Blame for code/verify_minimax.py

#!/usr/bin/env python3
"""Finite checks of exact minimax recovery for nonnegative Schur channels.

These computations supplement the mathematical proofs. They do not establish
the theorem or certify novelty. Only NumPy, SciPy, and the standard library are
required. No semidefinite optimizer or recovery-search claim is used.
"""

from __future__ import annotations

import argparse
import json
from pathlib import Path

import numpy as np
from scipy.optimize import minimize
import scipy


SEED = 20261102
TOLERANCE = 5.0e-7


def trace_norm_hermitian(matrix: np.ndarray) -> float:
    matrix = (matrix + matrix.conj().T) / 2
    return float(np.sum(np.abs(np.linalg.eigvalsh(matrix))))


def solve_simplex(correlation: np.ndarray) -> dict:
    """Return a feasible primal vector and a rigorous-form dual certificate.

The certificate expressions are mathematically valid for an exact feasible
vector; their floating-point evaluation is subject to roundoff. The optimizer
is never treated as a proof of optimality.
"""
    dimension = len(correlation)
    result = minimize(
        lambda q: float(q @ correlation @ q),
        np.full(dimension, 1.0 / dimension),
        jac=lambda q: 2 * correlation @ q,
        method="SLSQP",
        bounds=[(0.0, 1.0)] * dimension,
        constraints=[{"type": "eq", "fun": lambda q: float(np.sum(q) - 1),
                      "jac": lambda q: np.ones(dimension)}],
        options={"ftol": 1e-14, "maxiter": 2000},
    )
    assert result.success, result.message
    q = np.maximum(result.x, 0.0)
    q /= np.sum(q)
    upper = float(q @ correlation @ q)
    lower = float(2 * np.min(correlation @ q) - upper)
    gap = max(0.0, upper - lower)
    assert gap < TOLERANCE
    return {"q": q, "upper": upper, "lower": lower, "duality_gap": gap}


def random_recovery(dimension: int, input_dimension: int,
                    rng: np.random.Generator) -> list[np.ndarray]:
    kraus_count = max(3, (input_dimension + dimension - 1) // dimension + 1)
    gaussian = rng.normal(size=(kraus_count * dimension, input_dimension))
    gaussian = gaussian + 1j * rng.normal(size=gaussian.shape)
    isometry, _ = np.linalg.qr(gaussian, mode="reduced")
    kraus = [isometry[i * dimension:(i + 1) * dimension, :]
             for i in range(kraus_count)]
    assert np.linalg.norm(sum(r.conj().T @ r for r in kraus)
                          - np.eye(input_dimension)) < 1e-10
    return kraus


def random_recovery_fidelity(correlation: np.ndarray, q: np.ndarray,
                             rng: np.random.Generator) -> float:
    dimension = len(correlation)
    recovery = random_recovery(dimension, dimension, rng)
    diagonals = np.array([np.diag(kraus) for kraus in recovery])
    diagonal_gram = diagonals.T @ diagonals.conj()
    fidelity = float(np.real(q @ (correlation * diagonal_gram) @ q))
    return fidelity


def positive_correlation(dimension: int, branches: int,
                         rng: np.random.Generator) -> np.ndarray:
    vectors = rng.uniform(0.001, 1.0, (branches, dimension))
    vectors /= np.linalg.norm(vectors, axis=0, keepdims=True)
    return vectors.T @ vectors


def verify_channel(correlation: np.ndarray, rng: np.random.Generator,
                   name: str, require_nonnegative: bool = True) -> dict:
    dimension = len(correlation)
    assert np.max(np.abs(correlation - correlation.T)) < 1e-12
    assert np.max(np.abs(np.diag(correlation) - 1)) < 1e-12
    assert np.linalg.eigvalsh(correlation)[0] > -1e-10
    if require_nonnegative:
        assert np.min(correlation) > -1e-12
    solution = solve_simplex(correlation)
    q, upper, lower = solution["q"], solution["upper"], solution["lower"]
    difference_kernel = np.ones_like(correlation) - correlation
    worst_weighted_excess = 0.0
    for _ in range(16):
        p = rng.dirichlet(np.ones(dimension))
        weighted = np.sqrt(p)[:, None] * difference_kernel * np.sqrt(p)[None, :]
        norm = trace_norm_hermitian(weighted)
        worst_weighted_excess = max(worst_weighted_excess, norm - 2 * (1 - lower))
        assert norm <= 2 * (1 - lower) + TOLERANCE
        assert np.count_nonzero(np.linalg.eigvalsh(weighted) > 1e-10) <= 1
    weighted = np.sqrt(q)[:, None] * difference_kernel * np.sqrt(q)[None, :]
    extremal_error = abs(trace_norm_hermitian(weighted) - 2 * (1 - upper))
    assert extremal_error < TOLERANCE
    fidelity_excess = 0.0
    if require_nonnegative:
        for _ in range(4):
            fidelity = random_recovery_fidelity(correlation, q, rng)
            fidelity_excess = max(fidelity_excess, fidelity - upper)
            assert fidelity <= upper + TOLERANCE
    return {
        "kind": "nonnegative_schur" if require_nonnegative else "real_schur_fixed_channel_only",
        "name": name, "dimension": dimension,
        "simplex_minimum_upper": upper, "simplex_minimum_lower": lower,
        "duality_gap": solution["duality_gap"],
        "extremal_trace_norm_identity_error": extremal_error,
        "maximum_random_input_bound_excess": max(0.0, worst_weighted_excess),
        "maximum_random_recovery_fidelity_excess": max(0.0, fidelity_excess),
    }


def verify_flagged(dimension: int, branches: int,
                   rng: np.random.Generator) -> dict:
    probabilities = rng.uniform(0.0, 1.0, (branches, dimension))
    probabilities /= np.sum(probabilities, axis=0, keepdims=True)
    roots = np.sqrt(probabilities)
    correlation = roots.T @ roots
    solution = solve_simplex(correlation)
    q = solution["q"]
    noise, polar_recovery = [], []
    for b in range(branches):
        embedding = np.zeros((branches * dimension, dimension), dtype=complex)
        embedding[b * dimension:(b + 1) * dimension, :] = np.eye(dimension)
        noise.append(embedding @ np.diag(roots[b]))
        polar_recovery.append(embedding.conj().T)
    assert np.linalg.norm(sum(e.conj().T @ e for e in noise) - np.eye(dimension)) < 1e-10
    assert np.linalg.norm(sum(r.conj().T @ r for r in polar_recovery)
                          - np.eye(branches * dimension)) < 1e-10
    seed = rng.normal(size=(dimension, dimension)) + 1j * rng.normal(size=(dimension, dimension))
    rho = seed @ seed.conj().T
    rho /= np.trace(rho)
    noisy = sum(e @ rho @ e.conj().T for e in noise)
    recovered = sum(r @ noisy @ r.conj().T for r in polar_recovery)
    polar_channel_error = float(np.linalg.norm(recovered - correlation * rho))
    assert polar_channel_error < 1e-10
    maximum_fidelity_excess = 0.0
    for _ in range(4):
        recovery = random_recovery(dimension, branches * dimension, rng)
        fidelity = sum(abs(np.trace(np.diag(q) @ r @ e)) ** 2
                       for r in recovery for e in noise)
        maximum_fidelity_excess = max(maximum_fidelity_excess,
                                     float(fidelity - solution["upper"]))
        assert fidelity <= solution["upper"] + TOLERANCE
    return {"kind": "orthogonal_flagged_commuting_channel", "dimension": dimension,
            "branches": branches, "duality_gap": solution["duality_gap"],
            "polar_channel_identity_error": polar_channel_error,
            "maximum_random_recovery_fidelity_excess": max(0.0, maximum_fidelity_excess)}


def verify_binary(dimension: int, rng: np.random.Generator) -> dict:
    a, b = sorted(rng.uniform(-1, 1, 2))
    lambdas = np.concatenate(([a, b], rng.uniform(a, b, dimension - 2)))
    roots = np.sqrt(np.vstack(((1 - lambdas) / 2, (1 + lambdas) / 2)))
    correlation = roots.T @ roots
    solution = solve_simplex(correlation)
    fidelity = 0.5 * (np.sqrt((1 + a) * (1 + b))
                      + np.sqrt((1 - a) * (1 - b)))
    exact_error = 1 - fidelity
    qp_error = 2 * (1 - solution["upper"])
    assert abs(qp_error - exact_error) < TOLERANCE
    return {"kind": "binary_closed_formula", "dimension": dimension,
            "minimum_eigenvalue": float(a), "maximum_eigenvalue": float(b),
            "closed_form_diamond_error": float(exact_error),
            "closed_formula_error": abs(qp_error - exact_error),
            "duality_gap": solution["duality_gap"]}


def verify_tensor(rng: np.random.Generator) -> dict:
    first = positive_correlation(3, 4, rng)
    second = positive_correlation(4, 3, rng)
    left, right = solve_simplex(first), solve_simplex(second)
    product = np.kron(first, second)
    joint = solve_simplex(product)
    q_product = np.kron(left["q"], right["q"])
    expected = left["upper"] * right["upper"]
    tensor_error = abs(joint["upper"] - expected)
    assert tensor_error < TOLERANCE
    assert abs(q_product @ product @ q_product - expected) < 1e-12
    product_gap = max(0.0, 2 * (q_product @ product @ q_product
                             - np.min(product @ q_product)))
    assert product_gap < TOLERANCE
    return {"kind": "tensorization", "dimensions": [3, 4],
            "joint_simplex_minimum": joint["upper"],
            "product_of_simplex_minima": expected,
            "tensorization_error": tensor_error,
            "product_witness_duality_gap": float(product_gap),
            "duality_gap": joint["duality_gap"]}


def run_checks() -> dict:
    rng = np.random.default_rng(SEED)
    checks = []
    for d in (2, 3, 5, 8):
        checks.append(verify_channel(np.eye(d), rng, f"complete_dephasing_{d}"))
    checks.append(verify_channel(np.ones((5, 5)), rng, "identity_channel"))
    for d, p in ((3, 0.15), (4, 0.5), (6, 0.8), (7, 1.0)):
        correlation = (1 - p) * np.ones((d, d)) + p * np.eye(d)
        record = verify_channel(correlation, rng, f"equicorrelation_{d}_{p}")
        exact = 1 - p + p / d
        assert abs(record["simplex_minimum_upper"] - exact) < 1e-12
        checks.append(record)
    for repetition in range(7):
        d, branches = 3 + repetition % 4, 2 + repetition % 5
        checks.append(verify_channel(positive_correlation(d, branches, rng),
                                     rng, f"positive_gram_{repetition}"))
    for repetition in range(3):
        points = rng.normal(size=(6, 2))
        squared = np.sum((points[:, None, :] - points[None, :, :]) ** 2, axis=2)
        correlation = np.exp(-(0.2 + repetition) * squared)
        checks.append(verify_channel(correlation, rng, f"gaussian_phase_kernel_{repetition}"))
    cycle = np.eye(5)
    for i in range(5):
        cycle[i, (i + 1) % 5] = cycle[(i + 1) % 5, i] = 0.6
    checks.append(verify_channel(cycle, rng, "nonnegative_five_cycle_correlation"))
    for repetition in range(8):
        checks.append(verify_binary(2 + repetition % 5, rng))
    for repetition in range(8):
        checks.append(verify_flagged(2 + repetition % 4, 2 + repetition % 3, rng))
    for _ in range(8):
        checks.append(verify_tensor(rng))
    for repetition in range(4):
        vectors = rng.normal(size=(3, 4))
        vectors /= np.linalg.norm(vectors, axis=0, keepdims=True)
        checks.append(verify_channel(vectors.T @ vectors, rng,
                                     f"signed_real_gram_{repetition}", False))
    negative_phase = np.array([[1.0, -1.0], [-1.0, 1.0]])
    solution = solve_simplex(negative_phase)
    assert abs(solution["upper"]) < 1e-12
    assert np.linalg.norm(negative_phase * negative_phase - np.ones((2, 2))) < 1e-12
    checks.append({"kind": "negative_phase_counterexample_to_identity_optimality",
                   "fixed_channel_diamond_error": 2.0,
                   "phase_unitary_corrected_diamond_error": 0.0})
    complex_phase = np.array([[1, 1j], [-1j, 1]], dtype=complex)
    p = np.array([0.5, 0.5])
    weighted = np.sqrt(p)[:, None] * (np.ones((2, 2)) - complex_phase) * np.sqrt(p)[None, :]
    measured = trace_norm_hermitian(weighted)
    wrong_real_extension = 2 * (1 - float(np.real(p @ complex_phase @ p)))
    assert abs(measured - np.sqrt(2)) < 1e-12
    assert abs(measured - wrong_real_extension) > 0.4
    checks.append({"kind": "complex_phase_counterexample_to_real_distance_formula",
                   "fixed_channel_diamond_error": measured,
                   "invalid_real_formula_value": wrong_real_extension})
    return {
        "label": "Finite numerical checks; not a proof of universal optimality or a novelty certificate.",
        "seed": SEED, "numpy_version": np.__version__, "scipy_version": scipy.__version__,
        "tolerance": TOLERANCE, "all_checks_passed": True,
        "check_count": len(checks),
        "maximum_simplex_duality_gap": max(entry.get("duality_gap", 0.0) for entry in checks),
        "maximum_extremal_trace_norm_identity_error": max(entry.get("extremal_trace_norm_identity_error", 0.0) for entry in checks),
        "maximum_binary_closed_formula_error": max(entry.get("closed_formula_error", 0.0) for entry in checks),
        "maximum_tensorization_error": max(entry.get("tensorization_error", 0.0) for entry in checks),
        "checks": checks,
    }


def main() -> None:
    parser = argparse.ArgumentParser(description=__doc__)
    parser.add_argument("--output", type=Path,
                        default=Path(__file__).with_name("minimax_results.json"))
    arguments = parser.parse_args()
    results = run_checks()
    arguments.output.write_text(json.dumps(results, indent=2) + "\n", encoding="utf-8")
    print(json.dumps({key: value for key, value in results.items() if key != "checks"}, indent=2))


if __name__ == "__main__":
    main()