Uhlmann fidelities from tensor networks

Markus Hauru and Guifre Vidal

Phys. Rev. A 98, 042316 – Published 12 October 2018

Abstract

Given two states $| ψ 〉$ and $| ϕ 〉$ of a quantum many-body system, one may use the overlap or fidelity $| 〈 ψ | ϕ 〉 |$ to quantify how similar they are. To further resolve the similarity of $| ψ 〉$ and $| ϕ 〉$ in space, one can consider their reduced density matrices $ρ$ and $σ$ on various regions of the system and compute the Uhlmann fidelity $F (ρ, σ) = Tr \sqrt{\sqrt{ρ} σ \sqrt{ρ}}$ . In this paper, we show how computing such subsystem fidelities can be done efficiently in many cases when the two states are represented as tensor networks. Formulated using Uhlmann's theorem, such subsystem fidelities appear as natural quantities to extract for certain subsystems of matrix product states and tree tensor networks, and evaluating them is algorithmically simple and computationally affordable. We demonstrate the usefulness of evaluating subsystem fidelities with three example applications: studying local quenches, comparing critical and noncritical states, and quantifying convergence in tensor network simulations.

Received 5 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042316

Physics Subject Headings (PhySH)

Quantum quench

Density matrix renormalization group Lattice models in condensed matter Matrix product states Tensor network methods

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Markus Hauru^* and Guifre Vidal

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5 and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

^*markus@mhauru.org

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Above, the expectation value of the Pauli $Z$ operator, on the locally quenched state $| ψ (t) 〉$ , as a function of position $x$ and time $t$ . The model in question is the critical 1D Ising model, and the quench consisted of perturbing the ground state with $Z$ . This plot serves to merely illustrate the progression of the quench. Below, subsystem fidelities between the ground state $| E_{0} 〉$ and the quenched state $| ψ (t = 10) 〉$ as functions of position $x$ . The three different fidelities plotted are the half-system Uhlmann fidelities to the left (descending orange dots) and to the right (ascending green dots) of $x$ , and the two-site fidelity at $x$ (blue dots).
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Figure 2
Subsystem fidelities between the ground state $| E_{0} 〉$ of the infinite, critical Ising model and the locally quenched state $e^{i t H_{Ising}} Z_{0} | E_{0} 〉$ , as functions of position $x$ , at various times $t$ after the quench. The three different fidelities plotted are the half-system Uhlmann fidelities to the left (descending orange dots) and to the right (ascending green dots) of $x$ , and the two-site fidelity at $x$ (blue dots). A bond dimension of 50 MPS was used in the time evolution.
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Figure 3
Window fidelities between the ground states of the infinite, critical Ising model at the critical point $h = 1.0$ and slightly in the disordered phase $h = 1.01$ as a function of the size of the window. The monotonously decreasing blue lines are the fidelities, dotted line for the Uhlmann fidelity $F$ and solid line for the disjoint fidelity $F_{d}$ , with their axis on the left. The green lines are the discrete derivatives of the blue lines, with their axis on the right. The vertical gray line marks the correlation length of the $h = 1.01$ ground state. Bond dimensions of 50 MPSs were used to generate these results.
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Figure 4
Window fidelities of pairs of MPSs, representing the same ground state but with different bond dimensions of 10 and 20, as functions of the window size $| M |$ . In the optimal case these fidelities would be exactly 1, so the vertical axis is the difference $1 - fidelity$ , on a logarithmic scale. Dotted lines mark the Uhlmann fidelities $F$ , solid lines are disjoint fidelities $F_{d}$ . In both cases the model is the infinite Ising model, with the blue lines at the top being for ground states of the $h = 1.0$ critical Hamiltonian and the green ones at the bottom for $h = 1.05$ .
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Figure 5
Per-site window fidelities $F^{\frac{1}{| M |}}$ of two TTN states $| ψ_{m} 〉$ and $| ψ_{m - 10} 〉$ as a function of the number of iterations $m$ in an optimization algorithm. Different lines correspond to different window sizes $| M |$ , which furthermore relate to different layers of the TTN. The TTN used here consists of eight layers and is enforced to be translation and reflection invariant, meaning that the position of the window does not matter. The optimization is for the ground state of the critical Ising model, and the bond dimension of the TTN is 30.
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