toqito

The toqito package is an open source library for studying various objects in quantum information, namely, states, channels, and measurements. toqito focuses on providing numerical tools to study problems pertaining to entanglement theory, nonlocal games, and other aspects of quantum information that are often associated with computer science.

User Documentation

• Getting started
  • Installing
  • Contributing
  • Reporting Issues
  • Citing
• Introductory Tutorial
  • States
  • Channels
  • Measurements
• Additional Tutorials
  • Introductory Tutorial
  • Nonlocal Games
  • XOR Games
  • Extended Nonlocal Games
  • Quantum State Distinguishability
  • Quantum State Exclusion
  • Superdense Coding

• Contributing Guide
  • Making Changes
  • Testing
  • Code Style
  • References in Docstrings
  • Documentation
  • Adding a new feature
  • Additional Resources
• Code of Conduct
  • Our Pledge
  • Our Standards
  • Our Responsibilities
  • Scope
  • Enforcement
  • Attribution

API Reference

• API Reference
  • channel_metrics
    • Submodules
      • channel_metrics.channel_fidelity
        • Module Contents
          • Functions
            • channel_fidelity()
      • channel_metrics.completely_bounded_spectral_norm
        • Module Contents
          • Functions
            • completely_bounded_spectral_norm()
      • channel_metrics.completely_bounded_trace_norm
        • Module Contents
          • Functions
            • completely_bounded_trace_norm()
      • channel_metrics.diamond_norm
        • Module Contents
          • Functions
            • diamond_norm()
      • channel_metrics.fidelity_of_separability
        • Module Contents
          • Functions
            • fidelity_of_separability()
  • channel_ops
    • Submodules
      • channel_ops.apply_channel
        • Module Contents
          • Functions
            • apply_channel()
      • channel_ops.choi_to_kraus
        • Module Contents
          • Functions
            • choi_to_kraus()
      • channel_ops.dual_channel
        • Module Contents
          • Functions
            • dual_channel()
      • channel_ops.kraus_to_choi
        • Module Contents
          • Functions
            • kraus_to_choi()
      • channel_ops.partial_channel
        • Module Contents
          • Functions
            • partial_channel()
  • channel_props
    • Submodules
      • channel_props.choi_rank
        • Module Contents
          • Functions
            • choi_rank()
      • channel_props.is_completely_positive
        • Module Contents
          • Functions
            • is_completely_positive()
      • channel_props.is_herm_preserving
        • Module Contents
          • Functions
            • is_herm_preserving()
      • channel_props.is_positive
        • Module Contents
          • Functions
            • is_positive()
      • channel_props.is_quantum_channel
        • Module Contents
          • Functions
            • is_quantum_channel()
      • channel_props.is_trace_preserving
        • Module Contents
          • Functions
            • is_trace_preserving()
      • channel_props.is_unital
        • Module Contents
          • Functions
            • is_unital()
      • channel_props.is_unitary
        • Module Contents
          • Functions
            • is_unitary()
  • channels
    • Submodules
      • channels.choi
        • Module Contents
          • Functions
            • choi()
      • channels.dephasing
        • Module Contents
          • Functions
            • dephasing()
      • channels.depolarizing
        • Module Contents
          • Functions
            • depolarizing()
      • channels.partial_trace
        • Module Contents
          • Functions
            • partial_trace()
      • channels.partial_transpose
        • Module Contents
          • Functions
            • partial_transpose()
      • channels.realignment
        • Module Contents
          • Functions
            • realignment()
      • channels.reduction
        • Module Contents
          • Functions
            • reduction()
  • helper
    • Submodules
      • helper.channel_dim
        • Module Contents
          • Functions
            • _expand_dim()
            • channel_dim()
      • helper.cvx_kron
        • Module Contents
          • Functions
            • cvx_kron()
      • helper.expr_as_np_array
        • Module Contents
          • Functions
            • expr_as_np_array()
      • helper.np_array_as_expr
        • Module Contents
          • Functions
            • np_array_as_expr()
      • helper.npa_hierarchy
        • Module Contents
          • Functions
          • Attributes
            • Symbol
            • _gen_words()
            • _get_nonlocal_game_params()
            • _is_meas()
            • _is_meas_on_one_player()
            • _is_zero()
            • _parse()
            • _reduce()
            • npa_constraints()
      • helper.update_odometer
        • Module Contents
          • Functions
            • update_odometer()
  • matrices
    • Submodules
      • matrices.clock
        • Module Contents
          • Functions
            • clock()
      • matrices.cnot
        • Module Contents
          • Functions
            • cnot()
      • matrices.cyclic_permutation
        • Module Contents
          • Functions
            • cyclic_permutation_matrix()
      • matrices.fourier
        • Module Contents
          • Functions
            • fourier()
      • matrices.gell_mann
        • Module Contents
          • Functions
            • gell_mann()
      • matrices.gen_gell_mann
        • Module Contents
          • Functions
            • gen_gell_mann()
      • matrices.gen_pauli
        • Module Contents
          • Functions
            • gen_pauli()
      • matrices.hadamard
        • Module Contents
          • Functions
            • _hamming_distance()
            • hadamard()
      • matrices.pauli
        • Module Contents
          • Functions
            • pauli()
      • matrices.shift
        • Module Contents
          • Functions
            • shift()
      • matrices.standard_basis
        • Module Contents
          • Functions
            • standard_basis()
  • matrix_ops
    • Submodules
      • matrix_ops.calculate_vector_matrix_dimension
        • Module Contents
          • Functions
            • calculate_vector_matrix_dimension()
      • matrix_ops.inner_product
        • Module Contents
          • Functions
            • inner_product()
      • matrix_ops.outer_product
        • Module Contents
          • Functions
            • outer_product()
      • matrix_ops.tensor
        • Module Contents
          • Functions
            • tensor()
      • matrix_ops.unvec
        • Module Contents
          • Functions
            • unvec()
      • matrix_ops.vec
        • Module Contents
          • Functions
            • vec()
      • matrix_ops.vectors_from_gram_matrix
        • Module Contents
          • Functions
            • vectors_from_gram_matrix()
      • matrix_ops.vectors_to_gram_matrix
        • Module Contents
          • Functions
            • vectors_to_gram_matrix()
  • matrix_props
    • Submodules
      • matrix_props.has_same_dimension
        • Module Contents
          • Functions
            • has_same_dimension()
      • matrix_props.is_block_positive
        • Module Contents
          • Functions
            • is_block_positive()
      • matrix_props.is_circulant
        • Module Contents
          • Functions
            • is_circulant()
      • matrix_props.is_commuting
        • Module Contents
          • Functions
            • is_commuting()
      • matrix_props.is_density
        • Module Contents
          • Functions
            • is_density()
      • matrix_props.is_diagonal
        • Module Contents
          • Functions
            • is_diagonal()
      • matrix_props.is_diagonally_dominant
        • Module Contents
          • Functions
            • is_diagonally_dominant()
      • matrix_props.is_hermitian
        • Module Contents
          • Functions
            • is_hermitian()
      • matrix_props.is_idempotent
        • Module Contents
          • Functions
            • is_idempotent()
      • matrix_props.is_identity
        • Module Contents
          • Functions
            • is_identity()
      • matrix_props.is_linearly_independent
        • Module Contents
          • Functions
            • is_linearly_independent()
      • matrix_props.is_normal
        • Module Contents
          • Functions
            • is_normal()
      • matrix_props.is_orthonormal
        • Module Contents
          • Functions
            • is_orthonormal()
      • matrix_props.is_permutation
        • Module Contents
          • Functions
            • is_permutation()
      • matrix_props.is_positive_definite
        • Module Contents
          • Functions
            • is_positive_definite()
      • matrix_props.is_positive_semidefinite
        • Module Contents
          • Functions
            • is_positive_semidefinite()
      • matrix_props.is_projection
        • Module Contents
          • Functions
            • is_projection()
      • matrix_props.is_square
        • Module Contents
          • Functions
            • is_square()
      • matrix_props.is_symmetric
        • Module Contents
          • Functions
            • is_symmetric()
      • matrix_props.is_totally_positive
        • Module Contents
          • Functions
            • is_totally_positive()
      • matrix_props.is_unitary
        • Module Contents
          • Functions
            • is_unitary()
      • matrix_props.kp_norm
        • Module Contents
          • Functions
            • kp_norm()
      • matrix_props.majorizes
        • Module Contents
          • Functions
            • majorizes()
      • matrix_props.sk_norm
        • Module Contents
          • Functions
            • __lower_bound_sk_norm_randomized()
            • __target_is_proved()
            • sk_operator_norm()
      • matrix_props.trace_norm
        • Module Contents
          • Functions
            • trace_norm()
  • measurement_ops
    • Submodules
      • measurement_ops.measure
        • Module Contents
          • Functions
            • measure()
  • measurement_props
    • Submodules
      • measurement_props.is_povm
        • Module Contents
          • Functions
            • is_povm()
  • nonlocal_games
    • Submodules
      • nonlocal_games.extended_nonlocal_game
        • Module Contents
          • Classes
            • ExtendedNonlocalGame
              • ExtendedNonlocalGame.__optimize_alice()
              • ExtendedNonlocalGame.__optimize_bob()
              • ExtendedNonlocalGame.commuting_measurement_value_upper_bound()
              • ExtendedNonlocalGame.nonsignaling_value()
              • ExtendedNonlocalGame.quantum_value_lower_bound()
              • ExtendedNonlocalGame.unentangled_value()
      • nonlocal_games.nonlocal_game
        • Module Contents
          • Classes
            • NonlocalGame
              • NonlocalGame.__optimize_alice()
              • NonlocalGame.__optimize_bob()
              • NonlocalGame.classical_value()
              • NonlocalGame.commuting_measurement_value_upper_bound()
              • NonlocalGame.from_bcs_game()
              • NonlocalGame.nonsignaling_value()
              • NonlocalGame.quantum_value_lower_bound()
      • nonlocal_games.quantum_hedging
        • Module Contents
          • Classes
            • QuantumHedging
              • QuantumHedging.max_prob_outcome_a_dual()
              • QuantumHedging.max_prob_outcome_a_primal()
              • QuantumHedging.min_prob_outcome_a_dual()
              • QuantumHedging.min_prob_outcome_a_primal()
      • nonlocal_games.xor_game
        • Module Contents
          • Classes
            • XORGame
              • XORGame.classical_value()
              • XORGame.nonsignaling_value()
              • XORGame.quantum_value()
              • XORGame.to_nonlocal_game()
  • perms
    • Submodules
      • perms.antisymmetric_projection
        • Module Contents
          • Functions
            • antisymmetric_projection()
      • perms.perfect_matchings
        • Module Contents
          • Functions
            • perfect_matchings()
      • perms.perm_sign
        • Module Contents
          • Functions
            • perm_sign()
      • perms.permutation_operator
        • Module Contents
          • Functions
            • permutation_operator()
      • perms.permute_systems
        • Module Contents
          • Functions
            • permute_systems()
      • perms.swap
        • Module Contents
          • Functions
            • swap()
      • perms.swap_operator
        • Module Contents
          • Functions
            • swap_operator()
      • perms.symmetric_projection
        • Module Contents
          • Functions
            • symmetric_projection()
      • perms.unique_perms
        • Module Contents
          • Classes
          • Functions
            • UniqueElement
              • UniqueElement.occurrences
              • UniqueElement.value
            • perm_unique_helper()
            • unique_perms()
  • rand
    • Submodules
      • rand.random_circulant_gram
        • Module Contents
          • Functions
            • random_circulant_gram()
      • rand.random_density_matrix
        • Module Contents
          • Functions
            • random_density_matrix()
      • rand.random_ginibre
        • Module Contents
          • Functions
            • random_ginibre()
      • rand.random_povm
        • Module Contents
          • Functions
            • random_povm()
      • rand.random_state_vector
        • Module Contents
          • Functions
            • random_state_vector()
      • rand.random_states
        • Module Contents
          • Functions
            • random_states()
      • rand.random_unitary
        • Module Contents
          • Functions
            • random_unitary()
  • state_metrics
    • Submodules
      • state_metrics.bures_angle
        • Module Contents
          • Functions
            • bures_angle()
      • state_metrics.bures_distance
        • Module Contents
          • Functions
            • bures_distance()
      • state_metrics.fidelity
        • Module Contents
          • Functions
            • fidelity()
      • state_metrics.fidelity_of_separability
        • Module Contents
          • Functions
            • fidelity_of_separability()
      • state_metrics.helstrom_holevo
        • Module Contents
          • Functions
            • helstrom_holevo()
      • state_metrics.hilbert_schmidt
        • Module Contents
          • Functions
            • hilbert_schmidt()
      • state_metrics.hilbert_schmidt_inner_product
        • Module Contents
          • Functions
            • hilbert_schmidt_inner_product()
      • state_metrics.matsumoto_fidelity
        • Module Contents
          • Functions
            • matsumoto_fidelity()
      • state_metrics.sub_fidelity
        • Module Contents
          • Functions
            • sub_fidelity()
      • state_metrics.trace_distance
        • Module Contents
          • Functions
            • trace_distance()
  • state_ops
    • Submodules
      • state_ops.pure_to_mixed
        • Module Contents
          • Functions
            • pure_to_mixed()
      • state_ops.schmidt_decomposition
        • Module Contents
          • Functions
            • _operator_schmidt_decomposition()
            • schmidt_decomposition()
  • state_opt
    • Submodules
      • state_opt.optimal_clone
        • Module Contents
          • Functions
            • dual_problem()
            • optimal_clone()
            • primal_problem()
      • state_opt.ppt_distinguishability
        • Module Contents
          • Functions
            • dual_problem()
            • ppt_distinguishability()
            • primal_problem()
      • state_opt.state_distinguishability
        • Module Contents
          • Functions
            • _min_error_dual()
            • _min_error_primal()
            • state_distinguishability()
      • state_opt.state_exclusion
        • Module Contents
          • Functions
            • _min_error_dual()
            • _min_error_primal()
            • state_exclusion()
      • state_opt.state_helper
        • Module Contents
          • Functions
            • __is_probs_valid()
            • __is_states_valid()
      • state_opt.symmetric_extension_hierarchy
        • Module Contents
          • Functions
            • symmetric_extension_hierarchy()
  • state_props
    • Submodules
      • state_props.concurrence
        • Module Contents
          • Functions
            • concurrence()
      • state_props.entanglement_of_formation
        • Module Contents
          • Functions
            • entanglement_of_formation()
      • state_props.has_symmetric_extension
        • Module Contents
          • Functions
            • has_symmetric_extension()
      • state_props.in_separable_ball
        • Module Contents
          • Functions
            • in_separable_ball()
      • state_props.is_antidistinguishable
        • Module Contents
          • Functions
            • is_antidistinguishable()
      • state_props.is_distinguishable
        • Module Contents
          • Functions
            • is_distinguishable()
      • state_props.is_ensemble
        • Module Contents
          • Functions
            • is_ensemble()
      • state_props.is_mixed
        • Module Contents
          • Functions
            • is_mixed()
      • state_props.is_mutually_orthogonal
        • Module Contents
          • Functions
            • is_mutually_orthogonal()
      • state_props.is_mutually_unbiased_basis
        • Module Contents
          • Functions
            • is_mutually_unbiased_basis()
      • state_props.is_npt
        • Module Contents
          • Functions
            • is_npt()
      • state_props.is_ppt
        • Module Contents
          • Functions
            • is_ppt()
      • state_props.is_product
        • Module Contents
          • Functions
            • _is_product()
            • _operator_is_product()
            • is_product()
      • state_props.is_pure
        • Module Contents
          • Functions
            • is_pure()
      • state_props.is_separable
        • Module Contents
          • Functions
            • is_separable()
      • state_props.is_unextendible_product_basis
        • Module Contents
          • Functions
            • is_unextendible_product_basis()
      • state_props.l1_norm_coherence
        • Module Contents
          • Functions
            • l1_norm_coherence()
      • state_props.log_negativity
        • Module Contents
          • Functions
            • log_negativity()
      • state_props.negativity
        • Module Contents
          • Functions
            • negativity()
      • state_props.purity
        • Module Contents
          • Functions
            • purity()
      • state_props.schmidt_rank
        • Module Contents
          • Functions
            • _operator_schmidt_rank()
            • schmidt_rank()
      • state_props.sk_vec_norm
        • Module Contents
          • Functions
            • sk_vector_norm()
      • state_props.test_is_distinguishable
        • Module Contents
          • Functions
            • test_is_distinguishable()
      • state_props.von_neumann_entropy
        • Module Contents
          • Functions
            • von_neumann_entropy()
  • states
    • Submodules
      • states.basis
        • Module Contents
          • Functions
            • basis()
      • states.bb84
        • Module Contents
          • Functions
            • bb84()
      • states.bell
        • Module Contents
          • Functions
            • bell()
      • states.brauer
        • Module Contents
          • Functions
            • brauer()
      • states.breuer
        • Module Contents
          • Functions
            • breuer()
      • states.chessboard
        • Module Contents
          • Functions
            • chessboard()
      • states.domino
        • Module Contents
          • Functions
            • domino()
      • states.gen_bell
        • Module Contents
          • Functions
            • gen_bell()
      • states.ghz
        • Module Contents
          • Functions
            • ghz()
      • states.gisin
        • Module Contents
          • Functions
            • gisin()
      • states.horodecki
        • Module Contents
          • Functions
            • horodecki()
      • states.isotropic
        • Module Contents
          • Functions
            • isotropic()
      • states.max_entangled
        • Module Contents
          • Functions
            • max_entangled()
      • states.max_mixed
        • Module Contents
          • Functions
            • max_mixed()
      • states.mutually_unbiased_basis
        • Module Contents
          • Functions
            • _is_prime_power()
            • mutually_unbiased_basis()
      • states.pusey_barrett_rudolph
        • Module Contents
          • Functions
            • pusey_barrett_rudolph()
      • states.singlet
        • Module Contents
          • Functions
            • singlet()
      • states.tile
        • Module Contents
          • Functions
            • tile()
      • states.trine
        • Module Contents
          • Functions
            • trine()
      • states.w_state
        • Module Contents
          • Functions
            • w_state()
      • states.werner
        • Module Contents
          • Functions
            • werner()

References

• References