Quantum State Invariants API

Quantum State Invariants

graphcalc.quantum.invariants.entanglement_entropy(state: QuantumState, *, subsystems: Iterable[int], base: float = 2.0) → float[source]

Return the entanglement entropy across a bipartition for a pure state.

Parameters:

  • state (QuantumState) – Input quantum state.

  • subsystems (iterable of int) – Subsystems on one side of the bipartition.

  • base (float, default=2.0) – Logarithm base used in the entropy.

Notes

For a pure state, the entanglement entropy across a bipartition A | A^c is the von Neumann entropy of the reduced state on either side.

This function is only defined here for pure states. For mixed states, several inequivalent notions of entanglement measure exist, so this function raises ValueError.

graphcalc.quantum.invariants.fidelity(state1: QuantumState, state2: QuantumState) → float[source]

Return the Uhlmann fidelity between two quantum states.

Parameters:

Notes

The fidelity is defined by

F(rho, sigma) = (Tr(sqrt(sqrt(rho) sigma sqrt(rho))))^2.

This function returns a value in [0, 1] up to numerical tolerance. The input states must have the same total dimension.

graphcalc.quantum.invariants.linear_entropy(state: QuantumState) → float[source]

Return the linear entropy 1 - Tr(rho^2).

graphcalc.quantum.invariants.logarithmic_negativity(state: QuantumState, *, subsystems: Iterable[int]) → float[source]

Return the logarithmic negativity of a quantum state.

Parameters:

  • state (QuantumState) – Input quantum state.

  • subsystems (iterable of int) – Indices of subsystems on which to take the partial transpose.

Notes

The logarithmic negativity is

E_N(rho) = log_2 ||rho^Gamma||_1 = log_2(2 N(rho) + 1).

graphcalc.quantum.invariants.mutual_information(state: QuantumState, *, subsystems_a: Iterable[int], subsystems_b: Iterable[int], base: float = 2.0) → float[source]

Return the quantum mutual information between two subsystem collections.

Parameters:

  • state (QuantumState) – Input quantum state.

  • subsystems_a (iterable of int) – First subsystem collection.

  • subsystems_b (iterable of int) – Second subsystem collection.

  • base (float, default=2.0) – Logarithm base used in the entropy.

Notes

The quantum mutual information is defined by

I(A : B) = S(rho_A) + S(rho_B) - S(rho_AB),

where rho_A, rho_B, and rho_AB are the reduced states on the specified subsystem sets.

The subsystem sets must be disjoint.

graphcalc.quantum.invariants.negativity(state: QuantumState, *, subsystems: Iterable[int]) → float[source]

Return the entanglement negativity with respect to a partial transpose.

Parameters:

  • state (QuantumState) – Input quantum state.

  • subsystems (iterable of int) – Indices of subsystems on which to take the partial transpose.

Notes

If rho^Gamma denotes the partial transpose, then the negativity is

N(rho) = (||rho^Gamma||_1 - 1) / 2,

equivalently the sum of absolute values of the negative eigenvalues of rho^Gamma.

graphcalc.quantum.invariants.purity(state: QuantumState) → float[source]

Return the purity Tr(rho^2) of a quantum state.

graphcalc.quantum.invariants.rank(state: QuantumState) → int[source]

Return the rank of the density operator.

graphcalc.quantum.invariants.von_neumann_entropy(state: QuantumState, *, base: float = 2.0) → float[source]

Return the von Neumann entropy of a quantum state.

Parameters:

  • state (QuantumState) – Input quantum state.

  • base (float, default=2.0) – Logarithm base. Must be positive and not equal to 1.

Notes

The entropy is computed from the eigenvalues of the density operator: S(rho) = -Tr(rho log rho).