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.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`(rho, sigma)	Compute the Hilbert-Schmidt distance between two states [WikHS].
`toqito.state_metrics.hilbert_schmidt_inner_product`(...)	Compute the Hilbert-Schmidt inner product between two matrices [WikHSO].
`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.
`toqito.state_metrics.trace_norm`(rho)	Compute the trace norm of the state [WikTn].
`toqito.state_metrics.bures_distance`(rho_1, rho_2)	Compute the Bures distance of two density matrices [WikBures].
`toqito.state_metrics.matsumoto_fidelity`(rho, ...)	Compute the Matsumoto fidelity of two density matrices [Mat10].

Optimizations over Quantum States

`toqito.state_opt.optimal_clone`(states, probs)	Compute probability of counterfeiting quantum money [MVW12].
`toqito.state_opt.ppt_distinguishability`(states)	Compute probability of optimally distinguishing a state via PPT measurements [COS13].
`toqito.state_opt.state_distinguishability`(states)	Compute probability of state distinguishability [ELD03].
`toqito.state_opt.state_exclusion`(states[, ...])	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`(states)	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`(rho)	Compute entanglement-of-formation of a bipartite quantum state [WikEOF].
`toqito.state_props.has_symmetric_extension`(rho)	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_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.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.is_separable`	Check if state is separable.
`toqito.state_props.von_neumann_entropy`(rho)	Compute the von Neumann entropy of a density matrix [WikVent].
`toqito.state_props.sk_vector_norm`(rho[, k, dim])	Compute the S(k)-norm of a vector [NJDK09].

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[, ...])	Produce a maximally entangled bipartite pure state [WikEnt].
`toqito.states.max_mixed`(dim[, is_sparse])	Produce the maximally mixed state [AAR6].
`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].
`toqito.states.singlet`(dim)	Produce a generalized singlet state acting on two n-dimensional systems [Gsinglet].