toqito.state_opt.state_exclusion ================================ .. py:module:: toqito.state_opt.state_exclusion .. autoapi-nested-parse:: Calculates the probability of error of single state conclusive state exclusion. Module Contents --------------- .. py:function:: state_exclusion(vectors, probs = None, strategy = 'min_error', solver = 'cvxopt', primal_dual = 'dual', **kwargs) Compute probability of error of single state conclusive 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 at least one out of the "n" possible quantum states by applying a measurement. 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 exclusion. \[ \begin{equation} \begin{aligned} \text{minimize:} \quad & \sum_{i=1}^n p_i \langle M_i, \rho_i \rangle \\ \text{subject to:} \quad & \sum_{i=1}^n M_i = \mathbb{I}_{\mathcal{X}}, \\ & M_0, \ldots, M_n \in \text{Pos}(\mathcal{X}). \end{aligned} \end{equation} \] \[ \begin{equation} \begin{aligned} \text{maximize:} \quad & \text{Tr}(Y) \\ \text{subject to:} \quad & Y \preceq p_1\rho_1, \\ & Y \preceq p_2\rho_2, \\ & \vdots \\ & Y \preceq p_n\rho_n, \\ & Y \in\text{Herm}(\mathcal{X}). \end{aligned} \end{equation} \] For `strategy = "unambiguous"`, Bob never provides an incorrect answer, although it is possible that his answer is inconclusive. This function then yields the probability of an inconclusive outcome. In that case, this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct unambiguous quantum state distinguishability. \[ \begin{align*} \text{minimize:} \quad & \text{Tr}\left( \left(\sum_{i=1}^n p_i\rho_i\right)\left(\mathbb{I}-\sum_{i=1}^nM_i\right) \right) \\ \text{subject to:} \quad & \sum_{i=1}^nM_i \preceq \mathbb{I},\\ & M_1, \ldots, M_n \succeq 0, \\ & \langle M_1, \rho_1 \rangle, \ldots, \langle M_n, \rho_n \rangle =0 \end{align*} \] \[ \begin{align*} \text{maximize:} \quad & 1 - \text{Tr}(N) \\ \text{subject to:} \quad & a_1p_1\rho_1, \ldots, a_np_n\rho_n \succeq \sum_{i=1}^np_i\rho_i - N,\\ & N \succeq 0,\\ & a_1, \ldots, a_n \in\mathbb{R} \end{align*} \] !!! Note This function supports both pure states (vectors) and mixed states (density matrices). It is known that it is always possible to perfectly exclude pure states that are linearly dependent. Thus, calling this function on a set of states with this property will return 0. The conclusive state exclusion SDP is written explicitly in [@Bandyopadhyay_2014_Conclusive]. The problem of conclusive state exclusion was also thought about under a different guise in [@Pusey_2012_On]. .. rubric:: Examples Consider the following two Bell states \[ \begin{equation} \begin{aligned} u_0 &= \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), \\ u_1 &= \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right). \end{aligned} \end{equation} \] 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. ```python exec="1" source="above" import numpy as np from toqito.states import bell from toqito.state_opt import state_exclusion vectors = [bell(0), bell(1)] probs = [1/2, 1/2] print(np.around(state_exclusion(vectors, probs)[0], decimals=2)) ``` Unambiguous state exclusion for unbiased pure states. ```python exec="1" source="above" import numpy as np from toqito.state_opt import state_exclusion states = [np.array([[1.], [0.]]), np.array([[1.],[1.]]) / np.sqrt(2)] res, _ = state_exclusion(states, primal_dual="primal", strategy="unambiguous", abs_ipm_opt_tol=1e-7) print(np.around(res, decimals=2)) ``` State exclusion for mixed states. ```python exec="1" source="above" import numpy as np from toqito.state_opt import state_exclusion # Two mixed states rho1 = 0.7 * np.array([[1., 0.], [0., 0.]]) + 0.3 * np.eye(2) / 2 rho2 = 0.7 * np.array([[0., 0.], [0., 1.]]) + 0.3 * np.eye(2) / 2 states = [rho1, rho2] res, _ = state_exclusion(states, primal_dual="dual") print(np.around(res, decimals=2)) ``` !!! Note If you encounter a `ZeroDivisionError` or an `ArithmeticError` when using cvxopt as a solver (which is the default), you might want to set the `abs_ipm_opt_tol` option to a lower value (the default being `1e-8`) or to set the `cvxopt_kktsolver` option to `ldl`. See https://gitlab.com/picos-api/picos/-/issues/341 :param vectors: A list of states provided as vectors (for pure states) or density matrices (for mixed states). :param probs: Respective list of probabilities each state is selected. If no probabilities are provided, a uniform :param probability distribution is assumed.: :param strategy: Whether to perform minimal error or unambiguous discrimination task. Possible values are "min_error" :param and "unambiguous". Both strategies support pure and mixed states.: :param solver: Optimization option for `picos` solver. Default option is `solver_option="cvxopt"`. :param primal_dual: Option for the optimization problem. :param 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.