States¶
A quantum state is a density operator
\[\rho \in \text{D}(\mathcal{X})\]
where \(\mathcal{X}\) is a complex Euclidean space and where \(\text{D}(\cdot)\) represents the set of density matrices.
Distance Metrics for Quantum States¶
toqito.state_metrics.bures_angle(rho_1, …) |
Compute the Bures angle of two density matrices [WikBures]. |
toqito.state_metrics.bures_distance(rho_1, …) |
Compute the Bures distance of two density matrices [WikBures]. |
toqito.state_metrics.fidelity_of_separability(…) |
Define the first benchmark introduced in Appendix H of [Phil23]. |
toqito.state_metrics.fidelity(rho, sigma) |
Compute the fidelity of two density matrices [WikFid]. |
toqito.state_metrics.helstrom_holevo(rho, sigma) |
Compute the Helstrom-Holevo distance between density matrices [WikHeHo]. |
toqito.state_metrics.hilbert_schmidt_inner_product(…) |
Compute the Hilbert-Schmidt inner product between two matrices [WikHSO]. |
toqito.state_metrics.hilbert_schmidt(rho, sigma) |
Compute the Hilbert-Schmidt distance between two states [WikHS]. |
toqito.state_metrics.matsumoto_fidelity(rho, …) |
Compute the Matsumoto fidelity of two density matrices [Mat10]. |
toqito.state_metrics.sub_fidelity(rho, sigma) |
Compute the sub fidelity of two density matrices [MPHUZSub08]. |
toqito.state_metrics.trace_distance(rho, sigma) |
Compute the trace distance between density operators rho and sigma. |
Optimizations over Quantum States¶
toqito.state_opt.optimal_clone(states, …) |
Compute probability of counterfeiting quantum money [MVW12]. |
toqito.state_opt.ppt_distinguishability(…) |
Compute probability of optimally distinguishing a state via PPT measurements [COS13]. |
toqito.state_opt.state_distinguishability(…) |
Compute probability of state distinguishability [ELD03]. |
toqito.state_opt.state_exclusion(vectors, …) |
Compute probability of single state exclusion. |
toqito.state_opt.state_helper |
Helper functions for checking validity of states and probability vectors. |
toqito.state_opt.symmetric_extension_hierarchy(…) |
Compute optimal value of the symmetric extension hierarchy SDP [Nav08]. |
Operations on Quantum States¶
toqito.state_ops.pure_to_mixed(phi) |
Convert a state vector or density matrix to a density matrix. |
toqito.state_ops.schmidt_decomposition(rho, …) |
Compute the Schmidt decomposition of a bipartite vector [WikSD]. |
Properties of Quantum States¶
toqito.state_props.concurrence(rho) |
Calculate the concurrence of a bipartite state [WikCon]. |
toqito.state_props.entanglement_of_formation(…) |
Compute entanglement-of-formation of a bipartite quantum state [WikEOF]. |
toqito.state_props.has_symmetric_extension(…) |
Determine whether there exists a symmetric extension for a given quantum state. |
toqito.state_props.in_separable_ball(mat) |
Check whether an operator is contained in ball of separability [GB02]. |
toqito.state_props.is_ensemble(states) |
Determine if a set of states constitute an ensemble [WatEns18]. |
toqito.state_props.is_mixed(state) |
Determine if a given quantum state is mixed [WikMix]. |
toqito.state_props.is_mutually_orthogonal(…) |
Check if list of vectors are mutually orthogonal [WikOrthog]. |
toqito.state_props.is_mutually_unbiased_basis(…) |
Check if list of vectors constitute a mutually unbiased basis [WikMUB]. |
toqito.state_props.is_npt(mat, sys, dim, tol) |
Determine whether or not a matrix has negative partial transpose [WikPPT]. |
toqito.state_props.is_ppt(mat, sys, dim, tol) |
Determine whether or not a matrix has positive partial transpose [WikPPT]. |
toqito.state_props.is_product(rho, dim) |
Determine if a given vector is a product state [WikProdState]. |
toqito.state_props.is_pure(state) |
Determine if a given state is pure or list of states are pure [WikIsPure]. |
toqito.state_props.is_separable(state, dim, …) |
Determine if a given state (given as a density matrix) is a separable state [WikSepState]. |
toqito.state_props.l1_norm_coherence(rho) |
Compute the l1-norm of coherence of a quantum state [RPWL17]. |
toqito.state_props.log_negativity(rho, dim) |
Compute the log-negativity of a bipartite quantum state [WikLogNeg]. |
toqito.state_props.negativity(rho, dim) |
Compute the negativity of a bipartite quantum state [WikNeg]. |
toqito.state_props.purity(rho) |
Compute the purity of a quantum state [WikPurity]. |
toqito.state_props.schmidt_rank(rho, dim) |
Compute the Schmidt rank [WikSR]. |
toqito.state_props.sk_vector_norm(rho, k, dim) |
Compute the S(k)-norm of a vector [NJDK09]. |
toqito.state_props.von_neumann_entropy(rho) |
Compute the von Neumann entropy of a density matrix [WikVent]. |
Quantum States¶
toqito.states.basis(dim, pos) |
Obtain the ket of dimension dim [WikKet]. |
toqito.states.bell(idx) |
Produce a Bell state [WikBell]. |
toqito.states.brauer(dim, p_val) |
Produce all Brauer states [WikBrauer]. |
toqito.states.breuer(dim, lam) |
Produce a Breuer state [HPBreuer]. |
toqito.states.chessboard(mat_params, …) |
Produce a chessboard state [BP00]. |
toqito.states.domino(idx) |
Produce a domino state [CBDOM99], [UPB99]. |
toqito.states.gen_bell(k_1, k_2, dim) |
Produce a generalized Bell state [DL09]. |
toqito.states.ghz(dim, num_qubits, coeff) |
Generate a (generalized) GHZ state [GHZ07]. |
toqito.states.gisin(lambda_var, theta) |
Produce a Gisin state [GIS96]. |
toqito.states.horodecki(a_param, dim) |
Produce a Horodecki state [HOR], [CHR]. |
toqito.states.isotropic(dim, alpha) |
Produce a isotropic state [HH99]. |
toqito.states.max_entangled(dim, is_sparse, …) |
Produce a maximally entangled bipartite pure state [WikEnt]. |
toqito.states.max_mixed(dim, is_sparse) |
Produce the maximally mixed state [AAR6]. |
toqito.states.singlet(dim) |
Produce a generalized singlet state acting on two n-dimensional systems [Gsinglet]. |
toqito.states.tile(idx) |
Produce a Tile state [UPBTile99]. |
toqito.states.w_state(num_qubits, coeff) |
Produce a W-state [DVC00]. |
toqito.states.werner(dim, alpha) |
Produce a Werner state [Wer89]. |