toqito.state_opt.state_exclusion

toqito.state_opt.state_exclusion(states, probs=None, method='conclusive')

Compute probability of single state exclusion.

The quantum state exclusion 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 not given from the collection of states. State exclusion implies that ability to discard (with certainty) at least one out of the “n” possible quantum states by applying a measurement.

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

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

The conclusive state exclusion SDP is written explicitly in [BJOP14]. The problem of conclusive state exclusion was also thought about under a different guise in [PBR12].

For the unambiguous case, the following semidefinite program that provides the optimal probability with which Bob can conduct quantum state exclusion.

\[\begin{split}\begin{align*} \text{maximize:} \quad & \sum_{i=0}^n \sum_{j=0}^n \langle M_i, \rho_j \rangle \\ \text{subject to:} \quad & \sum_{i=0}^n M_i \leq \mathbb{I},\\ & \text{Tr}(\rho_i M_i) = 0, \quad \quad \forall 1 \leq i \leq n, \\ & M_0, \ldots, M_n \geq 0 \end{align*}\end{split}\]

Examples

Consider the following two Bell states

\[\begin{split}\begin{equation} \begin{aligned} u_0 &= \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), \\ u_1 &= \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right). \end{aligned} \end{equation}\end{split}\]

For the corresponding density matrices \(\rho_0 = u_0 u_0^*\) and \(\rho_1 = u_1 u_1^*\), we may construct a set

\[\rho = \{\rho_0, \rho_1 \}\]

such that

\[p = \{1/2, 1/2\}.\]

It is not possible to conclusively exclude either of the two states. We can see that the result of the function in toqito yields a value of \(0\) as the probability for this to occur.

>>> from toqito.state_opt import state_exclusion
>>> from toqito.states import bell
>>> import numpy as np
>>> rho1 = bell(0) * bell(0).conj().T
>>> rho2 = bell(1) * bell(1).conj().T
>>>
>>> states = [rho1, rho2]
>>> probs = [1/2, 1/2]
>>>
>>> state_exclusion(states, probs, "conclusive")
1.6824720366950206e-09

Consider the following two Bell states

\[\begin{split}\begin{equation} \begin{aligned} u_0 &= \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \\ u_1 &= \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right). \end{aligned} \end{equation}\end{split}\]

For the corresponding density matrices \(\rho_0 = u_0 u_0^*\) and \(\rho_1 = u_1 u_1^*\), we may construct a set

\[\rho = \{\rho_0, \rho_1 \}\]

such that

\[p = \{1/2, 1/2\}.\]

It is not possible to unambiguously exclude either of the two states. We can see that the result of the function in toqito yields a value of \(0\) as the probability for this to occur.

>>> from toqito.state_opt import state_exclusion
>>> from toqito.states import bell
>>> import numpy as np
>>> rho1 = bell(0) * bell(0).conj().T
>>> rho2 = bell(1) * bell(1).conj().T
>>>
>>> states = [rho1, rho2]
>>> probs = [1/2, 1/2]
>>>
>>> state_exclusion(states, probs, "unambiguous")
-7.250173600116328e-18

References

[PBR12]

“On the reality of the quantum state” Pusey, Matthew F., Barrett, Jonathan, and Rudolph, Terry. Nature Physics 8.6 (2012): 475-478. arXiv:1111.3328

[BJOP14]

“Conclusive exclusion of quantum states” Bandyopadhyay, Somshubhro, Jain, Rahul, Oppenheim, Jonathan, Perry, Christopher Physical Review A 89.2 (2014): 022336. arXiv:1306.4683

Parameters:

• states – A list of states provided as either matrices or vectors.

• probs – Respective list of probabilities each state is selected.

• method – Exclusion method (either conclusive or unambiguous.

Returns:

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