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|toqito> 1.1.3 documentation
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|toqito> 1.1.3 documentation
  • Getting started
  • Contributing Guide
  • Code of Conduct
  • Examples
    • Basic Tutorials
      • Superdense Coding
      • Introductory Tutorial
    • Quantum States
      • Quantum state distinguishability
      • The Pusey-Barrett-Rudolph (PBR) Theorem
      • Antidistinguishability of Circulant States and the Eigenvalue Criterion
      • Quantum state exclusion
      • Quantum classification, factor width, \(k\)-incoherence
      • Equiangular States and the Antidistinguishability Threshold
      • The Pretty Good and Pretty Bad Measurements
    • Nonlocal Games
      • Calculating the quantum and classical value of a two-player XOR game
      • Nonlocal games
    • Extended Nonlocal Games
      • Extended nonlocal games
      • Modeling Bit Commitment Binding Failure
      • An extended nonlocal game with quantum advantage
      • The CHSH extended nonlocal game
      • The BB84 extended nonlocal game
  • API Reference
    • API Reference
      • toqito
        • toqito.channel_metrics
          • toqito.channel_metrics.channel_distinguishability
          • toqito.channel_metrics.channel_fidelity
          • toqito.channel_metrics.channel_measured_relative_entropy
          • toqito.channel_metrics.completely_bounded_spectral_norm
          • toqito.channel_metrics.completely_bounded_trace_norm
          • toqito.channel_metrics.diamond_distance
          • toqito.channel_metrics.fidelity_of_separability
        • toqito.channel_ops
          • toqito.channel_ops.apply_channel
          • toqito.channel_ops.choi_to_kraus
          • toqito.channel_ops.complementary_channel
          • toqito.channel_ops.dual_channel
          • toqito.channel_ops.kraus_to_choi
          • toqito.channel_ops.natural_representation
          • toqito.channel_ops.partial_channel
        • toqito.channel_props
          • toqito.channel_props.channel_dim
          • toqito.channel_props.choi_rank
          • toqito.channel_props.is_completely_positive
          • toqito.channel_props.is_extremal
          • toqito.channel_props.is_herm_preserving
          • toqito.channel_props.is_positive
          • toqito.channel_props.is_quantum_channel
          • toqito.channel_props.is_trace_preserving
          • toqito.channel_props.is_unital
          • toqito.channel_props.is_unitary
        • toqito.channels
          • toqito.channels.amplitude_damping
          • toqito.channels.bitflip
          • toqito.channels.choi
          • toqito.channels.dephasing
          • toqito.channels.depolarizing
          • toqito.channels.ldot_channel
          • toqito.channels.pauli_channel
          • toqito.channels.phase_damping
          • toqito.channels.realignment
          • toqito.channels.reduction
        • toqito.helper
          • toqito.helper.expr_as_np_array
          • toqito.helper.np_array_as_expr
          • toqito.helper.update_odometer
        • toqito.matrices
          • toqito.matrices.cnot
          • toqito.matrices.comparison
          • toqito.matrices.cyclic_permutation_matrix
          • toqito.matrices.fourier
          • toqito.matrices.gell_mann
          • toqito.matrices.gen_gell_mann
          • toqito.matrices.gen_pauli
          • toqito.matrices.gen_pauli_x
          • toqito.matrices.gen_pauli_z
          • toqito.matrices.hadamard
          • toqito.matrices.pauli
          • toqito.matrices.standard_basis
        • toqito.matrix_ops
          • toqito.matrix_ops.calculate_vector_matrix_dimension
          • toqito.matrix_ops.null_space
          • toqito.matrix_ops.partial_trace
          • toqito.matrix_ops.partial_transpose
          • toqito.matrix_ops.perturb_vectors
          • toqito.matrix_ops.tensor
          • toqito.matrix_ops.tensor_comb
          • toqito.matrix_ops.tensor_unravel
          • toqito.matrix_ops.to_density_matrix
          • toqito.matrix_ops.unvec
          • toqito.matrix_ops.vectors_from_gram_matrix
          • toqito.matrix_ops.vectors_to_gram_matrix
        • toqito.matrix_props
          • toqito.matrix_props.commutant
          • toqito.matrix_props.factor_width
          • toqito.matrix_props.has_same_dimension
          • toqito.matrix_props.is_absolutely_k_incoherent
          • toqito.matrix_props.is_anti_hermitian
          • toqito.matrix_props.is_block_positive
          • toqito.matrix_props.is_circulant
          • toqito.matrix_props.is_commuting
          • toqito.matrix_props.is_density
          • toqito.matrix_props.is_diagonal
          • toqito.matrix_props.is_diagonally_dominant
          • toqito.matrix_props.is_hermitian
          • toqito.matrix_props.is_idempotent
          • toqito.matrix_props.is_identity
          • toqito.matrix_props.is_k_incoherent
          • toqito.matrix_props.is_ldoi
          • toqito.matrix_props.is_linearly_independent
          • toqito.matrix_props.is_nonnegative
          • toqito.matrix_props.is_normal
          • toqito.matrix_props.is_orthonormal
          • toqito.matrix_props.is_permutation
          • toqito.matrix_props.is_positive
          • toqito.matrix_props.is_positive_definite
          • toqito.matrix_props.is_positive_semidefinite
          • toqito.matrix_props.is_projection
          • toqito.matrix_props.is_pseudo_hermitian
          • toqito.matrix_props.is_pseudo_unitary
          • toqito.matrix_props.is_rank_one
          • toqito.matrix_props.is_square
          • toqito.matrix_props.is_stochastic
          • toqito.matrix_props.is_symmetric
          • toqito.matrix_props.is_totally_positive
          • toqito.matrix_props.is_unitary
          • toqito.matrix_props.kp_norm
          • toqito.matrix_props.majorizes
          • toqito.matrix_props.mutual_coherence
          • toqito.matrix_props.positive_semidefinite_rank
          • toqito.matrix_props.sk_norm
          • toqito.matrix_props.spark
          • toqito.matrix_props.trace_norm
        • toqito.measurement_ops
          • toqito.measurement_ops.measure
        • toqito.measurement_props
          • toqito.measurement_props.is_povm
        • toqito.measurements
          • toqito.measurements.pretty_bad_measurement
          • toqito.measurements.pretty_good_measurement
        • toqito.nonlocal_games
          • toqito.nonlocal_games.binary_constraint_system_game
          • toqito.nonlocal_games.extended_nonlocal_game
          • toqito.nonlocal_games.nonlocal_game
          • toqito.nonlocal_games.quantum_hedging
          • toqito.nonlocal_games.xor_game
        • toqito.perms
          • toqito.perms.antisymmetric_projection
          • toqito.perms.perfect_matchings
          • toqito.perms.perm_sign
          • toqito.perms.permutation_operator
          • toqito.perms.permute_systems
          • toqito.perms.swap
          • toqito.perms.swap_operator
          • toqito.perms.symmetric_projection
          • toqito.perms.unique_perms
          • toqito.perms.vec
        • toqito.rand
          • toqito.rand.random_circulant_gram_matrix
          • toqito.rand.random_density_matrix
          • toqito.rand.random_ginibre
          • toqito.rand.random_orthonormal_basis
          • toqito.rand.random_povm
          • toqito.rand.random_psd_operator
          • toqito.rand.random_state_vector
          • toqito.rand.random_states
          • toqito.rand.random_unitary
        • toqito.state_metrics
          • toqito.state_metrics.bures_angle
          • toqito.state_metrics.bures_distance
          • toqito.state_metrics.fidelity
          • toqito.state_metrics.fidelity_of_separability
          • toqito.state_metrics.helstrom_holevo
          • toqito.state_metrics.hilbert_schmidt
          • toqito.state_metrics.hilbert_schmidt_inner_product
          • toqito.state_metrics.matsumoto_fidelity
          • toqito.state_metrics.measured_relative_entropy
          • toqito.state_metrics.sub_fidelity
          • toqito.state_metrics.trace_distance
        • toqito.state_ops
          • toqito.state_ops.normalize
          • toqito.state_ops.schmidt_decomposition
        • toqito.state_opt
          • toqito.state_opt.bell_inequality_max
          • toqito.state_opt.bell_notation_conversions
          • toqito.state_opt.npa_hierarchy
          • toqito.state_opt.optimal_clone
          • toqito.state_opt.ppt_distinguishability
          • toqito.state_opt.state_distinguishability
          • toqito.state_opt.state_exclusion
          • toqito.state_opt.state_helper
          • toqito.state_opt.symmetric_extension_hierarchy
        • toqito.state_props
          • toqito.state_props.abs_ppt_constraints
          • toqito.state_props.common_quantum_overlap
          • toqito.state_props.concurrence
          • toqito.state_props.entanglement_of_formation
          • toqito.state_props.has_symmetric_extension
          • toqito.state_props.in_separable_ball
          • toqito.state_props.is_abs_ppt
          • toqito.state_props.is_antidistinguishable
          • toqito.state_props.is_distinguishable
          • toqito.state_props.is_ensemble
          • toqito.state_props.is_mixed
          • toqito.state_props.is_mutually_orthogonal
          • toqito.state_props.is_mutually_unbiased_basis
          • toqito.state_props.is_npt
          • toqito.state_props.is_ppt
          • toqito.state_props.is_product
          • toqito.state_props.is_pure
          • toqito.state_props.is_separable
          • toqito.state_props.is_sic_povm
          • toqito.state_props.is_unextendible_product_basis
          • toqito.state_props.l1_norm_coherence
          • toqito.state_props.learnability
          • toqito.state_props.log_negativity
          • toqito.state_props.negativity
          • toqito.state_props.purity
          • toqito.state_props.renyi_entropy
          • toqito.state_props.schmidt_rank
          • toqito.state_props.sk_vec_norm
          • toqito.state_props.von_neumann_entropy
        • toqito.states
          • toqito.states.basis
          • toqito.states.bb84
          • toqito.states.bell
          • toqito.states.brauer
          • toqito.states.breuer
          • toqito.states.chessboard
          • toqito.states.dicke
          • toqito.states.domino
          • toqito.states.gen_bell
          • toqito.states.ghz
          • toqito.states.gisin
          • toqito.states.horodecki
          • toqito.states.isotropic
          • toqito.states.max_entangled
          • toqito.states.max_mixed
          • toqito.states.mutually_unbiased_basis
          • toqito.states.pusey_barrett_rudolph
          • toqito.states.singlet
          • toqito.states.tile
          • toqito.states.trine
          • toqito.states.w_state
          • toqito.states.werner
    • toqito
      • toqito.channel_metrics
        • toqito.channel_metrics.channel_distinguishability
        • toqito.channel_metrics.channel_fidelity
        • toqito.channel_metrics.channel_measured_relative_entropy
        • toqito.channel_metrics.completely_bounded_spectral_norm
        • toqito.channel_metrics.completely_bounded_trace_norm
        • toqito.channel_metrics.diamond_distance
        • toqito.channel_metrics.fidelity_of_separability
      • toqito.channel_ops
        • toqito.channel_ops.apply_channel
        • toqito.channel_ops.choi_to_kraus
        • toqito.channel_ops.complementary_channel
        • toqito.channel_ops.dual_channel
        • toqito.channel_ops.kraus_to_choi
        • toqito.channel_ops.natural_representation
        • toqito.channel_ops.partial_channel
      • toqito.channel_props
        • toqito.channel_props.channel_dim
        • toqito.channel_props.choi_rank
        • toqito.channel_props.is_completely_positive
        • toqito.channel_props.is_extremal
        • toqito.channel_props.is_herm_preserving
        • toqito.channel_props.is_positive
        • toqito.channel_props.is_quantum_channel
        • toqito.channel_props.is_trace_preserving
        • toqito.channel_props.is_unital
        • toqito.channel_props.is_unitary
      • toqito.channels
        • toqito.channels.amplitude_damping
        • toqito.channels.bitflip
        • toqito.channels.choi
        • toqito.channels.dephasing
        • toqito.channels.depolarizing
        • toqito.channels.ldot_channel
        • toqito.channels.pauli_channel
        • toqito.channels.phase_damping
        • toqito.channels.realignment
        • toqito.channels.reduction
      • toqito.helper
        • toqito.helper.expr_as_np_array
        • toqito.helper.np_array_as_expr
        • toqito.helper.update_odometer
      • toqito.matrices
        • toqito.matrices.cnot
        • toqito.matrices.comparison
        • toqito.matrices.cyclic_permutation_matrix
        • toqito.matrices.fourier
        • toqito.matrices.gell_mann
        • toqito.matrices.gen_gell_mann
        • toqito.matrices.gen_pauli
        • toqito.matrices.gen_pauli_x
        • toqito.matrices.gen_pauli_z
        • toqito.matrices.hadamard
        • toqito.matrices.pauli
        • toqito.matrices.standard_basis
      • toqito.matrix_ops
        • toqito.matrix_ops.calculate_vector_matrix_dimension
        • toqito.matrix_ops.null_space
        • toqito.matrix_ops.partial_trace
        • toqito.matrix_ops.partial_transpose
        • toqito.matrix_ops.perturb_vectors
        • toqito.matrix_ops.tensor
        • toqito.matrix_ops.tensor_comb
        • toqito.matrix_ops.tensor_unravel
        • toqito.matrix_ops.to_density_matrix
        • toqito.matrix_ops.unvec
        • toqito.matrix_ops.vectors_from_gram_matrix
        • toqito.matrix_ops.vectors_to_gram_matrix
      • toqito.matrix_props
        • toqito.matrix_props.commutant
        • toqito.matrix_props.factor_width
        • toqito.matrix_props.has_same_dimension
        • toqito.matrix_props.is_absolutely_k_incoherent
        • toqito.matrix_props.is_anti_hermitian
        • toqito.matrix_props.is_block_positive
        • toqito.matrix_props.is_circulant
        • toqito.matrix_props.is_commuting
        • toqito.matrix_props.is_density
        • toqito.matrix_props.is_diagonal
        • toqito.matrix_props.is_diagonally_dominant
        • toqito.matrix_props.is_hermitian
        • toqito.matrix_props.is_idempotent
        • toqito.matrix_props.is_identity
        • toqito.matrix_props.is_k_incoherent
        • toqito.matrix_props.is_ldoi
        • toqito.matrix_props.is_linearly_independent
        • toqito.matrix_props.is_nonnegative
        • toqito.matrix_props.is_normal
        • toqito.matrix_props.is_orthonormal
        • toqito.matrix_props.is_permutation
        • toqito.matrix_props.is_positive
        • toqito.matrix_props.is_positive_definite
        • toqito.matrix_props.is_positive_semidefinite
        • toqito.matrix_props.is_projection
        • toqito.matrix_props.is_pseudo_hermitian
        • toqito.matrix_props.is_pseudo_unitary
        • toqito.matrix_props.is_rank_one
        • toqito.matrix_props.is_square
        • toqito.matrix_props.is_stochastic
        • toqito.matrix_props.is_symmetric
        • toqito.matrix_props.is_totally_positive
        • toqito.matrix_props.is_unitary
        • toqito.matrix_props.kp_norm
        • toqito.matrix_props.majorizes
        • toqito.matrix_props.mutual_coherence
        • toqito.matrix_props.positive_semidefinite_rank
        • toqito.matrix_props.sk_norm
        • toqito.matrix_props.spark
        • toqito.matrix_props.trace_norm
      • toqito.measurement_ops
        • toqito.measurement_ops.measure
      • toqito.measurement_props
        • toqito.measurement_props.is_povm
      • toqito.measurements
        • toqito.measurements.pretty_bad_measurement
        • toqito.measurements.pretty_good_measurement
      • toqito.nonlocal_games
        • toqito.nonlocal_games.binary_constraint_system_game
        • toqito.nonlocal_games.extended_nonlocal_game
        • toqito.nonlocal_games.nonlocal_game
        • toqito.nonlocal_games.quantum_hedging
        • toqito.nonlocal_games.xor_game
      • toqito.perms
        • toqito.perms.antisymmetric_projection
        • toqito.perms.perfect_matchings
        • toqito.perms.perm_sign
        • toqito.perms.permutation_operator
        • toqito.perms.permute_systems
        • toqito.perms.swap
        • toqito.perms.swap_operator
        • toqito.perms.symmetric_projection
        • toqito.perms.unique_perms
        • toqito.perms.vec
      • toqito.rand
        • toqito.rand.random_circulant_gram_matrix
        • toqito.rand.random_density_matrix
        • toqito.rand.random_ginibre
        • toqito.rand.random_orthonormal_basis
        • toqito.rand.random_povm
        • toqito.rand.random_psd_operator
        • toqito.rand.random_state_vector
        • toqito.rand.random_states
        • toqito.rand.random_unitary
      • toqito.state_metrics
        • toqito.state_metrics.bures_angle
        • toqito.state_metrics.bures_distance
        • toqito.state_metrics.fidelity
        • toqito.state_metrics.fidelity_of_separability
        • toqito.state_metrics.helstrom_holevo
        • toqito.state_metrics.hilbert_schmidt
        • toqito.state_metrics.hilbert_schmidt_inner_product
        • toqito.state_metrics.matsumoto_fidelity
        • toqito.state_metrics.measured_relative_entropy
        • toqito.state_metrics.sub_fidelity
        • toqito.state_metrics.trace_distance
      • toqito.state_ops
        • toqito.state_ops.normalize
        • toqito.state_ops.schmidt_decomposition
      • toqito.state_opt
        • toqito.state_opt.bell_inequality_max
        • toqito.state_opt.bell_notation_conversions
        • toqito.state_opt.npa_hierarchy
        • toqito.state_opt.optimal_clone
        • toqito.state_opt.ppt_distinguishability
        • toqito.state_opt.state_distinguishability
        • toqito.state_opt.state_exclusion
        • toqito.state_opt.state_helper
        • toqito.state_opt.symmetric_extension_hierarchy
      • toqito.state_props
        • toqito.state_props.abs_ppt_constraints
        • toqito.state_props.common_quantum_overlap
        • toqito.state_props.concurrence
        • toqito.state_props.entanglement_of_formation
        • toqito.state_props.has_symmetric_extension
        • toqito.state_props.in_separable_ball
        • toqito.state_props.is_abs_ppt
        • toqito.state_props.is_antidistinguishable
        • toqito.state_props.is_distinguishable
        • toqito.state_props.is_ensemble
        • toqito.state_props.is_mixed
        • toqito.state_props.is_mutually_orthogonal
        • toqito.state_props.is_mutually_unbiased_basis
        • toqito.state_props.is_npt
        • toqito.state_props.is_ppt
        • toqito.state_props.is_product
        • toqito.state_props.is_pure
        • toqito.state_props.is_separable
        • toqito.state_props.is_sic_povm
        • toqito.state_props.is_unextendible_product_basis
        • toqito.state_props.l1_norm_coherence
        • toqito.state_props.learnability
        • toqito.state_props.log_negativity
        • toqito.state_props.negativity
        • toqito.state_props.purity
        • toqito.state_props.renyi_entropy
        • toqito.state_props.schmidt_rank
        • toqito.state_props.sk_vec_norm
        • toqito.state_props.von_neumann_entropy
      • toqito.states
        • toqito.states.basis
        • toqito.states.bb84
        • toqito.states.bell
        • toqito.states.brauer
        • toqito.states.breuer
        • toqito.states.chessboard
        • toqito.states.dicke
        • toqito.states.domino
        • toqito.states.gen_bell
        • toqito.states.ghz
        • toqito.states.gisin
        • toqito.states.horodecki
        • toqito.states.isotropic
        • toqito.states.max_entangled
        • toqito.states.max_mixed
        • toqito.states.mutually_unbiased_basis
        • toqito.states.pusey_barrett_rudolph
        • toqito.states.singlet
        • toqito.states.tile
        • toqito.states.trine
        • toqito.states.w_state
        • toqito.states.werner
  • References

Project Links

  • GitHub
  • PyPI
  • Discord Channel
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Basic Tutorials¶

The following rendered tutorials demonstrate introductory concepts:

  • Introductory Tutorial

  • Superdense Coding

Superdense Coding

Superdense Coding

Introductory Tutorial

Introductory Tutorial
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Superdense Coding
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Examples
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Last updated on Feb 27, 2026