resources in `n` and the error `epsilon`, and exponential resources in `|t|`." The same abstract states the consequence: "Vidal's time-dependent density matrix renormalisation group will require only polynomial resources to simulate 1D quantum spin systems for logarithmic `|t|`."

**Evidence class: Established.** Conditions: finite 1D chain of `n` spins; nearest-neighbor local Hamiltonian `H = sum_j h_j`, with `||h_j||` bounded independently of `n`; time-dependent terms are allowed; finite-range interactions can be handled by blocking a constant number of sites; no translation invariance is assumed. The proof uses Lieb-Robinson bounds to cut the chain and patch the cut with a unitary supported near the boundary.

**Evidence class: Established.** The quantitative decomposition is an approximate quantum cellular automaton (QCA) decomposition. Osborne writes

```text
e^{itH} = (tensor_j U_{Omega_j}(t))(tensor_j V_{Omega'_j}(t)) + O(epsilon)
```

where the blocks have size

```text
|Omega| = O(c0 |t| + c1 log(n/epsilon)).
```

The resulting finitely correlated/operator-MPS representation has auxiliary dimension at most `2^{2|Omega|}`. Therefore the global propagator approximation has storage polynomial in `n/epsilon` and exponential in `|t|`. If `|t| = O(log n)`, this remains polynomial in `n`.

**Evidence class: Established.** Osborne's proof contains the local-light-cone mechanism but states the main theorem globally. The `log(n/epsilon)` term comes from controlling the error of a global propagator decomposition over `O(n/|Omega|)` patches. Since an operator-norm approximation to the propagator implies correct expectation values for all bounded observables and all initial states, it is enough for local observables, but it is more stringent than asking for one local observable to fixed error.

**Evidence class: Your own inference.** For a fixed local observable and fixed local error, the same Lieb-Robinson patching logic suggests replacing `log(n/epsilon)` by `log(1/epsilon)` plus a light-cone radius proportional to `|t|`, because one need not union-bound over the whole chain. I did not find Osborne 2006 stating this local-observable-only cost as a separate theorem; the paper's explicit theorem is for the full propagator.

## 2. Schuch, Wolf, Verstraete, Cirac 2008: Entropy Scaling vs MPS Simulability

**Paper.** Norbert Schuch, Michael M. Wolf, Frank Verstraete, J. Ignacio Cirac, "Entropy scaling and simulability by Matrix Product States," Phys. Rev. Lett. 100, 030504 (2008), arXiv:0705.0292. URL: https://arxiv.org/abs/0705.0292 DOI: https://doi.org/10.1103/PhysRevLett.100.030504

**Evidence class: Established.** Abstract quote: "We investigate the relation between the scaling of block entropies and the efficient simulability by Matrix Product States (MPS), and clarify the connection both for von Neumann and Renyi entropies." The key warning is also in the abstract: "Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPS."

**Evidence class: Established.** Their definition of efficient MPS approximability is global trace-norm approximation: for every fixed error `delta`, there must be MPS bond dimension `D(N) = poly_delta(N)` such that `||psi_N - phi_{N,D}||_tr <= delta`. This is a full-state/full-observable guarantee, because they note that this bounds the error in any non-extensive observable by `delta`.

**Evidence class: Established.** Positive direction: if there exist constants `c,c' > 0` and `0 <= alpha < 1` such that all block Renyi entropies obey

```text
S_alpha(rho_N^L) <= c log N + c'
```

then the family can be efficiently approximated by MPS. They attribute the proof route to Verstraete and Cirac 2006 and place it in their Table I map.

**Evidence class: Established.** Negative directions: if von Neumann block entropy is linear for a block of length `L >= eta N`, `S(rho_N^L) >= c L`, then no polynomial-bond-dimension MPS can approximate globally for sufficiently small fixed trace error. They derive

```text
log D(N) >= eta(c - delta log(d)/2) N - 1.
```

Also, if some Renyi entropy with `alpha > 1` grows faster than logarithmically, `S_alpha(rho_N^L) >= c L^kappa` with `L >= eta N`, then MPS approximation needs superpolynomial/exponential-in-`N^kappa` bond dimension.

**Evidence class: Established.** The paper explicitly gives examples on both sides of the "undetermined" region: states with linearly growing Renyi entropies for `alpha < 1` that are approximable, states with sublinear algebraic von Neumann entropy that are approximable, and states with strict bounded von Neumann entropy that are not approximable. This is the precise sense in which "large entropy" may be simulable and "area law" may still be impossible: it depends on the entropy order and on the Schmidt tail, not only on von Neumann entropy.

**Evidence class: Established.** Their application to time evolution gives a concrete entanglement-barrier statement. They write that for a critical Ising Hamiltonian acting on a product state, there is evidence and a rigorous lower bound

```text
S(rho_N^L(t)) >= 4t/(3 pi) + O(log t), for t <= e L/4.
```

Plugging this into their MPS lower bound with `L = 4t/e` gives exponential-in-time required bond dimension for any trace error `delta < 2e/(3 pi) ~= 0.58`.

## 3. Entanglement Barrier and `t_max ~ log chi`

**Paper.** Pasquale Calabrese and John Cardy, "Evolution of Entanglement Entropy in One-Dimensional Systems," J. Stat. Mech. P04010 (2005), arXiv:cond-mat/0503393. URL: https://arxiv.org/abs/cond-mat/0503393 DOI: https://doi.org/10.1088/1742-5468/2005/04/P04010

**Evidence class: Established.** Abstract quote: "In general the entanglement entropy increases linearly with time `t` up to `t=l/2v`, after which it saturates at a value proportional to `l`, the coefficient depending on the initial state." This is the standard quench picture behind the MPS time barrier.

**Paper.** Guifre Vidal, "Efficient classical simulation of slightly entangled quantum computations," Phys. Rev. Lett. 91, 147902 (2003), arXiv:quant-ph/0301063. URL: https://arxiv.org/abs/quant-ph/0301063 DOI: https://doi.org/10.1103/PhysRevLett.91.147902

**Evidence class: Established.** Abstract quote: "The evolution of a pure state of n qubits can be simulated by using computational resources that grow linearly in n and exponentially in the entanglement." This gives the operational bridge from entanglement growth to cost growth.

**Paper.** Guifre Vidal, "Efficient simulation of one-dimensional quantum many-body systems," Phys. Rev. Lett. 93, 040502 (2004), arXiv:quant-ph/0310089. URL: https://arxiv.org/abs/quant-ph/0310089 DOI: https://doi.org/10.1103/PhysRevLett.93.040502

**Evidence class: Established.** Abstract quote: "The efficiency of the scheme depends on the amount of the entanglement involved in the simulated evolution." This is the TEBD/tDMRG condition.

**Paper.** Norbert Schuch, Michael M. Wolf, Kristan G. H. Vollbrecht, J. Ignacio Cirac, "On entropy growth and the hardness of simulating time evolution," New J. Phys. 10, 033032 (2008), arXiv:0801.2078. URL: https://arxiv.org/abs/0801.2078 DOI: https://doi.org/10.1088/1367-2630/10/3/033032

**Evidence class: Established.** Abstract quote: they give "a rigorous proof of the observation that a time independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework" and that this "implies that every global approximation of the evolution requires exponential resources for any MPS based method." This paper is the clean rigorous version of the entanglement-barrier statement.

**Paper.** Tomaz Prosen and Marko Znidaric, "Is efficiency of classical simulations of quantum dynamics related to integrability?" Phys. Rev. E 75, 015202(R) (2007), arXiv:quant-ph/0608057. URL: https://arxiv.org/abs/quant-ph/0608057 DOI: https://doi.org/10.1103/PhysRevE.75.015202

**Evidence class: Established numerical evidence.** Abstract quote: "We find that rank generally increases exponentially, `D_epsilon(t) proportional exp(const t)`, unless the system is integrable in which case we find polynomial increase." In the paper's central result, for a non-integrable tilted-field Ising chain `H_C = H(1,1)`, they report `D_epsilon(t) ~ exp(h_q t)` with `h_q = 1.10`, independent of error, initial local/extensive observable, and `n` once `n` is large enough. For integrable transverse Ising `H_R = H(0,2)`, they find local finite-index operators saturate, local infinite-index operators grow linearly, and extensive infinite-index operators likely grow quadratically.

**Evidence class: Established plus Your own inference.** A pure MPS across one cut has entropy `S <= log chi` (base matching the logarithm convention). If entanglement grows as `S(t) = s t + O(log t)`, then full-state MPS fidelity requires `log chi >= s t + O(log t)`, so

```text
t_max(chi) <= (log chi)/s + lower-order corrections.
```

Using Schuch et al. 2008's critical-Ising bound for fixed trace error `delta`, the explicit lower bound is

```text
log D >= (4/(3 pi) - 2 delta/e) t + O(log t),  delta < 2e/(3 pi).
```

Thus, in that rigorous example, accessible time grows at most logarithmically in bond dimension. This is the exact form I found closest to the requested `t_max ~ log chi` relation.

## 4. Counter-Literature: Local Observables and Transport Beyond the State

### Operator-space methods and integrable exceptions

**Paper.** Tomaz Prosen and Marko Znidaric 2007, above.

**Evidence class: Established numerical evidence.** This is already counter-literature in operator space: for integrable dynamics, local Heisenberg operators can be efficient even when typical pure states are not. The method represents observables/density operators as MPOs and measures the needed MPO rank `D_epsilon(t)`. What it gets right: Heisenberg evolution of local/extensive observables or high-temperature thermal states within a fidelity/truncation threshold. What it gives up: generic non-integrable systems still show exponential MPO-rank growth, and thermal-state fidelity at fixed `t,D` decreases with `n` in the thermodynamic limit.

**Paper.** Iztok Pizorn and Tomaz Prosen, "Operator space entanglement entropy in transverse Ising chain," Phys. Rev. A 76, 032316 (2007), arXiv:0706.2480. URL: https://arxiv.org/abs/0706.2480 DOI: https://doi.org/10.1103/PhysRevA.76.032316

**Evidence class: Established.** Abstract quote: "We introduce a new measure of entanglement in the space of operators and show, for transverse Ising spin 1/2 chain, that the simulation of observables, contrary to simulation of typical pure quantum states, is efficient for initial local operators." They prove/compute saturation of operator-space entanglement entropy (OSEE) for finite-index Majorana operators and logarithmic growth for infinite-index operators.

**Paper.** Iztok Pizorn and Tomaz Prosen, "Operator Space Entanglement Entropy in XY Spin Chains," Phys. Rev. B 79, 184416 (2009), arXiv:0903.2432. URL: https://arxiv.org/abs/0903.2432 DOI: https://doi.org/10.1103/PhysRevB.79.184416

**Evidence class: Established.** Abstract quote: "We show that in the homogeneous Heisenberg XY spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time." They report prefactors changing from `1/3` to `2/3` at a transition, and disorder-triggered saturation. What it gets right: local-operator evolution in free/integrable chains. What it gives up: it does not cover generic chaotic dynamics.

**Null result.** I searched for the requested "Wang & Xiang" connection to operator-space entanglement and did not find a Wang-Xiang OSEE paper. Searches included `Wang Xiang operator space entanglement`, `Wang Xiang time-dependent density matrix renormalization group finite temperature`, and related variants. What I found instead: X. Wang and T. Xiang, "Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantum systems," Phys. Rev. B 56, 5061 (1997), and Luo, Xiang, Wang's 2003 comment on tDMRG. Those are relevant to finite-temperature DMRG history, but I did not find them establishing the local-observable/operator-entanglement claims requested here.

### DMT

**Paper.** Christopher David White, Michael Zaletel, Roger S. K. Mong, Gil Refael, "Quantum dynamics of thermalizing systems," Phys. Rev. B 97, 035127 (2018), arXiv:1707.01506. URL: https://arxiv.org/abs/1707.01506 DOI: https://doi.org/10.1103/PhysRevB.97.035127

**Evidence class: Established method, physically motivated rather than worst-case theorem.** Abstract quote: "We introduce a method `DMT' for approximating density operators of 1D systems that, when combined with a standard framework for time evolution (TEBD), makes possible simulation of the dynamics of strongly thermalizing systems to arbitrary times." They emphasize the standard barrier: "MPS methods can only treat large systems - but only when the systems have little entanglement entropy" and "for thermalizing systems, whose entanglement entropy grows linearly with time, MPS methods can only treat short-time behavior."

**Evidence class: Established method.** DMT truncates matrix product density operators while preserving trace and reduced density matrices on short regions around the cut, hence preserving expectation values of local conserved quantities. It targets local observables/hydrodynamic modes in ETH systems, including energy-density Fourier modes, not global fidelity to a pure state. What it gives up: it deliberately discards long-range information and does not converge monotonically to the exact global state at fixed small bond dimension; the authors note accuracy can saturate with increasing bond dimension because Frobenius-norm remnants still over-weight long operators.

### DAOE

**Paper.** Tibor Rakovszky, C. W. von Keyserlingk, Frank Pollmann, "Dissipation-assisted operator evolution method for capturing hydrodynamic transport," Phys. Rev. B 105, 075131 (2022), arXiv:2004.05177. URL: https://arxiv.org/abs/2004.05177 DOI: https://doi.org/10.1103/PhysRevB.105.075131

**Evidence class: Established numerical method with extrapolation.** Abstract quote: "DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators." Also: "the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times" and "By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation."

**Evidence class: Established.** The target quantities are high-temperature dynamical density correlations, mean-square displacement, and spin/energy diffusion constants. The artificial dissipator damps Pauli strings longer than a cutoff `ell_*`; it has exact MPO bond dimension `ell_*+1`. The authors explicitly state generic unitary operator entanglement grows linearly, `S_vN[q_j(t)] proportional t`, requiring MPO bond dimension exponential in `t`; with nonzero dissipation, operator entanglement peaks and decreases, permitting finite `chi` to long times. What it gives up: the evolved operator is not the true Heisenberg operator; physical results are obtained by extrapolating `gamma -> 0`, and the argument depends on suppression of backflow from long strings to conserved densities.

### TDVP beyond the barrier

**Paper.** Eyal Leviatan, Frank Pollmann, Jens H. Bardarson, David A. Huse, Ehud Altman, "Quantum thermalization dynamics with Matrix-Product States," arXiv:1702.08894 (2017). URL: https://arxiv.org/abs/1702.08894

**Evidence class: Serious speculation backed by numerical evidence.** Abstract quote: "Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle." They extract transport coefficients such as energy diffusion constants at small bond dimension. What it gives up: the state vector after the entanglement barrier is not globally accurate; TDVP is treated as an effective dynamics on the low-bond-dimension MPS manifold that preserves energy and can capture thermalizing local physics.

**Paper.** Jutho Haegeman et al., "Time-dependent variational principle for quantum lattices," Phys. Rev. Lett. 107, 070601 (2011), arXiv:1103.0936. URL: https://arxiv.org/abs/1103.0936 DOI: https://doi.org/10.1103/PhysRevLett.107.070601

**Evidence class: Established algorithmic framework.** Abstract quote: the TDVP procedure "explicitly preserves all symmetries and conservation laws" and has "low computational complexity." The 2016 unifying paper is Jutho Haegeman et al., "Unifying time evolution and optimization with matrix product states," Phys. Rev. B 94, 165116 (2016), arXiv:1408.5056. URL: https://arxiv.org/abs/1408.5056

### MPDO, finite-temperature, and dissipation methods

**Paper.** Frank Verstraete, Juan J. Garcia-Ripoll, J. Ignacio Cirac, "Matrix Product Density Operators: Simulation of finite-T and dissipative systems," Phys. Rev. Lett. 93, 207204 (2004), arXiv:cond-mat/0406426. URL: https://arxiv.org/abs/cond-mat/0406426 DOI: https://doi.org/10.1103/PhysRevLett.93.207204

**Evidence class: Established method.** Abstract quote: "We show how to simulate numerically both the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium." It represents density operators as MPDOs and uses variational truncation. What it gets right: finite-temperature and Lindblad/dissipative 1D dynamics when operator entanglement remains controlled. What it gives up: no general proof that arbitrary dissipative dynamics stays bounded in operator-space entanglement.

**Paper.** Michael Zwolak and Guifre Vidal, "Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm," Phys. Rev. Lett. 93, 207205 (2004), arXiv:cond-mat/0406440. URL: https://arxiv.org/abs/cond-mat/0406440 DOI: https://doi.org/10.1103/PhysRevLett.93.207205

**Evidence class: Established method.** This is the parallel MPDO/superoperator algorithm line for mixed states. It supports the idea that mixed-state/operator representations can be compact even when pure-state entanglement is not.

## 5. Hardness Results Cutting the Other Way

**Evidence class: Established.** Estimating a single local measurement after a polynomial-size quantum circuit is BQP-complete by the definition of circuit-model BQP: the output bit is a measurement of one qubit. Thus, a general polynomial-time classical algorithm for all local observables after all local quantum evolutions would imply `BQP` is contained in classical polynomial time with the same error model.

**Paper.** Seth Lloyd, "Universal Quantum Simulators," Science 273, 1073-1078 (1996), arXiv:quant-ph/9603028. URL: https://arxiv.org/abs/quant-ph/9603028 DOI: https://doi.org/10.1126/science.273.5278.1073

**Evidence class: Established.** Abstract quote from arXiv: "We suggest that quantum computers can solve quantum many-body problems that are impracticable to solve on a classical computer." Lloyd gives the standard product-formula route by which local quantum dynamics simulates local quantum dynamics efficiently. This is a quantum upper bound, not by itself a BQP-hardness theorem for a fixed Hamiltonian family.

**Paper.** Andrew M. Childs, David Gosset, Zak Webb, "Universal quantum walks and adiabatic algorithms by 1D Hamiltonians," arXiv:0802.1207 (2008). URL: https://arxiv.org/abs/0802.1207

**Evidence class: Established.** Abstract quote: "We construct a family of time-independent nearest-neighbor Hamiltonians coupling eight-state systems on a 1D ring that enables universal quantum computation." It further states: "Our results demonstrate that simulating 1D chains of spin-7/2 particles is BQP-hard, and indeed BQP-complete because the outputs of decision problems can be encoded in the outputs of such simulations." This is the direct local-Hamiltonian dynamics hardness result closest to the present question.

**Paper.** Dorit Aharonov and Amnon Ta-Shma, "Adiabatic Quantum State Generation and Statistical Zero Knowledge," arXiv:quant-ph/0301023; STOC 2003; SIAM J. Comput. later version. URL: https://arxiv.org/abs/quant-ph/0301023

**Evidence class: Established.** Abstract quote: they define "adiabatic state generation" and prove it "is equivalent to state generation in the standard quantum computing model." This supports the broader Feynman-Kitaev/adibatic equivalence line: local Hamiltonian evolution/state generation can encode quantum computation.

**Paper.** Shengyu Zhang, "Several natural BQP-Complete problems," arXiv:quant-ph/0606179 (2006). URL: https://arxiv.org/abs/quant-ph/0606179

**Evidence class: Established.** Abstract quote: "We present several BQP-complete problems, including Local Hamiltonian Eigenvalue Sampling and Phase Estimation Sampling." This is not exactly "local observable after time evolution," but it reinforces that natural Hamiltonian linear-algebra tasks are BQP-complete.

**Evidence class: Your own inference.** What this hardness does and does not say: it rules out a uniform, worst-case, polynomial-time classical algorithm for arbitrary local Hamiltonians, polynomial times, and inverse-polynomial accuracy local measurements unless BQP collapses. It does not rule out: fixed finite `n`; fixed short `t`; high-temperature hydrodynamic observables; noisy circuits; integrable/free models; ETH-motivated approximations; or empirical separations between `chi_observable` and `chi_fidelity` in small exact diagonalization.

## 6. Mixed States, Dissipation, Noise, and Area Laws

**Paper.** Michael M. Wolf, Frank Verstraete, Matthew B. Hastings, J. Ignacio Cirac, "Area laws in quantum systems: mutual information and correlations," Phys. Rev. Lett. 100, 070502 (2008), arXiv:0704.3906. URL: https://arxiv.org/abs/0704.3906 DOI: https://doi.org/10.1103/PhysRevLett.100.070502

**Evidence class: Established.** Abstract quote: "the information contained in part of a system in thermal equilibrium obeys an area law." This is a mixed-state area law in terms of mutual information, not a statement that every mixed state or every open-system trajectory has bounded MPO/OSEE.

**Paper.** Andras Molnar, Norbert Schuch, Frank Verstraete, J. Ignacio Cirac, "Approximating Gibbs states of local Hamiltonians efficiently with PEPS," Phys. Rev. B 91, 045138 (2015), arXiv:1406.2973. URL: https://arxiv.org/abs/1406.2973 DOI: https://doi.org/10.1103/PhysRevB.91.045138

**Evidence class: Established.** Abstract quote: they derive `D = (N/epsilon)^{O(beta)}` and state: "This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians." In 1D, PEPS reduces to MPO/MPS-type tensor networks. This is about equilibrium Gibbs states at finite temperature, not arbitrary real-time open dynamics.

**Paper.** Yichen Huang, "Locally accurate matrix product approximation to thermal states," Science Bulletin 66, 2456 (2021), arXiv:2106.03854. URL: https://arxiv.org/abs/2106.03854 DOI: https://doi.org/10.1016/j.scib.2021.08.011

**Evidence class: Established.** Abstract quote: "We prove that a thermal state at constant inverse temperature beta has a matrix product representation with bond dimension `e^{tilde O(sqrt(beta log(1/epsilon)))}` such that all local properties are approximated to accuracy `epsilon`." This is directly about local observables being cheaper than global state approximation for thermal states.

**Paper.** Mario Berta, Fernando G. S. L. Brandao, Christian Majenz, Mark M. Wilde, "Thermal States as Convex Combinations of Matrix Product States," Phys. Rev. B 98, 235154 (2018), arXiv:1709.07423. URL: https://arxiv.org/abs/1709.07423 DOI: https://doi.org/10.1103/PhysRevB.98.235154

**Evidence class: Established.** Abstract quote: "We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of matrix product states." They connect this to METTS and hydrodynamic time-dependent MPS algorithms.

**Paper.** Kyungjoo Noh, Liang Jiang, Bill Fefferman, "Efficient classical simulation of noisy random quantum circuits in one dimension," Quantum 4, 318 (2020), arXiv:2003.13163. URL: https://arxiv.org/abs/2003.13163 DOI: https://doi.org/10.22331/q-2020-09-11-318

**Evidence class: Established numerical and heuristic analysis.** Abstract quote: they use MPO entanglement entropy because it "determines the cost of classical MPO simulation" and show that above a characteristic system size "the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size." They also find the cost increases exponentially as error rate decreases. This is a clean noisy-circuit case where decoherence bounds MPO entanglement and makes system-size scaling benign.

**Evidence class: Your own inference.** The mixed-state/open-system literature supports a qualified principle: thermalization, noise, and dissipation often suppress the information that makes global pure states hard, so local observables and thermal properties may admit bounded-bond-dimension descriptions. It does not establish a universal theorem that all open/decohering systems obey bounded operator-space entanglement indefinitely. Counterexamples can be engineered with weak/no dissipation, protected coherent subspaces, or dynamics that stores computationally hard information before noise removes it.

## Null Results and Search Notes

**Evidence class: Established search result.** I found no paper that states the broad claim "local observables are always far cheaper than full-state fidelity for quantum dynamics" as a theorem. The literature separates into: (1) worst-case hardness, (2) global-MPS entanglement barriers, (3) rigorous finite-time/light-cone/operator approximations, and (4) physically motivated algorithms for ETH, hydrodynamics, integrability, thermal states, and noisy circuits.

**Evidence class: Established search result.** I did not find a single canonical paper giving a universal measured law `t_max = c log chi` for all TEBD/MPS simulations. The exact relation follows immediately from `S <= log chi` plus linear entanglement growth, and Schuch et al. 2008 give the rigorous critical-Ising lower-bound version with explicit constants. Prosen and Znidaric 2007 give direct measured `D_epsilon(t) ~ exp(1.10 t)` for a chaotic tilted-field Ising chain in operator/MPO simulations.

**Evidence class: Established search result.** I did not find a Wang-Xiang operator-space-entanglement result matching the prompt. The closest relevant thread is finite-temperature transfer-matrix DMRG and earlier tDMRG commentary, while the operator-space entanglement literature is primarily Prosen, Znidaric, Pizorn, Zanardi, Dubail, and later operator-growth papers.

## VERDICT FOR THE GATE

The claim "local observables are far cheaper than full-state fidelity in quantum dynamics" is **known but under-quantified**, not a single established textbook theorem in that generality. It is established in important regimes: finite light cones for finite times, thermal/Gibbs local properties, noisy 1D circuits, integrable operator evolution, hydrodynamic/ETH-motivated methods, and transport-coefficient algorithms; but worst-case local observable prediction remains BQP-hard. The novelty gate should therefore not be "local cheaper than global" broadly, but rather any precise quantitative law comparing `chi_observable` and `chi_fidelity` for a specified model, time window, observable class, accuracy metric, and failure criterion.

## WHAT WOULD BE NEW

**Evidence class: Your own inference.** A small exact-diagonalization study would not newly establish the entanglement barrier or the qualitative fact that local observables can converge after global fidelity fails; those are already in the literature above.

**Evidence class: Your own inference.** It could add value if it defines an operational, reproducible pair of minimal bond dimensions, for example `chi_observable(t, epsilon, O)` versus `chi_fidelity(t, epsilon)` on the same exact wavefunction data, and measures their separation across chaotic/integrable/noisy/thermalizing cases.

**Evidence class: Your own inference.** It would be most plausibly new if it produces a scaling law or diagnostic not already explicit in DMT/DAOE/TDVP/OSEE work, such as a finite-size collapse for `chi_observable`, a relation to operator entanglement/backflow, or a model-independent empirical threshold where local observables stabilize while global Schmidt truncation error remains large.