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Metrological resources

The metrological resources that QuanEstimation can calculate are spin squeezing and the minimum time to reach the given target. The spin squeezing can be calculated via the function: 

SpinSqueezing(rho, basis="Dicke", output="KU")

SpinSqueezing(rho; basis="Dicke", output="KU")

rho represents the density matrix of the state. In this function, the basis of the state can be Dicke basis or the original basis of each spin, which can be adjusted by setting basis="Dicke" or basis="Pauli". The variable output represents the type of spin squeezing calculation. output="KU" represents the spin squeezing defined by Kitagawa and Ueda [1] and output="WBIMH" calculates the spin squeezing defined by Wineland et al. [2].

Example 4.1
In this example, QuTip [3,4] is used to generate spin coherent state.

from quanestimation import *
import numpy as np
from qutip import spin_coherent

# generation of spin coherent state with QuTip
j = 2
theta = 0.5*np.pi
phi = 0.5*np.pi
rho_CSS = spin_coherent(j, theta, phi, type='dm').full()
xi = SpinSqueezing(rho_CSS, basis="Dicke", output="KU")

using QuanEstimation
using SparseArrays

# generation of the coherent spin state
j, theta, phi = 2, 0.5pi, 0.5pi
Jp = Matrix(spdiagm(1=>[sqrt(j*(j+1)-m*(m+1)) for m in j:-1:-j][2:end]))
Jm = Jp'
psi0 = exp(0.5*theta*exp(im*phi)*Jm - 0.5*theta*exp(-im*phi)*Jp)*
       QuanEstimation.basis(Int(2*j+1), 1)
rho = psi0*psi0'
xi = QuanEstimation.SpinSqueezing(rho; basis="Dicke", output="KU")

Calculation of the minimum time to reach a given precision limit with

TargetTime(f, tspan, func, *args, **kwargs)
where f is the given value of the objective function and tspan is the time length for the evolution. func represents the function for calculating the objective function, *args and **kwargs are the corresponding input parameters and the keyword arguments.

TargetTime(f, tspan, func, args...; kwargs...)
where f is the given value of the objective function and tspan is the time length for the evolution. func represents the function for calculating the objective function, args... and kwargs... are the corresponding input parameters and the keyword arguments.

Example 4.2
In this example, the free evolution Hamiltonian of a single qubit system is \(H_0=\frac{1}{2} \omega \sigma_3\) with \(\omega\) the frequency and \(\sigma_3\) a Pauli matrix. The dynamics of the system is governed by \begin{align} \partial_t\rho=-i[H_0, \rho], \end{align}

where \(\rho\) is the parameterized density matrix. The probe state is taken as \(|+\rangle\langle+|\) with \(|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\). Here \(|0\rangle\) \((|1\rangle)\) is the eigenstate of \(\sigma_3\) with respect to the eigenvalue \(1\) \((-1)\).

from quanestimation import *
import numpy as np

# initial state
rho0 = 0.5*np.array([[1., 1.], [1., 1.]])
# free Hamiltonian
omega = 1.0
sz = np.array([[1., 0.], [0., -1.]])
H0 = 0.5*omega*sz
# derivative of the free Hamiltonian on omega
dH = [0.5*sz]
# time length for the evolution
tspan = np.linspace(0., 50., 2000)
# dynamics
dynamics = Lindblad(tspan, rho0, H0, dH)
rho, drho = dynamics.expm()
# the value of the objective function
f = 20.0
t = TargetTime(f, tspan, QFIM, rho, drho)

using QuanEstimation

# initial state
rho0 = 0.5*ones(2, 2)
# free Hamiltonian
omega = 1.0
sx = [0. 1.; 1. 0.0im]
sy = [0. -im; im 0.]
sz = [1. 0.0im; 0. -1.]
H0 = 0.5*omega*sz
# derivative of the free Hamiltonian on omega
dH = [0.5*sz]
# time length for the evolution
tspan = range(0., 50., length=2000)
# dynamics
rho, drho = QuanEstimation.expm(tspan, rho0, H0, dH)
drho = [drho[i][1] for i in 1:2000]
# the value of the objective function
f = 20
t = QuanEstimation.TargetTime(f, tspan, QuanEstimation.QFIM, rho, drho)

Bibliography

[1] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).

[2] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Spin squeezing and reduced quantum noise in spectroscopy, Phys. Rev. A 46, R6797(R) (1992).

[3] J. R. Johansson, P. D. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comp. Phys. Comm. 183, 1760 (2012).

[4] J. R. Johansson, P. D. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comp. Phys. Comm. 184, 1234 (2013).