Equiangular States and the Antidistinguishability Threshold

In this tutorial, we explore a sharp threshold for the antidistinguishability of a special class of quantum states known as equiangular states. We will numerically verify a tight bound presented in the paper by Johnson et.al [1] and visualize the “sharp cliff” where this property changes.

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

Antidistinguishability Threshold for Equiangular States

A set of \(n\) pure states \(\{|\psi_0\rangle, \ldots, |\psi_{n-1}\rangle\}\) is called equiangular if the absolute value of the inner product between any two distinct states is a constant, i.e., \(|\langle \psi_i | \psi_j \rangle| = \gamma\) for all \(i \neq j\).

Johnston et.al [1] introduced a simple and powerful necessary condition for a set of states to be antidistinguishable.

According to Corollary 4.2 from [1], when \(n \geq 2\), \(S = \{| \psi_0\rangle, \ldots, |\psi_{n-1}\rangle\}\) is not anstidistinguishable if the following condition is satisfied.

\[|\langle \psi_i | \psi_j \rangle| > \frac{n-2}{n-1} \quad \forall \ i \neq j,\]

Crucially, Example 3.3 in the paper demonstrates that this bound is tight. That is, a set of equiangular states with an inner product exactly equal to the threshold \(\gamma = \frac{n-2}{n-1}\) is antidistinguishable. We can use |toqito⟩ to verify this sharp transition.

Numerical Verification

To demonstrate the tightness of this bound, we follow Example 3.3 from the paper [1]. The Gram matrix for a set of \(n\) equiangular states is given by

\[G = (1 - \gamma) I + \gamma J,\]

where \(I\) is the identity matrix and \(J\) is the all-ones matrix.

We will verify the threshold for the \(n=4\) case, where the critical inner product is \(\gamma_{\text{crit}} = (4-2)/(4-1) = 2/3\). Our verification plan is as follows:

1. Construct the Gram matrix \(G_{\text{at}}\) for \(\gamma = \gamma_{\text{crit}}\).

2. Construct a second Gram matrix \(G_{\text{above}}\) for \(\gamma\) slightly greater than \(\gamma_{\text{crit}}\).

3. For each Gram matrix, use |toqito⟩ to generate a corresponding set of state vectors.

4. For each set of states, compute the minimum probability of error for state exclusion using the state_exclusion() function.

5. Confirm that the states at the threshold are antidistinguishable (error probability is \(0\)) and the states above it are not (error probability is > \(0\)).

import numpy as np

from toqito.matrix_ops import vectors_from_gram_matrix
from toqito.state_opt import state_exclusion

# Define parameters for n=4.
n = 4
gamma_crit = (n - 2) / (n - 1)
# Use a larger epsilon to make the effect numerically obvious.
epsilon = 0.01

print(f"For n={n}, the critical threshold is γ = {gamma_crit:.4f}")

# 1. Construct and test the Gram matrix AT the threshold.
gamma_at = gamma_crit
gram_at = (1 - gamma_at) * np.identity(n) + gamma_at * np.ones((n, n))
states_at = vectors_from_gram_matrix(gram_at)
opt_val_at, _ = state_exclusion(states_at)
is_ad_at = np.isclose(opt_val_at, 0)

print(f"\nFor γ = {gamma_at:.4f} (at threshold):")
print(f"  - Optimal error probability is {opt_val_at:.2e}")
print(f"  - Is the set antidistinguishable? {is_ad_at} (as expected)")

# 2. Construct and test the Gram matrix slightly ABOVE the threshold.
gamma_above = gamma_crit + epsilon
gram_above = (1 - gamma_above) * np.identity(n) + gamma_above * np.ones((n, n))
states_above = vectors_from_gram_matrix(gram_above)
opt_val_above, _ = state_exclusion(states_above)
is_ad_above = np.isclose(opt_val_above, 0)

print(f"\nFor γ = {gamma_above:.4f} (above threshold):")
print(f"  - Optimal error probability is {opt_val_above:.2e}")
print(f"  - Is the set antidistinguishable? {is_ad_above} (as expected)")

For n=4, the critical threshold is γ = 0.6667

For γ = 0.6667 (at threshold):
  - Optimal error probability is 5.11e-11
  - Is the set antidistinguishable? True (as expected)

For γ = 0.6767 (above threshold):
  - Optimal error probability is 7.58e-05
  - Is the set antidistinguishable? False (as expected)

Antidistinguishability and (n-1)-Incoherence

The core theoretical result of (Theorem 3.2) [1] is that a set of \(n\) pure states is antidistinguishable if and only if its Gram matrix is \((n-1)\)-incoherent. Our numerical results above, obtained by solving the state exclusion SDP, implicitly verify this property for the Gram matrix as well.

Visualizing the Threshold

We can make this “sharp cliff” even clearer by plotting the optimal error probability of state exclusion against the inner product \(\gamma\). To match the style of Figure 2 from [1], we will plot this for several values of \(n\).

The value returned by state_exclusion() is the optimal probability of error. The plot should show this probability lifting off from \(0\) precisely at the threshold \(\gamma_{\text{crit}} = (n-2)/(n-1)\) for each respective \(n\).

A Gram matrix for equiangular states is positive semidefinite (and thus physically valid) if and only if \(-1/(n-1) \leq \gamma \leq 1\).

import matplotlib.pyplot as plt

from toqito.matrix_props import is_positive_semidefinite

fig, ax = plt.subplots(figsize=(8, 5), dpi=100)
gamma_range = np.linspace(0, 0.999, 101)
n_vals_to_plot = [2, 3, 4, 5, 10]

for n_val in n_vals_to_plot:
    error_probs = []
    gamma_crit_n = (n_val - 2) / (n_val - 1)

    for gamma in gamma_range:
        # The Gram matrix is only PSD in a specific range.
        if gamma < -1 / (n_val - 1):
            error_probs.append(np.nan)
            continue

        gram_matrix = (1 - gamma) * np.identity(n_val) + gamma * np.ones((n_val, n_val))

        if is_positive_semidefinite(gram_matrix):
            states = vectors_from_gram_matrix(gram_matrix)
            opt_val, _ = state_exclusion(states)
            # The returned optimal value is the error probability.
            error_probs.append(opt_val)
        else:
            # If not PSD, it's not a valid Gram matrix for a state ensemble.
            error_probs.append(np.nan)

    ax.plot(gamma_range, error_probs, label=f"$n={n_val}$ ($\\gamma_{{crit}} \\approx {gamma_crit_n:.2f}$)")


ax.set_xlabel("Inner Product $\\gamma = |\\langle \\psi_i | \\psi_j \\rangle|$", fontsize=12)
ax.set_ylabel("Optimal Exclusion Error Probability", fontsize=12)
ax.set_title("Antidistinguishability Threshold for Equiangular States", fontsize=14)
ax.legend(fontsize=10)
ax.grid(True)
ax.set_ylim(bottom=-0.01)
plt.tight_layout()
plt.show()

This plot, which numerically reproduces the results from Figure 2 of [1], shows that the optimal probability of error is exactly \(0\) for all \(\gamma \leq (n-2)/(n-1)\), indicating that the states are perfectly antidistinguishable. The moment \(\gamma\) exceeds this value, the probability becomes non-zero, meaning perfect exclusion is no longer possible.

References

[1] (1,2,3,4,5,6,7)

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