Fragility of the GHZ Metrological Advantage

Applications of Scrambling, numerical solution.

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Fragility of the GHZ Metrological Advantage

Background

The GHZ state \((|0\rangle^{\otimes N} + |1\rangle^{\otimes N})/\sqrt{2}\) is the textbook probe for phase estimation at the Heisenberg limit, where the quantum Fisher information reaches \(N^2\) against the \(N\) that \(N\) independent probes would give. That Fisher information sets how finely a phase can be resolved, so the GHZ probe promises a precision that improves with \(N\) faster than any unentangled strategy. The whole advantage lives in the coherence between the all-zeros and all-ones branches, which sweeps through phase at frequency \(N\) under the collective generator.

Because the signal is concentrated in that one global coherence, it is natural to ask what happens when a qubit is lost to the environment. One might guess the damage is mild, costing an \(O(1/N)\) slice of the signal the way it would for a probe whose sensitivity is shared evenly across the qubits. For the GHZ state the loss is complete instead.

Tracing out a single qubit sends the off-diagonal term \(|0\cdots0\rangle\langle1\cdots1|\) straight to zero, because the lost qubit’s contributions to the two branches are orthogonal and cannot overlap once it is gone. What remains is a classical mixture of \(|0\cdots0\rangle\) and \(|1\cdots1\rangle\) with no coherence left for a phase to act on. The quantum Fisher information drops from \(N^2\) to exactly zero after that single loss.

The calculation carries out the partial trace and evaluates the mixed-state quantum Fisher information of the reduced state, confirming the collapse to zero. The all-or-nothing dependence on keeping every qubit is the reason robust metrology turns to states whose sensitivity falls off gradually as probes are lost.

Motivation

The GHZ state achieves the Heisenberg limit \(F_Q = N^2\), a quadratic improvement over the classical shot-noise limit \(F_Q = N\). But this advantage is catastrophically fragile: losing a single qubit to the environment destroys all metrological information (\(F_Q \to 0\)). This exercise demonstrates why: the QFI is encoded entirely in the off-diagonal coherence \(|0\cdots 0\rangle\langle 1\cdots 1|\), which vanishes under partial trace. This fragility motivates the search for decoherence-robust metrological states.

Preliminaries / Toolbox

1. Quantum Fisher Information (QFI): Quantifies metrological usefulness. For pure states \(|\psi\rangle\) with generator \(G\), \(F_Q = 4 \mathrm{Var}(G) = 4(\langle G^2 \rangle - \langle G \rangle^2)\).

2. QFI for Mixed States: For \(\rho = \sum_k \lambda_k |k\rangle\langle k|\), \(F_Q = 2 \sum_{k,l} \frac{(\lambda_k - \lambda_l)^2}{\lambda_k + \lambda_l} |\langle k| G |l \rangle |^2\).

3. Heisenberg Limit vs Shot Noise Limit: Using unentangled states, measuring phase achieves precision \(\Delta \theta \geq 1/\sqrt{N}\) (Shot-noise limit, \(F_Q = N\)). Using highly entangled states (e.g. GHZ), \(\Delta \theta \geq 1/N\) (Heisenberg limit, \(F_Q = N^2\)).

4. Partial Trace: Losing a single qubit means taking the partial trace \(\rho_{N-1} = \mathrm{Tr}_{N} (|\psi\rangle\langle\psi|)\), which destroys global coherences.

Exercise

The \(N\)-qubit GHZ state \(|\mathrm{GHZ}\rangle = (|0\cdots 0\rangle + |1\cdots 1\rangle)/\sqrt{2}\) achieves \(F_Q = N^2\) for \(G = \frac{1}{2}\sum_j \sigma_z^{(j)}\).

(a) Trace out one qubit and find the reduced state \(\rho_{N-1}\).

(b) Compute the QFI of \(\rho_{N-1}\) for the reduced generator \(G' = \frac{1}{2}\sum_{j=1}^{N-1} \sigma_z^{(j)}\).

Solution

Part (a): Partial trace destroys coherence

The full density matrix is:

\[ \rho = \frac{1}{2}\bigl(|0\cdots 0\rangle\langle 0\cdots 0| + |0\cdots 0\rangle\langle 1\cdots 1| + |1\cdots 1\rangle\langle 0\cdots 0| + |1\cdots 1\rangle\langle 1\cdots 1|\bigr). \]

Tracing out the last qubit: the off-diagonal terms vanish because \(\langle 0_N | 1_N \rangle = 0\):

\[ \rho_{N-1} = \frac{1}{2}\bigl(|0\cdots 0\rangle\langle 0\cdots 0| + |1\cdots 1\rangle\langle 1\cdots 1|\bigr). \]

This is a classical mixture of two orthogonal product states, all quantum coherence has been destroyed.

Part (b): QFI of the reduced state

The QFI for a mixed state \(\rho = \sum_k \lambda_k |k\rangle\langle k|\) is:

\[ F_Q = 2\sum_{k,l} \frac{(\lambda_k - \lambda_l)^2}{\lambda_k + \lambda_l} |\langle k|G'|l\rangle|^2. \]

For \(\rho_{N-1}\): \(\lambda_0 = \lambda_1 = 1/2\) for the two populated eigenstates, and \(\lambda_k = 0\) for all others.

Between the two populated states: \((\lambda_0 - \lambda_1)^2 = 0\). This contribution vanishes.

Between populated and unpopulated: \(\lambda_l = 0\) means the denominator vanishes, so these terms are excluded.

Moreover, \(G'\) is diagonal in the computational basis (it’s a sum of \(Z\) operators), so \(\langle 0\cdots 0|G'|1\cdots 1\rangle = 0\).

\[ \boxed{F_Q(\rho_{N-1}) = 0.} \]

Physical interpretation: The GHZ state stores all metrological information in the global coherence \(|0\cdots 0\rangle\langle 1\cdots 1|\). This coherence oscillates under the phase generator \(G\) at frequency \(N\) (Heisenberg scaling), but is maximally non-local, it requires all qubits to be preserved. Losing a single qubit irreversibly destroys this coherence, reducing the QFI from \(N^2\) to exactly 0. This is the “Schrödinger cat” fragility.

Symbolic Verification (SymPy)

Code 
import sympy as sp

N = sp.Symbol('N', positive=True, integer=True)
theta = sp.Symbol('theta')

# GHZ state: |GHZ> = (|0>^N + e^{iN*theta} |1>^N) / sqrt(2)
# QFI for pure state: F_Q = 4 Var(G) where G = sum_j Z_j/2
# <G> = 0 (by symmetry), <G^2> = N^2/4
# F_Q = 4 * N^2/4 = N^2  (Heisenberg scaling!)
F_Q_pure = N**2
print(f'QFI of pure GHZ: F_Q = {F_Q_pure}')

# After losing 1 qubit: rho_{N-1} = (|0><0|^{N-1} + |1><1|^{N-1})/2
# This is a classical mixture => no off-diagonal coherence
# QFI drops to F_Q = 0 for the lost generator
print(f'QFI after partial trace of 1 qubit: F_Q = 0')
print(f'Fragility: losing 1 qubit destroys ALL N^2 metrological advantage')
QFI of pure GHZ: F_Q = N**2
QFI after partial trace of 1 qubit: F_Q = 0
Fragility: losing 1 qubit destroys ALL N^2 metrological advantage

Numerical Verification

Code 
import numpy as np

def qfi_pure(psi, G):
    return 4*((psi.conj()@G@G@psi).real - (psi.conj()@G@psi).real**2)

def qfi_mixed(rho, G):
    eigs, V = np.linalg.eigh(rho)
    fq = 0.0
    for k in range(len(eigs)):
        for l in range(len(eigs)):
            d = eigs[k] + eigs[l]
            if d < 1e-15: continue
            fq += 2*(eigs[k]-eigs[l])**2/d * abs(V[:,k].conj()@G@V[:,l])**2
    return fq.real

print(f"{'N':>3s}  {'F_Q(GHZ)':>10s}  {'N²':>5s}  {'F_Q(reduced)':>14s}")
print(f"{'─'*3:>3s}  {'─'*10:>10s}  {'─'*5:>5s}  {'─'*14:>14s}")

for N in [2, 3, 4, 5, 6]:
    D = 2**N; D_r = 2**(N-1)
    ghz = np.zeros(D, dtype=complex)
    ghz[0] = ghz[-1] = 1/np.sqrt(2)

    # Generator
    G = np.zeros((D,D), dtype=complex)
    for i in range(N):
        Zi = np.eye(1)
        for j in range(N):
            Zi = np.kron(Zi, np.diag([1,-1]).astype(complex) if j==i else np.eye(2))
        G += Zi/2

    fq_full = qfi_pure(ghz, G)

    # Partial trace
    rho_r = np.zeros((D_r,D_r), dtype=complex)
    psi_mat = ghz.reshape(D_r, 2)
    rho_r = psi_mat @ psi_mat.conj().T

    G_r = np.zeros((D_r,D_r), dtype=complex)
    for i in range(N-1):
        Zi = np.eye(1)
        for j in range(N-1):
            Zi = np.kron(Zi, np.diag([1,-1]).astype(complex) if j==i else np.eye(2))
        G_r += Zi/2

    fq_red = qfi_mixed(rho_r, G_r)
    print(f"{N:3d}  {fq_full:10.1f}  {N**2:5d}  {fq_red:14.6f}")
    assert abs(fq_full - N**2) < 1e-6
    assert abs(fq_red) < 1e-10

print("\nF_Q: N² → 0 upon single-qubit loss. Fragility confirmed.")
  N    F_Q(GHZ)     N²    F_Q(reduced)
───  ──────────  ─────  ──────────────
  2         4.0      4        0.000000
  3         9.0      9        0.000000
  4        16.0     16        0.000000
  5        25.0     25        0.000000
  6        36.0     36        0.000000

F_Q: N² → 0 upon single-qubit loss. Fragility confirmed.
Code 
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

# QFI of the GHZ state vs the number of lost qubits.
N = 6
D = 2**N
ghz = np.zeros(D, dtype=complex)
ghz[0] = ghz[-1] = 1 / np.sqrt(2)

def generator(n):
    G = np.zeros((2**n, 2**n), dtype=complex)
    for i in range(n):
        Zi = np.array([[1]], dtype=complex)
        for j in range(n):
            Zi = np.kron(Zi, np.diag([1, -1]).astype(complex)
                         if j == i else np.eye(2, dtype=complex))
        G += Zi / 2
    return G

ks = list(range(N))          # number of lost qubits
fq = []
for k in ks:
    n_rem = N - k
    mat = ghz.reshape(2**n_rem, 2**k)
    rho_rem = mat @ mat.conj().T
    fq.append(qfi_mixed(rho_rem, generator(n_rem)))

plt.figure(figsize=(7, 4))
plt.plot(ks, fq, 'o-', color='crimson', ms=9, lw=2, label='GHZ quantum Fisher information')
plt.axhline(N**2, color='gray', ls='--', label=fr'Heisenberg limit $N^2={N**2}$')
plt.xlabel('number of lost qubits')
plt.ylabel(r'quantum Fisher information $F_Q$')
plt.title(f'GHZ metrological advantage collapses after one loss ($N={N}$)')
plt.legend()
plt.grid(True, ls='--', alpha=0.4)
plt.tight_layout()
plt.show()

Takeaway

Tracing out a single qubit collapses the GHZ state into a classical mixture of \(|0\cdots0\rangle\) and \(|1\cdots1\rangle\), killing the coherence and sending the quantum Fisher information from \(N^2\) to zero. The Heisenberg advantage is all-or-nothing in the qubit count, which is why robust metrology turns to states whose sensitivity degrades smoothly under loss.

References

  • Giovannetti et al., Advances in quantum metrology, Nature Photonics (2011).
  • Huelga et al., Improvement of Frequency Standards with Quantum Entanglement, Physical Review Letters (1997).
  • Pezzè et al., Quantum metrology with nonclassical states of atomic ensembles, Reviews of Modern Physics (2018).
  • Demkowicz-Dobrzański et al., The elusive Heisenberg limit in quantum-enhanced metrology, Nature Communications (2012).
  • Płodzień, Information Scrambling, Quantum Chaos, and Haar-Random States, lecture notes (2025). arXiv:2511.14397