We show how local interactions in a partial differential equation (PDE) model lead to eigenfunctions with favorable low-rank properties. To this end, we utilize ideas from quantum entanglement of multi-particle spin systems. We begin by analyzing the connection between entropy area laws and low-rank approximability. The characterization for 1D chains such as Matrix Product States (MPS) / Tensor Trains (TT) is rather extensive though incomplete. We then show that a Nearest Neighbor Interaction (NNI) PDE operator has eigenfunctions that are approximately separable in a certain sense. Under a further assumption on the approximand, we show that this implies a constant entropy bound. To the best of our knowledge, the presented approach is the first systematic treatment of low-rank approximability for general PDEs. The assumptions are general enough to include a large class of models (in, e.g., quantum mechanics) and have a physical interpretation. The weak points are the aforementioned assumption on the approximand and that the validity is limited to MPS/TT formats.
We show how local bounded interactions in an unbounded Hamiltonian lead to eigenfunctions with favorable low-rank properties. To this end, we utilize ideas from quantum entanglement of multi-particle spin systems. We begin by analyzing the connection between entropy area laws and low-rank approximability. The characterization for 1D chains such as Matrix Product States (MPS) / Tensor Trains (TT) is rather extensive though incomplete. We then show that a Nearest Neighbor Interaction (NNI) Hamiltonian has eigenfunctions that are approximately separable in a certain sense. Under a further assumption on the approximand, we show that this implies a constant entropy bound. To the best of our knowledge, this work is the first analysis of low-rank approximability for unbounded Hamiltonians. Moreover, it extends previous results on entanglement entropy area laws to unbounded operators. The assumptions include a variety of self-adjoint operators and have a physical interpretation. The weak points are the aforementioned assumption on the approximand and that the validity is limited to MPS/TT formats.
We describe a family of convex continuous functions
EF,ε,
ε∈(0,1], on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems) obtained by applying the Mora-Piani-Briegel construction of
ε-measure (proposed in [Phys. Rev. A 109, 052413]) to the Entanglement of Formation (EoF) with the use of the trace norm distance in the role of basic metric. These functions are non-increasing in
ε and converge pointwise to the EoF as
ε→0+. We conducted a detailed analysis of properties of the functions
EF,ε, using (among others) the relationship between the family
{EF,ε}ε∈[0,1] and the family
{EFλ(ρ)}λ>0 of Moreau-Yosida approximations of the EoF explored in [arXiv:2609.30246].
We obtain a variational expression for the functions
EF,ε (which for
ε=0 turns into the standard variational expression for the EoF). We also describe the weak subadditivity property of the functions
EF,ε and analyse their continuity properties.
The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46, 2632-2658],[arXiv:2404.07978]) allow us to obtain easily computable upper bounds on the rate of uniform convergence of the function
EF,ε to the EoF as
ε→0+ on the sets of states with bounded rank/energy of one of the marginal states.
Applicability of the obtained results to other entanglement measures, in particular, to the relative entropy of entanglement, is discussed.
As an auxiliary result, it is shown that for any state of an infinite-dimensional bipartite quantum system, there exists a nearest separable state w.r.t. the trace norm distance. The non-uniqueness of such a state is also shown.
Encoding classical data in a quantum state is a key prerequisite of many quantum algorithms. Recently, matrix product state (MPS) methods emerged as the most promising approach for constructing shallow quantum circuits approximating input functions, including probability distributions, with only linear number of gates. We derive rigorous asymptotic expansions for the decay of entanglement across bonds in the MPS representation depending on the smoothness of the input function, real or complex. We also consider the dependence of the entanglement on localization properties and function support. Based on these analytical results, we construct an improved MPS-based algorithm yielding shallow and accurate encoding quantum circuits. By using tensor cross interpolation, we are able to construct utility-scale quantum circuits in a compute- and memory-efficient way. We validate our methods by loading heavy-tailed distributions, including Lévy, important in finance, but they apply to any smooth function inputs. We test the performance of the resulting quantum circuits by executing and sampling from them on IBM quantum devices, for up to 156 qubits.
Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE
α-Rényi entropies. Specifically, we prove that a volume law scaling for
α≥1 implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the
α<1 entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form
Lφ(ρ)=tr(φ(dρ))/d, where
ρ is an unknown
d-dimensional quantum state and
φ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating
Lφ(ρ) from properties of
φ, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting.
The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is
Θ(εlog2(d)max{1,εlog2(d)}d2+ε2log2(d)), and the sample complexity of spectrum estimation is
Θ(ε2log2(d)max{1,ε2log2(d)}d2). In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.