Antidistinguishability of Circulant States and the Eigenvalue Criterion

In this tutorial, we investigate the antidistinguishability of a special class of quantum states known as circulant states. We will numerically verify a necessary and sufficient condition based on the eigenvalues of the states’ Gram matrix, as presented in the paper by Johnston et al. [1].

This tutorial builds upon the concepts introduced in the Quantum state exclusion tutorial.

Eigenvalue Criterion for Circulant States

A set of \(n\) pure states is called circulant if its Gram matrix is circulant. A matrix is circulant if each of its rows is a cyclic shift of the row above it. Such sets of states have a high degree of symmetry and appear in various quantum information contexts.

A key result from (Theorem 5.1) [1] provides a simple and exact criterion for determining if a circulant set is antidistinguishable, based solely on the eigenvalues of its Gram matrix.

The theorem states that a set of \(n\) states with a circulant Gram matrix \(G\) is antidistinguishable if and only if its eigenvalues \(\lambda_0 \ge \lambda_1 \ge \cdots \ge \lambda_{n-1}\) satisfy the following inequality:

\[\sqrt{\lambda_0} \le \sum_{j=1}^{n-1} \sqrt{\lambda_j}\]

This gives us a direct analytical test that is much more efficient than solving a full semidefinite program (SDP). We can use |toqito⟩ to verify this equivalence.

Numerical Verification

Our plan to verify this theorem is as follows:

1. Generate a random circulant Gram matrix \(G\) using random_circulant_gram_matrix().

2. Compute its eigenvalues and perform the analytical check using the inequality from the theorem.

3. Generate the corresponding set of state vectors from \(G\) using vectors_from_gram_matrix().

4. Perform a numerical check by calling the high-level function is_antidistinguishable() to directly verify the property.

5. Confirm that the analytical and numerical checks yield the same conclusion.

import numpy as np

from toqito.matrix_ops import vectors_from_gram_matrix
from toqito.rand import random_circulant_gram_matrix
from toqito.state_props import is_antidistinguishable

# 1. Define parameters and generate a random circulant Gram matrix.
n = 5
# Use a seed for reproducibility.
seed = 42

print(f"Generating a random {n}x{n} circulant Gram matrix (seed={seed})...")
gram_matrix = random_circulant_gram_matrix(n, seed=seed)

# 2. Perform the analytical check based on the eigenvalue criterion.
# Use 'eigvalsh' for Hermitian matrices; it's faster and returns real eigenvalues.
eigenvalues = np.linalg.eigvalsh(gram_matrix)
# Sort eigenvalues in descending order.
eigenvalues = np.sort(eigenvalues)[::-1]
lambda_0 = eigenvalues[0]
other_lambdas = eigenvalues[1:]

# The analytical check from the theorem:
lhs = np.sqrt(lambda_0)
# The sum of the square roots of the other eigenvalues.
# Use np.maximum to avoid numerical precision errors leading to sqrt of tiny negative numbers.
rhs = np.sum(np.sqrt(np.maximum(0, other_lambdas)))
analytical_is_ad = lhs <= rhs

print("\nANALYTICAL CHECK (from Theorem 5.1 of Johnston et al.):")
print(f"  sqrt(λ₀) = {lhs:.4f}")
print(f"  Σ sqrt(λⱼ) for j>0 = {rhs:.4f}")
print(f"  Is sqrt(λ₀) <= Σ sqrt(λⱼ)? {analytical_is_ad}")
print(f"  Conclusion: The set SHOULD BE antidistinguishable: {analytical_is_ad}")

# 3. Generate states from the Gram matrix for the numerical check.
states = vectors_from_gram_matrix(gram_matrix)

# 4. Perform the numerical check using |toqito⟩'s high-level function.
numerical_is_ad = is_antidistinguishable(states)

print("\nNUMERICAL CHECK (via is_antidistinguishable function):")
print(f"  Conclusion: The set IS antidistinguishable: {numerical_is_ad}")

# 5. Verify that both methods agree.
print("\n------------------------------------------------------")
print(f"Do the analytical and numerical results agree? {analytical_is_ad == numerical_is_ad}")
print("------------------------------------------------------")

Generating a random 5x5 circulant Gram matrix (seed=42)...

ANALYTICAL CHECK (from Theorem 5.1 of Johnston et al.):
  sqrt(λ₀) = 0.8820
  Σ sqrt(λⱼ) for j>0 = 2.7943
  Is sqrt(λ₀) <= Σ sqrt(λⱼ)? True
  Conclusion: The set SHOULD BE antidistinguishable: True

NUMERICAL CHECK (via is_antidistinguishable function):
  Conclusion: The set IS antidistinguishable: True

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Do the analytical and numerical results agree? True
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The results from both the analytical eigenvalue criterion and the numerical check using |toqito⟩’s helper function agree, providing a concrete verification of Theorem 5.1 from [1]. This demonstrates how a deep theoretical result can provide a powerful and efficient shortcut for a problem that would otherwise require a more computationally intensive optimization.

References

[1] (1,2,3)

Nathaniel Johnston, Vincent Russo, and Jamie Sikora. Tight bounds for antidistinguishability and circulant sets of pure quantum states. Quantum, 9:1622, 2025.