90 research outputs found
Generic Entanglement Entropy for Quantum States with Symmetry
When a quantum pure state is drawn uniformly at random from a Hilbert space,
the state is typically highly entangled. This property of a random state is
known as generic entanglement of quantum states and has been long investigated
from many perspectives, ranging from the black hole science to quantum
information science. In this paper, we address the question of how symmetry of
quantum states changes the properties of generic entanglement. More
specifically, we study bipartite entanglement entropy of a quantum state that
is drawn uniformly at random from an invariant subspace of a given symmetry. We
first extend the well-known concentration formula to the one applicable to any
subspace and then show that 1. quantum states in the subspaces associated with
an axial symmetry are still highly entangled, though it is less than that of
the quantum states without symmetry, 2. quantum states associated with the
permutation symmetry are significantly less entangled, and 3. quantum states
with translation symmetry are as entangled as the generic one. We also
numerically investigate the phase-transition behavior of the distribution of
generic entanglement, which indicates that the phase transition seems to still
exist even when random states have symmetry.Comment: ver 1: 8 pages, 2 figures, ver 2: substantially updated, 19 pages,
and 2 figure
Thermal states of random quantum many-body systems
We study a distribution of thermal states given by random Hamiltonians with a
local structure. We show that the ensemble of thermal states monotonically
approaches the unitarily invariant ensemble with decreasing temperature if all
particles interact according to a single random interaction and achieves a
state -design at temperature . For the system where the random
interactions are local, we show that the ensemble achieves a state -design.
We then provide numerical evidence indicating that the ensemble undergoes a
phase transition at finite temperature.Comment: 5 pages, 4 figure
Generating a state -design by diagonal quantum circuits
We investigate protocols for generating a state -design by using a fixed
separable initial state and a diagonal-unitary -design in the computational
basis, which is a -design of an ensemble of diagonal unitary matrices with
random phases as their eigenvalues. We first show that a diagonal-unitary
-design generates a -approximate state -design, where is
the number of qubits. We then discuss a way of improving the degree of
approximation by exploiting non-diagonal gates after applying a
diagonal-unitary -design. We also show that it is necessary and sufficient
to use -qubit gates with random phases to generate a
diagonal-unitary -design by diagonal quantum circuits, and that each
multi-qubit diagonal gate can be replaced by a sequence of multi-qubit
controlled-phase-type gates with discrete-valued random phases. Finally, we
analyze the number of gates for implementing a diagonal-unitary -design by
{\it non-diagonal} two- and one-qubit gates. Our results provide a concrete
application of diagonal quantum circuits in quantum informational tasks.Comment: ver. 1: 15 pages, 1 figures. ver.2: 16 pages, 2 figures, major
changes, we corrected a mistake, which slightly changes a main conclusion,
added a new result, and improved a presentation. ver.3: 11 pages, 2 figures,
published versio
Decoupling with random diagonal unitaries
We investigate decoupling, one of the most important primitives in quantum
Shannon theory, by replacing the uniformly distributed random unitaries
commonly used to achieve the protocol, with repeated applications of random
unitaries diagonal in the Pauli- and - bases. This strategy was recently
shown to achieve an approximate unitary -design after a number of
repetitions of the process, which implies that the strategy gradually achieves
decoupling. Here, we prove that even fewer repetitions of the process achieve
decoupling at the same rate as that with the uniform ones, showing that rather
imprecise approximations of unitary -designs are sufficient for decoupling.
We also briefly discuss efficient implementations of them and implications of
our decoupling theorem to coherent state merging and relative thermalisation.Comment: 26 pages, 3 figures. v2: 19 pages, 3 figures, both results and
presentations are improved. One conjecture in the previous version was
proven. v3: 16 pages, 1 figure. v4: doi links are added, published versio
Hayden-Preskill Recovery in Hamiltonian Systems
The key to understanding complex quantum systems is information scrambling,
originally proposed in the Hayden-Preskill recovery. The Hayden-Preskill
recovery refers to the phenomena in which localized information is spread over
the entire system and becomes accessible from any small subsystem. While this
phenomena is well-understood in random unitary models, it has been hardly
explored in Hamiltonian systems. In this Letter, we investigate the information
recovery for various time-independent Hamiltonians, including chaotic spin
chains and Sachdev-Ye-Kitaev (SYK) models. We show that information recovery is
possible in certain, but not all, chaotic models, which highlightes that the
information recovery differs from other concepts, such as quantum chaos based
on energy statistics and the saturation of out-of-time-ordered correlators
(OTOCs) for local observables. We further demonstrate that information recovery
serves as a powerful tool to probe transitions that originates from the changes
of information-theoretic properties of the dynamics.Comment: 8 pages, 5 figures, Supplemental Materials (12 pages, 10 figures
Measurement-Based Quantum Computation on Symmetry Breaking Thermal States
We consider measurement-based quantum computation (MBQC) on thermal states of
the interacting cluster Hamiltonian containing interactions between the cluster
stabilizers that undergoes thermal phase transitions. We show that the
long-range order of the symmetry breaking thermal states below a critical
temperature drastically enhance the robustness of MBQC against thermal
excitations. Specifically, we show the enhancement in two-dimensional cases and
prove that MBQC is topologically protected below the critical temperature in
three-dimensional cases. The interacting cluster Hamiltonian allows us to
perform MBQC even at a temperature an order of magnitude higher than that of
the free cluster Hamiltonian.Comment: 8 pages, 7 figure
Simulating typical entanglement with many-body Hamiltonian dynamics
We study the time evolution of the amount of entanglement generated by one
dimensional spin-1/2 Ising-type Hamiltonians composed of many-body
interactions. We investigate sets of states randomly selected during the time
evolution generated by several types of time-independent Hamiltonians by
analyzing the distributions of the amount of entanglement of the sets. We
compare such entanglement distributions with that of typical entanglement,
entanglement of a set of states randomly selected from a Hilbert space with
respect to the unitarily invariant measure. We show that the entanglement
distribution obtained by a time-independent Hamiltonian can simulate the
average and standard deviation of the typical entanglement, if the Hamiltonian
contains suitable many-body interactions. We also show that the time required
to achieve such a distribution is polynomial in the system size for certain
types of Hamiltonians.Comment: Revised, 11 pages, 7 figure
Thermal robustness of multipartite entanglement of the 1-D spin 1/2 XY model
We study the robustness of multipartite entanglement of the ground state of
the one-dimensional spin 1/2 XY model with a transverse magnetic field in the
presence of thermal excitations, by investigating a threshold temperature,
below which the thermal state is guaranteed to be entangled. We obtain the
threshold temperature based on the geometric measure of entanglement of the
ground state. The threshold temperature reflects three characteristic lines in
the phase diagram of the correlation function. Our approach reveals a region
where multipartite entanglement at zero temperature is high but is thermally
fragile, and another region where multipartite entanglement at zero temperature
is low but is thermally robust.Comment: Revised, 11 pages, 7 figure
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