state_opt.state_distinguishability

Calculates the probability of optimally distinguishing quantum states.

Functions

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

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.

_unambiguous_primal(vectors[, probs, solver])

Solve the primal problem for unambiguous quantum state distinguishability SDP.

_unambiguous_dual(vectors[, probs, solver])

Solve the dual problem for unambiguous quantum state distinguishability SDP.

Module Contents

state_opt.state_distinguishability.state_distinguishability(vectors, probs=None, strategy='min_error', solver='cvxopt', primal_dual='dual', **kwargs)

Compute probability of state distinguishability [1].

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

\[\rho = \{ \rho_1, \ldots, \rho_n \},\]

as well as a list of corresponding probabilities

\[p = \{ p_1, \ldots, p_n \}.\]

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.

For strategy = "min_error", this is the default method that yields the minimal probability of error for Bob.

In that case, this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct quantum state distinguishability.

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

For strategy = "unambiguous", Bob never provides an incorrect answer, although it is possible that his answer is inconclusive.

In that case, this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct unambiguous quantum state distinguishability.

\[\begin{split}\begin{align*} \text{maximize:} \quad & \mathbf{p} \cdot \mathbf{q} \\ \text{subject to:} \quad & \Gamma - Q \geq 0,\\ & \mathbf{q} \geq 0 \end{align*}\end{split}\]
\[\begin{split}\begin{align*} \text{minimize:} \quad & \text{Tr}(\Gamma Z) \\ \text{subject to:} \quad & z_i + p_i + \text{Tr}\left(F_iZ\right)=0,\\ & Z, z \geq 0 \end{align*}\end{split}\]

where \(\mathbf{p}\) is the vector whose \(i\)-th coordinate contains the probability that the state is prepared in state \(\left|\psi_i\right\rangle\), \(\Gamma\) is the Gram matrix of \(\left|\psi_1\right\rangle,\cdots,\left|\psi_n\right\rangle\) and \(F_i\) is \(-|i\rangle\langle i|\).

Warning

Note that it only makes sense to distinguish unambiguously when the pure states are linearly independent. Calling this function on a set of states that doesn’t verify this property will return 0.

Examples

Minimal-error state distinguishability for the Bell states (which are perfectly distinguishable).

>>> 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, _ = state_distinguishability(vectors=states, probs=probs, primal_dual="dual")
>>> np.around(res, decimals=2)
np.float64(1.0)

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]])

Unambiguous state distinguishability for unbiased states.

>>> from toqito.state_opt import state_distinguishability
>>> import numpy as np
>>> states = [np.array([[1.], [0.]]), np.array([[1.],[1.]]) / np.sqrt(2)]
>>> probs = [1 / 2, 1 / 2]
>>> res, _ = state_distinguishability(vectors=states, probs=probs, primal_dual="primal", strategy="unambiguous")
>>> np.around(res, decimals=2)
np.float64(0.29)

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.

[2] (1,2)

Tathagata Gupta, Shayeef Murshid, and Somshubhro Bandyopadhyay. Unambiguous discrimination of sequences of quantum states. Physical Review A, May 2024. URL: http://dx.doi.org/10.1103/PhysRevA.109.052222, doi:10.1103/physreva.109.052222.

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.

  • strategy (str) – Whether to perform unambiguous or minimal error discrimination task. Possible values are “min_error” and “unambiguous”. Default option is strategy=”min_error”.

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

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

  • kwargs – Additional arguments to pass to picos’ solve method.

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:

tuple[float, list[picos.HermitianVariable] | tuple[picos.HermitianVariable] | tuple[picos.RealVariable]]

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

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', **kwargs)

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]]

state_opt.state_distinguishability._unambiguous_primal(vectors, probs=None, solver='cvxopt', **kwargs)

Solve the primal problem for unambiguous quantum state distinguishability SDP.

Implemented according to Equation (5) of [2]:.

Parameters:

  • vectors (list[numpy.ndarray])

  • probs (list[float])

  • solver (str)

Return type:

tuple[float, tuple[picos.RealVariable]]

state_opt.state_distinguishability._unambiguous_dual(vectors, probs=None, solver='cvxopt', **kwargs)

Solve the dual problem for unambiguous quantum state distinguishability SDP.

Implemented according to Equation (5) of [2].

Parameters:

  • vectors (list[numpy.ndarray])

  • probs (list[float])

  • solver (str)

Return type:

tuple[float, tuple[picos.HermitianVariable]]