state_opt.state_distinguishability

State distinguishability.

Module Contents

Functions

state_distinguishability(vectors[, probs, solver, ...])

Compute probability of state distinguishability [1].

_min_error_primal(vectors, dim[, probs, solver])

Find the primal problem for minimum-error quantum state distinguishability SDP.

_min_error_dual(vectors, dim[, probs, solver])

Find the dual problem for minimum-error quantum state distinguishability SDP.
state_opt.state_distinguishability.state_distinguishability(vectors, probs=None, solver='cvxopt', primal_dual='dual')

Compute probability of state distinguishability [1].

The “quantum state distinguishability” problem involves a collection of \(n\) quantum states

\[\rho = \{ \rho_0, \ldots, \rho_n \},\]

as well as a list of corresponding probabilities

\[p = \{ p_0, \ldots, p_n \}.\]

Alice chooses \(i\) with probability \(p_i\) and creates the state \(\rho_i\). Bob wants to guess which state he was given from the collection of states.

This function implements the following semidefinite program that provides the optimal probability with which Bob can conduct quantum state distinguishability.

\[\begin{split}\begin{align*} \text{maximize:} \quad & \sum_{i=0}^n p_i \langle M_i, \rho_i \rangle \\ \text{subject to:} \quad & M_0 + \ldots + M_n = \mathbb{I},\\ & M_0, \ldots, M_n \geq 0. \end{align*}\end{split}\]

Examples

State distinguishability for the Bell states (which are perfectly distinguishable).

>>> from toqito.states import bell
>>> from toqito.state_opt import state_distinguishability
>>> states = [bell(0), bell(1), bell(2), bell(3)]
>>> probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
>>> res, _ = state_distinguishability(vectors=states, probs=probs, primal_dual="dual")
>>> '%.2f' % res
'1.00'

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

Note that if we are just interested in obtaining the optimal value, it is computationally less intensive to compute the dual problem over the primal problem. However, the primal problem does allow us to extract the explicit measurement operators which may be of interest to us.

>>> import numpy as np
>>> from toqito.states import bell
>>> from toqito.state_opt import state_distinguishability
>>> states = [bell(0), bell(1), bell(2), bell(3)]
>>> probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]
>>> res, measurements = state_distinguishability(vectors=states, probs=probs, primal_dual="primal")
>>> np.around(measurements[0], decimals=5)  
array([[ 0.5+0.j,  0. +0.j, -0. -0.j,  0.5-0.j],
       [ 0. -0.j,  0. +0.j, -0. +0.j,  0. -0.j],
       [-0. +0.j, -0. -0.j,  0. +0.j, -0. +0.j],
       [ 0.5+0.j,  0. +0.j, -0. -0.j,  0.5+0.j]])

References

[1] (1,2)

Y.C. Eldar. A semidefinite programming approach to optimal unambiguous discrimination of quantum states. IEEE Transactions on Information Theory, 49(2):446–456, Feb 2003. URL: http://dx.doi.org/10.1109/TIT.2002.807291, doi:10.1109/tit.2002.807291.

Parameters:

  • vectors (list[numpy.ndarray]) – A list of states provided as vectors.

  • probs (list[float]) – Respective list of probabilities each state is selected. If no probabilities are provided, a uniform probability distribution is assumed.

  • solver (str) – Optimization option for picos solver. Default option is solver_option=”cvxopt”.

  • primal_dual (str) – Option for the optimization problem. Default option is “dual”.

Returns:

The optimal probability with which Bob can guess the state he was not given from states along with the optimal set of measurements.

Return type:

float

state_opt.state_distinguishability._min_error_primal(vectors, dim, probs=None, solver='cvxopt')

Find the primal problem for minimum-error quantum state distinguishability SDP.

Parameters:

  • vectors (list[numpy.ndarray])

  • dim (int)

  • probs (list[float])

  • solver (str)

Return type:

tuple[float, list[picos.HermitianVariable]]

state_opt.state_distinguishability._min_error_dual(vectors, dim, probs=None, solver='cvxopt')

Find the dual problem for minimum-error quantum state distinguishability SDP.

Parameters:

  • vectors (list[numpy.ndarray])

  • dim (int)

  • probs (list[float])

  • solver (str)

Return type:

tuple[float, list[picos.HermitianVariable]]