arXiv:quant-ph/0603115 · Physical Review A

Fast rate estimation of a unitary operation in SU(d)

Jonas Kahn

Published Feb 23, 2007• 48 citations

Abstract

We give an explicit procedure based on entangled input states for estimating a $\mathrm{SU}(d)$ operation $U$ with rate of convergence $1∕{N}^{2}$ when sending $N$ particles through the device. We prove that this rate is optimal. We also evaluate the constant $C$ such that the asymptotic risk is $C∕{N}^{2}$. However, other strategies might yield a better constant $C$.

DOI: 10.1103/physreva.75.022326

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arXiv:quant-ph/0603115 · Physical Review A

Fast rate estimation of a unitary operation in SU(d)

Jonas Kahn

Published Feb 23, 2007• 48 citations

arXiv HTML PDF

Abstract

We give an explicit procedure based on entangled input states for estimating a

\mathrm{SU}(d)

operation

U

with rate of convergence

1∕{N}^{2}

when sending

N

particles through the device. We prove that this rate is optimal. We also evaluate the constant

C

such that the asymptotic risk is

C∕{N}^{2}

. However, other strategies might yield a better constant

C

DOI: 10.1103/physreva.75.022326

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Manuel A. Ballester

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We discuss upper bounds on the rate at which unitary evolution governed by a nonlocal Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian $H$ coupling two finite dimensional particles $A$ and $B$ , the entangling rate is shown to be upper bounded by $c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}$, where $d$ is the smallest dimension of the interacting particles, $\ensuremath{\Vert}H\ensuremath{\Vert}$ is the operator norm of $H$ , and $c$ is a constant close to 1. Under certain restrictions on the initial state we prove an analogous upper bound for the ancilla-assisted entangling rate with a constant $c$ that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate---a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.We discuss upper bounds on the rate at which unitary evolution governed by a nonlocal Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian $H$ coupling two finite dimensional particles $A$ and $B$, the entangling rate is shown to be upper bounded by $c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}$, where $d$ is the smallest dimension of the interacting particles, $\ensuremath{\Vert}H\ensuremath{\Vert}$ is the operator norm of $H$, and $c$ is a constant close to 1. Under certain restrictions on the initial state we prove an analogous upper bound for the ancilla-assisted entangling rate with a constant $c$ that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate---a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.

arXiv:0704.0964Physical Review ANov 21, 200764 citations

We give an explicit procedure based on entangled input states for estimating a

\mathrm{SU}(d)

operation

U

with rate of convergence

1∕{N}^{2}

when sending

N

particles through the device. We prove that this rate is optimal. We also evaluate the constant

C

such that the asymptotic risk is

C∕{N}^{2}

. However, other strategies might yield a better constant

C

Asymptotically optimal unitary estimation in

\mathrm{SU}(3)

by the analysis of graph Laplacian

Unitary estimation is the task to estimate an unknown unitary operator

U\in\mathrm{SU}(d)

with

n

queries to the corresponding unitary operation, and its accuracy is evaluated by an estimation fidelity. We show that the optimal asymptotic fidelity of

3

-dimensional unitary estimation is given by

F_\mathrm{est}(n,d=3) = 1-\frac{56π^2}{9n^2} + O(n^{-3})

by the analysis of the graph Laplacian based on the finite element method. We also show the lower bound on the fidelity of

d

-dimensional unitary estimation for an arbitrary

d

given by

F_\mathrm{est}(n,d) \geq 1- \frac{(d+1)(d-1)(3d-2)(3d-1)}{6n^2} + O(n^{-3})

achieving the best known lower bound and tight scaling with respect to

n

and

d

. This lower bound is derived based on the unitary estimation protocol shown in [J. Kahn, Phys. Rev. A 75, 022326, 2007].

We discuss upper bounds on the rate at which unitary evolution governed by a nonlocal Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian

H

coupling two finite dimensional particles

A

and

B

, the entangling rate is shown to be upper bounded by $c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}$, where

d

is the smallest dimension of the interacting particles, $\ensuremath{\Vert}H\ensuremath{\Vert}$ is the operator norm of

H

, and

c

is a constant close to 1. Under certain restrictions on the initial state we prove an analogous upper bound for the ancilla-assisted entangling rate with a constant

c

that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate---a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.