| """ | |
| Multi-qubit partial trace, von Neumann entropy, and mutual information. | |
| Nothing like this existed anywhere in the package before: the only prior | |
| partial trace (dashboard_core/state_visuals.py's private | |
| `_reduced_density_matrix`) is single-qubit-only and uses the *opposite*, | |
| little-endian convention (qubit 0 = least significant bit). Everything | |
| here uses this package's own convention instead, matching | |
| dense_evolution.observables/pauli_hamiltonian_to_matrix: qubit 0 is the | |
| *most* significant bit of the basis-state index. Do not mix the two -- | |
| reusing dashboard_core's helper here would silently transpose which | |
| qubits get traced out. | |
| Originated in research/wormhole_syk.py (a traversable-wormhole-inspired | |
| quantum teleportation reproduction) -- promoted here because these are | |
| generic quantum-information utilities, not specific to that experiment. | |
| Any state can have a subsystem's reduced density matrix, entropy, or the | |
| mutual information between two subsystems computed with these three | |
| functions; the wormhole work needed all three because the physically | |
| meaningful readout there (a message injected into one system showing up | |
| correlated with a reference qubit) is *not* visible in any single-qubit | |
| expectation value -- see mutual_information's docstring. | |
| """ | |
| import numpy as np | |
| __all__ = ['partial_trace', 'von_neumann_entropy', 'mutual_information'] | |
| def partial_trace(state, n_qubits, keep_qubits): | |
| """Reduced density matrix on `keep_qubits`, tracing out the rest. | |
| Parameters | |
| ---------- | |
| state : np.ndarray | |
| A pure statevector of length 2**n_qubits. | |
| n_qubits : int | |
| keep_qubits : list[int] | |
| Qubit indices (this package's MSB-first convention) to keep. | |
| Returns | |
| ------- | |
| np.ndarray | |
| Density matrix of shape (2**len(keep_qubits), 2**len(keep_qubits)). | |
| """ | |
| keep_qubits = sorted(keep_qubits) | |
| trace_qubits = [q for q in range(n_qubits) if q not in keep_qubits] | |
| psi = np.transpose(np.asarray(state).reshape([2] * n_qubits), keep_qubits + trace_qubits) | |
| keep_dim, trace_dim = 2 ** len(keep_qubits), 2 ** len(trace_qubits) | |
| psi = psi.reshape(keep_dim, trace_dim) | |
| return psi @ psi.conj().T | |
| def von_neumann_entropy(rho): | |
| """S(rho) = -Tr(rho log rho), computed from rho's eigenvalues. Nearly- | |
| zero eigenvalues (which a numerically pure/near-pure state produces, | |
| and which are mathematically forbidden from being exactly negative | |
| for a real density matrix but can land at a tiny negative float) are | |
| clipped before the log rather than raising or propagating a NaN.""" | |
| eigs = np.clip(np.linalg.eigvalsh(rho).real, 1e-14, None) | |
| return float(-np.sum(eigs * np.log(eigs))) | |
| def mutual_information(state, n_qubits, qubits_a, qubits_b): | |
| """I(A:B) = S(A) + S(B) - S(A union B), the standard quantum mutual | |
| information between two disjoint subsystems of a pure global state. | |
| Why this and not a single-qubit expectation value: a qubit entangled | |
| in a Bell pair (or more generally, maximally mixed on its own) has a | |
| marginal <Z> of exactly 0 regardless of what operation was applied to | |
| its partner -- this is the no-signaling theorem, not a measurement | |
| limitation, and no amount of clever circuit design around a | |
| single-qubit readout can get around it. Mutual information *can* | |
| reveal correlations a marginal expectation value structurally cannot, | |
| because it depends on the *joint* state of A and B, not either one | |
| alone. Verified in tests/unit/test_entropy.py against the exact textbook | |
| value for a Bell pair (I = 2*ln(2), maximal) and a GHZ state. | |
| """ | |
| s_a = von_neumann_entropy(partial_trace(state, n_qubits, qubits_a)) | |
| s_b = von_neumann_entropy(partial_trace(state, n_qubits, qubits_b)) | |
| s_ab = von_neumann_entropy(partial_trace(state, n_qubits, list(qubits_a) + list(qubits_b))) | |
| return s_a + s_b - s_ab | |