Fast tensor disentangling algorithm

Many recent tensor network algorithms apply unitary operators to parts of a tensor network in order to reduce entanglement. However, many of the previously used iterative algorithms to minimize entanglement can be slow. We introduce an approximate, fast, and simple algorithm to optimize disentangling unitary tensors. Our algorithm is asymptotically faster than previous iterative algorithms and often results in a residual entanglement entropy that is within 10 to 40% of the minimum. For certain input tensors, our algorithm returns an optimal solution. When disentangling order-4 tensors with equal bond dimensions, our algorithm achieves an entanglement spectrum where nearly half of the singular values are zero. We further validate our algorithm by showing that it can efficiently disentangle random 1D states of qubits.

Entanglement, non-hermiticity, and duality

Usually duality process keeps energy spectrum invariant. In this paper, we provide a duality, which keeps entanglement spectrum invariant, in order to diagnose quantum entanglement of non-Hermitian non-interacting fermionic systems. We limit our attention to non-Hermitian systems with a complete set of biorthonormal eigenvectors and an entirely real energy spectrum. The original system has a reduced density matrix \rho_\mathrm{o}ρo and the real space is partitioned via a projecting operator \mathcal{R}_{\mathrm o}ℛo. After dualization, we obtain a new reduced density matrix \rho_{\mathrm{d}}ρd and a new real space projector \mathcal{R}_{\mathrm d}ℛd. Remarkably, entanglement spectrum and entanglement entropy keep invariant. Inspired by the duality, we defined two types of non-Hermitian models, upon \mathcal R_{\mathrm{o}}ℛo is given. In type-I exemplified by the "non-reciprocal model'', there exists at least one duality such that \rho_{\mathrm{d}}ρd is Hermitian. In other words, entanglement information of type-I non-Hermitian models with a given \mathcal{R}_{\mathrm{o}}ℛo is entirely controlled by Hermitian models with \mathcal{R}_{\mathrm{d}}ℛd. As a result, we are allowed to apply known results of Hermitian systems to efficiently obtain entanglement properties of type-I models. On the other hand, the duals of type-II models, exemplified by "non-Hermitian Su-Schrieffer-Heeger model’’, are always non-Hermitian. For the practical purpose, the duality provides a potentially computation route to entanglement of non-Hermitian systems. Via connecting different models, the duality also sheds lights on either trivial or nontrivial role of non-Hermiticity played in quantum entanglement, paving the way to potentially systematic classification and characterization of non-Hermitian systems from the entanglement perspective.

Entanglement spectrum of geometric states

The reduced density matrix of a given subsystem, denoted by ρA, contains the information on subregion duality in a holographic theory. We may extract the information by using the spectrum (eigenvalue) of the matrix, called entanglement spectrum in this paper. We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state ρA,m associated with an eigenvalue λ for some examples, including a single interval and two intervals in vacuum state of 2D CFTs. We find there exists a microcanonical ensemble state with λ0 which can be seen as an approximate state of ρA. The parameter λ0 is obtained in the two examples. For a general geometric state, the approximate microcanonical ensemble state also exists. The parameter λ0 is associated with the entanglement entropy of A and Rényi entropy in the limit n → ∞. As an application of the above conclusion we reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs as conditions of Holevo information. We show the constraints on the eigenstates. Finally, we point out some unsolved problems and their significance on understanding the geometric states.

Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics

Clifford circuits are insufficient for universal quantum computation or creating tt-designs with t\ge 4t≥4. While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum of a time evolved random product state is: the entanglement levels are Poisson-distributed for circuits restricted to the Clifford gate-set, while the levels follow Wigner-Dyson statistics when universal gates are used. In this paper we show, using finite-size scaling analysis of different measures of level spacing statistics, that in the thermodynamic limit, inserting a single T (\pi/8)(π/8) gate in the middle of a random Clifford circuit is sufficient to alter the entanglement spectrum from a Poisson to a Wigner-Dyson distribution.