Continuous Matrix Product Research Articles

Tradeoff relations in open quantum dynamics via Robertson, Maccone–Pati, and Robertson–Schrödinger uncertainty relations

Abstract The Heisenberg uncertainty relation, together with Robertson’s generalisation, serves as a fundamental concept in quantum mechanics, showing that noncommutative pairs of observables cannot be measured precisely. In this study, we explore the Robertson-type uncertainty relations to demonstrate their effectiveness in establishing a series of thermodynamic uncertainty relations and quantum speed limits in open quantum dynamics. The derivation utilises a scaled continuous matrix product state representation that maps the time evolution of the quantum continuous measurement to the time evolution of the system and field. Specifically, we consider the Maccone–Pati uncertainty relation, a refinement of the Robertson uncertainty relation, to derive thermodynamic uncertainty relations and quantum speed limits. These newly derived relations, which use a state orthogonal to the initial state, yield bounds that are tighter than previously known bounds. Moreover, we consider the Robertson–Schrödinger uncertainty, which extends the Robertson uncertainty relation. Our findings not only reinforce the significance of the Robertson-type uncertainty relations, but also expand its applicability in identifying uncertainty relations in open quantum dynamics.

Unveiling quantum phases in quasi-one-dimensional dipolar gases using continuous matrix product states

Unveiling quantum phases in quasi-one-dimensional dipolar gases using continuous matrix product states

Exact solution to quantum dynamical activity.

The quantum dynamical activity constitutes a thermodynamic cost in trade-off relations such as the quantum speed limit and the quantum thermodynamic uncertainty relation. However, calculating the quantum dynamical activity has been a challenge. In this paper, we present the exact solution for the quantum dynamical activity by deploying the continuous matrix product state method. Moreover, using the derived exact solution, we determine the upper bound of the dynamical activity, which comprises the standard deviation of the system Hamiltonian and jump operators. We confirm the exact solution and the upper bound by performing numerical simulations.

Entanglement renormalization of the class of continuous matrix product states

Entanglement renormalization of the class of continuous matrix product states

Unifying speed limit, thermodynamic uncertainty relation and Heisenberg principle via bulk-boundary correspondence

The bulk-boundary correspondence provides a guiding principle for tackling strongly correlated and coupled systems. In the present work, we apply the concept of the bulk-boundary correspondence to thermodynamic bounds described by classical and quantum Markov processes. Using the continuous matrix product state, we convert a Markov process to a quantum field, such that jump events in the Markov process are represented by the creation of particles in the quantum field. Introducing the time evolution of the continuous matrix product state, we apply the geometric bound to its time evolution. We find that the geometric bound reduces to the speed limit relation when we represent the bound in terms of the system quantity, whereas the same bound reduces to the thermodynamic uncertainty relation when expressed based on quantities of the quantum field. Our results show that the speed limits and thermodynamic uncertainty relations are two aspects of the same geometric bound.

Universal Scaling of Klein Bottle Entropy near Conformal Critical Points.

We show that the Klein bottle entropy [H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.261603] for conformal field theories perturbed by a relevant operator is a universal function of the dimensionless coupling constant. The universal scaling of the Klein bottle entropy near criticality provides an efficient approach to extract the scaling dimension of lattice operators via data collapse. As paradigmatic examples, we validate the universal scaling of the Klein bottle entropy for Ising and Z_{3} parafermion conformal field theories with various perturbations using numerical simulation with continuous matrix product operator approach.

Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

We study the structure of the continuous matrix product operator (cMPO)[1] for the transverse field Ising model (TFIM). We prove TFIM’s cMPO is solvable and has the form . is a non-local free fermionic Hamiltonian on a ring with circumference β, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of is determined analytically. At the critical point, our results verify the state–operator-correspondence[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO’s ground state.

Quantum coherent states of interacting Bose-Fermi mixtures in one dimension

We study two-component atomic gas mixtures in one dimension involving both bosons and fermions. When the inter-species interaction is attractive, we report a rich variety of coherent ground-state phases that vary with the intrinsic and relative strength of the interactions. We avoid any artifacts of lattice discretization by developing a novel implementation of a continuous matrix product state ansatz for mixtures and priorly demonstrate the validity of our approach on the integrable point that exists for mixtures with equal masses and interactions (Lai-Yang model) where we find that the ansatz correctly and systematically converges towards the exact results.

Efficient simulation of quantum many-body thermodynamics by tailoring a zero-temperature tensor network

Various successful approaches using numerical annealing and renormalization groups have been conceived to study the thermodynamics of strongly correlated systems where perturbation or expansion theories fail to work. As the process of lowering the temperatures is usually involved in different manners, these approaches in general become much less efficient or accurate at the low temperatures. In this work, we propose to access the finite-temperature properties from the tensor network (TN) representing the zero-temperature partition function. We propose to ``scissor'' a finite part from such an infinite-size TN and ``stitch'' it to possess the periodic boundary condition along the imaginary-time direction. We dub this approach ``TN tailoring.'' Exceptional accuracy is achieved with a fine-tuning process, surpassing the previous methods including the linearized tensor renormalization group [W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, Phys. Rev. Lett. 106, 127202 (2011)], the continuous matrix product operator [W. Tang, H.-H. Tu, and L. Wang, Phys. Rev. Lett. 125, 170604 (2020)], etc. High efficiency is demonstrated, where the time cost is nearly independent of the target temperature, including the extremely low temperatures. The proposed idea can be extended to higher-dimensional systems of bosons and fermions.

Variational Optimization of Continuous Matrix Product States.

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

Tensor network simulation of the ( 1+1 )-dimensional O(3) nonlinear σ -model with θ=π term

We perform a tensor network simulation of the ($1+1$)-dimensional $O(3)$ nonlinear $\ensuremath{\sigma}$-model with $\ensuremath{\theta}=\ensuremath{\pi}$ term. Within the Hamiltonian formulation, this field theory emerges as the finite-temperature partition function of a modified quantum rotor model decorated with magnetic monopoles. Using the monopole harmonics basis, we derive the matrix representation for this modified quantum rotor model, which enables tensor network simulations. We employ our recently developed continuous matrix product operator method [Tang et al., Phys. Rev. Lett. 125, 170604 (2020)] to study the finite-temperature properties of this model and reveal its massless nature. The central charge as a function of the coupling constant is directly extracted in our calculations and compared with field theory predictions.

Variational method in relativistic quantum field theory without cutoff

The variational method is a powerful approach to solve many-body quantum problems nonperturbatively. However, in the context of relativistic quantum field theory, it needs to meet three seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In practice, variational methods break one of the three, which translates into the need to have an IR or UV cutoff. In this letter, I introduce a relativistic modification of continuous matrix product states that satisfies the three requirements jointly in $1+1$ dimensions. I apply it to the self-interacting scalar field, without UV cutoff and directly in the thermodynamic limit. Numerical evidence suggests the error decreases faster than any power law in the number of parameters, while the cost remains only polynomial.

Relativistic continuous matrix product states for quantum fields without cutoff

I introduce a modification of continuous matrix product states (CMPS) that makes them adapted to relativistic quantum field theories (QFT). These relativistic CMPS can be used to solve genuine 1+1 dimensional QFT without UV cutoff and directly in the thermodynamic limit. The main idea is to work directly in the basis that diagonalizes the free part of the model considered, which allows to fit its short distance behavior exactly. This makes computations slightly less trivial than with standard CMPS. However, they remain feasible and I present all the steps needed for the optimization. The asymptotic cost as a function of the bond dimension remains the same as for standard CMPS. I illustrate the method on the self-interacting scalar field, a.k.a. the $\phi^4_2$ model. Aside from providing unequaled precision in the continuum, the numerical results obtained are truly variational, and thus provide rigorous energy upper bounds.

Continuous Matrix Product Operator Approach to Finite Temperature Quantum States.

We present an algorithm for studying quantum systems at finite temperature using continuous matrix product operator representation. The approach handles both short-range and long-range interactions in the thermodynamic limit without incurring any time discretization error. Moreover, the approach provides direct access to physical observables including the specific heat, local susceptibility, and local spectral functions. After verifying the method using the prototypical quantum XXZ chains, we apply it to quantum Ising models with power-law decaying interactions and on the infinite cylinder, respectively. The approach offers predictions that are relevant to experiments in quantum simulators and the nuclear magnetic resonance spin-lattice relaxation rate.

Generalized ansatz for continuous matrix product states

Recently it was shown that continuous Matrix Product States (cMPS) cannot express the continuum limit state of any Matrix Product State (MPS), according to a certain natural definition of the latter. The missing element is a projector in the transfer matrix of the MPS. Here we provide a generalised ansatz of cMPS that is capable of expressing the continuum limit of any MPS. It consists of a sum of cMPS with different boundary conditions, each attached to an ancilla state. This new ansatz can be interpreted as the concatenation of a state which is at the closure of the set of cMPS together with a standard cMPS. The former can be seen as a cMPS in the thermodynamic limit, or with matrices of unbounded norm. We provide several examples and discuss the result.

Continuous Tensor Network States for Quantum Fields

We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions $d\geq 2$. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the $N$-point functions of local observables. While we discuss mostly the continuous tensor network states extending Projected Entangled Pair States (PEPS), we propose a generalization bearing similarities with the continuum Multi-scale Entanglement Renormalization Ansatz (cMERA).

Tangent-space methods for uniform matrix product states

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

Continuum limits of matrix product states

We determine which translationally invariant matrix product states have a continuum limit, that is, which can be considered as discretized versions of states defined in the continuum. To do this, we analyze a fine-graining renormalization procedure in real space, characterize the set of limiting states of its flow, and find that it strictly contains the set of continuous matrix product states. We also analyze which states have a continuum limit after a finite number of coarse-graining renormalization steps. We give several examples of states with and without the different kinds of continuum limits.

Continuous matrix product states for nonrelativistic quantum fields: A lattice algorithm for inhomogeneous systems

By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\Psi\rangle$, directly in the continuum, of the ground state of non-relativistic quantum field theories. This construction works both for translation invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian $H$, we consider a sequence of discretized Hamiltonians $\{H(\epsilon_{\alpha})\}_{\alpha=1,2,\cdots,p}$ on increasingly finer lattices with lattice spacing $\epsilon_1 > \epsilon_2 > \cdots > \epsilon_p$. We first use energy minimization to optimize an MPS approximation $|\Psi(\epsilon_1)\rangle$ for the ground state of $H(\epsilon_1)$. Given the MPS $|\Psi(\epsilon_{\alpha})\rangle$ optimized for the ground state of $H(\epsilon_{\alpha})$, we use it to initialize the energy minimization for Hamiltonian $H(\epsilon_{\alpha+1})$, resulting in the optimized MPS $|\Psi(\epsilon_{\alpha+1})\rangle$. By iteration we produce an optimized MPS $|\Psi(\epsilon_{p})\rangle$ for the ground state of $H(\epsilon_p)$, from which we finally extract the cMPS approximation $|\Psi\rangle$ for the ground state of $H$. Two key ingredients of our proposal are: (i) a procedure to discretize $H$ into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\rangle$ and one boson $|1\rangle$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.

Particle emission from open quantum systems

In this work, we discuss connections between different theoretical physics communities and their works, all related to systems that act as sources of particles such as photons, phonons, or electrons. Our interest is to understand how a low-dimensional quantum system driven by coherent fields, e.g. a two-level system, Jaynes-Cummings system, or photon pair source driven by a laser pulse, emits photons into a waveguide. Of particular relevance to solid-state sources is that we provide a way to include dissipation into the formalism for temporal-mode quantum optics. We will discuss the connections between temporal-mode quantum optics, scattering matrices, quantum stochastic calculus, continuous matrix product states and operators, and very traditional quantum optical concepts such as the Mandel photon counting formula and the Lindblad form of the quantum-optical master equation. We close with an example of how our formalism relates to single-photon sources with dephasing.