toqito.state_opt.state_distinguishability

toqito.state_opt.state_distinguishability(states, probs=None, dist_method='min-error')

Compute probability of state distinguishability [ELD03].

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.

One can specify the distinguishability method using the dist_method argument.

For dist_method = "min-error", this is the default method that yields the probability of distinguishing quantum states that minimize the probability of error.

For dist_method = "unambiguous", Alice and Bob never provide an incorrect answer, although it is possible that their answer is inconclusive.

When dist_method = "min-error", this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct quantum state distinguishability.

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

When dist_method = "unambiguous", this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct unambiguous quantum state distinguishability.

\[\begin{split}\begin{align*} \text{maximize:} \quad & \sum_{i=0}^n p_i \langle M_i, \rho_i \rangle \\ \text{subject to:} \quad & M_0 + \ldots + M_{n+1} = \mathbb{I},\\ & \langle M_i, \rho_j \rangle = 0, \quad 1 \leq i, j \leq n, \quad i \not= j, \\ & M_0, \ldots, M_n \geq 0. \end{align*}\end{split}\]

Examples

State distinguishability for two state density matrices. In this example, the states \(|0\rangle\) and \(|1\rangle\) are orthogonal and therefore perfectly distinguishable.

>>> from toqito.states import basis
>>> from toqito.state_opt import state_distinguishability
>>> e_0, e_1 = basis(2, 0), basis(2, 1)
>>> e_00 = e_0 * e_0.conj().T
>>> e_11 = e_1 * e_1.conj().T
>>> states = [e_00, e_11]
>>> probs = [1 / 2, 1 / 2]
>>> res = state_distinguishability(states, probs)
1.000

References

[ELD03]

Eldar, Yonina C. “A semidefinite programming approach to optimal unambiguous discrimination of quantum states.” IEEE Transactions on information theory 49.2 (2003): 446-456. https://arxiv.org/abs/quant-ph/0206093

Parameters:

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

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

• dist_method – Method of distinguishing to use.

Returns:

The optimal probability with which Bob can distinguish the state.