The GHZ State and All-or-Nothing Nonlocality

Bell's theorem beats local realism the way a casino beats a gambler: on average, over many runs, the house edge shows up. The CHSH correlator creeps past its classical ceiling of $2$ only after you accumulate statistics, and a stubborn skeptic can always mutter about detector flukes and finite samples. In 1989 Daniel Greenberger, Michael Horne and Anton Zeilinger found something sharper. Take three entangled qubits instead of two, and quantum mechanics and local realism no longer merely differ on average — they predict exactly opposite outcomes for one specific measurement, with certainty. Local realism says a particular product of spins must be $+1$ ; quantum mechanics guarantees $-1$ . A single ideal run settles it. No inequality, no margin, no statistics. This is why the GHZ argument is called "all-or-nothing." It converts the EPR programme — the idea that a value you can predict without disturbing a system must have been real all along — into a flat logical contradiction. Each of the three observers can predict their own outcome from the distant results of the other two, so by EPR's own criterion every outcome was predetermined. Yet those predetermined values, assembled into the all- $X$ experiment, cannot exist: the arithmetic of $\pm1$ forbids it. Where EPR argued quantum mechanics was incomplete, GHZ shows that completing it with local elements of reality is impossible. You are ready for this because you have met the singlet's perfect anti-correlations and the Bell/CHSH inequality; GHZ is what happens when three-body entanglement replaces two-body correlation and a deterministic clash replaces a statistical one. Mastering it gives you the cleanest refutation of local realism in all of physics, the workhorse resource of multi-party quantum protocols, and a first taste of why tripartite entanglement has a richer structure than the bipartite kind.

Key takeaways

The GHZ state $|\text{GHZ}\rangle=(|000\rangle-|111\rangle)/\sqrt2$ is a three-qubit maximally entangled state on which four commuting spin-products have definite values.
Quantum mechanics fixes $XXX=-1$ and $XYY=YXY=YYX=+1$ with certainty; these are joint eigenvalue equations, not averages.
Any local assignment of pre-existing $\pm1$ values is forced by the three " $+1$ " equations to predict $XXX=+1$ — the exact opposite of the quantum $-1$ .
Unlike Bell/CHSH, the refutation is deterministic: a single perfect run gives opposite predictions with probability one, so local realism dies "all-or-nothing," not statistically.
GHZ entanglement is genuinely tripartite yet fragile — trace out any one qubit and the survivors share only classical correlation (concurrence $0$ ), unlike the robust W state.
Key idea: for the GHZ state the local-realist and quantum predictions for $X_1X_2X_3$ are deterministically opposite — $+1$ versus $-1$ — so one ideal measurement, not a statistical inequality, refutes local hidden variables.

The core insight

For the GHZ state the local-realist and quantum predictions for the product $X_1X_2X_3$ are deterministically opposite ( $+1$ vs $-1$ ): a single perfect measurement run, not a statistical inequality, refutes local hidden variables.

Where this lives today

The GHZ state graduated from paradox to standard equipment. Trapped-ion and superconducting processors now assemble GHZ states of tens of qubits as benchmarks of genuine multipartite entanglement, and quantum metrology exploits them directly: an $n$ -qubit GHZ probe reaches Heisenberg-limited phase sensitivity $\Delta\phi\sim1/n$ , beating the standard quantum limit $1/\sqrt n$ that unentangled probes are stuck with (a factor of $\sqrt{10}\approx3.2$ improvement already at ten qubits). The same rigid three-party correlations power quantum secret sharing, where a message is split so that all parties must cooperate to read it, and multiparty "conference key" distribution. The deeper legacy is conceptual. GHZ made "all-or-nothing" nonlocality a template: from it grew the Kochen–Specker and Peres–Mermin state-independent contradictions, the study of which multipartite entangled states are "genuinely" $n$ -body, and the SLOCC classification that sorts three qubits into the inequivalent GHZ and W families. When you see a modern claim that some resource exhibits nonlocality "without inequalities," you are looking at a descendant of the 1989 argument.

The physical picture

Picture three observers — call them Alice, Bob and Carol — each holding one qubit of a shared triplet, sitting in three distant labs. Each freely chooses to measure either $X$ (the Pauli- $x$ spin component) or $Y$ (Pauli- $y$ ). They will compare notes afterward. Quantum mechanics makes two ironclad promises. First: whenever exactly two of them measure $Y$ and the third measures $X$ , the three $\pm1$ outcomes multiply to $+1$ — always, with no exceptions. Second: when all three measure $X$ , the outcomes multiply to $-1$ — again with certainty. Now run EPR's reasoning. Because each observer can predict their own $X$ -outcome perfectly from the other two observers' $Y$ -outcomes (which are made in distant labs and cannot influence them), that $X$ -outcome must be an "element of reality," a value the qubit carried all along. The same holds for every observer and both settings. So local realism grants each qubit a fixed little table: a value $m_x^{(i)}=\pm1$ for $X_i$ and a value $m_y^{(i)}=\pm1$ for $Y_i$ , decided before anyone measures. Here is the trap. Feed those fixed values into the all- $X$ experiment. The three "two- $Y$ " promises force the product $m_x^{(1)}m_x^{(2)}m_x^{(3)}$ to equal $+1$ (each $Y$ -value appears twice and squares away). But quantum mechanics guarantees the all- $X$ product is $-1$ . The predetermined values cannot exist. Nothing here is statistical — it is arithmetic over $\{+1,-1\}$ , and it fails by a sign.

Figure

The all-or-nothing clash: three settings, one impossible product

Figure

The Mermin margin: quantum value sits far above the local ceiling

The core equations

The GHZ state. The three-qubit resource is a balanced superposition of "all-up" and "all-down," carrying a relative minus sign. The sign is not cosmetic: it is exactly what routes the paradox onto the all- $X$ product. Written in the computational basis,

|\text{GHZ}\rangle=\frac{1}{\sqrt2}\big(|000\rangle-|111\rangle\big) \qquad (1)

where: $|0\rangle,|1\rangle$ are the single-qubit $Z$ -eigenstates (spin- $z$ up/down); $|abc\rangle\equiv|a\rangle_1\otimes|b\rangle_2\otimes|c\rangle_3$ . Reading it physically: the state is a coherent "all three the same" superposition — measuring $Z$ on any qubit instantly fixes the other two, but phase-sensitive $X,Y$ measurements reveal the genuinely quantum correlations below.

The spin observables. Each observer's meter is one of two Pauli operators, each with spectrum $\{+1,-1\}$ :

X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix} \qquad (2)

where: $X=\sigma_x$ , $Y=\sigma_y$ ; $X_i$ denotes $X$ on qubit $i$ and the identity elsewhere. The two matter because $Y|0\rangle=i|1\rangle$ and $Y|1\rangle=-i|0\rangle$ each drag in a factor of $i$ — and two factors of $i$ make the crucial $-1$ .

The four certain products. On $|\text{GHZ}\rangle$ the three "two- $Y$ , one- $X$ " products and the all- $X$ product are simultaneous eigenvalue equations — all four operators commute, so the state has definite values for every one at once:

X_1Y_2Y_3=Y_1X_2Y_3=Y_1Y_2X_3=+1,\qquad X_1X_2X_3=-1 \qquad (3)

where: each equation means " $|\text{GHZ}\rangle$ is an eigenstate of that operator with the eigenvalue shown." The physical content: any observer can predict their own outcome with certainty from the two distant partners — the hallmark that triggers EPR's "element of reality" claim.

The operator identity — read the sign carefully. Multiplying the three two- $Y$ operators, and collecting the factors qubit by qubit ( $Y^2=\mathbb1$ , operators on different qubits commute), gives a single-sign relation between the operators themselves:

(X_1Y_2Y_3)(Y_1X_2Y_3)(Y_1Y_2X_3)=-\,X_1X_2X_3 \qquad (4)

where: the minus is essential and easy to lose. On qubit 2 the factors appear in the order $Y_2X_2Y_2=-X_2$ (since $YXY=-X$ ), while qubits 1 and 3 give $+X$ ; the product is $X_1\otimes(-X_2)\otimes X_3=-X_1X_2X_3$ . This operator identity is the quantum consistency check: with eigenvalues, $(-1)=-(+1)(+1)(+1)$ ✓. It is also where local realism dies, because a local model must use the eigenvalue equations $(3)$ directly, and those alone force the clash.

Common trapThe base's operator identity dropped a sign — and it matters

Several textbook presentations (and this article's AllQFT source) write the operator identity as $X_1X_2X_3=(X_1Y_2Y_3)(Y_1X_2Y_3)(Y_1Y_2X_3)$ — with a plus sign. That is wrong: the correct relation is $X_1X_2X_3=-(X_1Y_2Y_3)(Y_1X_2Y_3)(Y_1Y_2X_3)$ , because $Y_2X_2Y_2=-X_2$ contributes a minus on the middle qubit. With the plus version the operator eigenvalues would read $-1\overset{?}{=}(+1)^3=+1$ inside quantum mechanics itself — an internal contradiction, not a QM-vs-LHV one. The GHZ paradox does not rely on this operator identity at all: it comes from the three eigenvalue equations $(3)$ fed into a local value assignment. So the argument survives the typo, but the identity you write down should carry the minus. (Verified: $XYY\cdot YXY\cdot YYX=-XXX$ symbolically; `verify_aq-qm-ghz-nonlocality.py`.)

Derivation AThe GHZ all-or-nothing contradiction

We turn EPR's "elements of reality" into a deterministic clash with quantum mechanics. The strategy: take the four certain quantum eigenvalues of equation $(3)$ as given (they are proved by direct action of the Pauli operators on $|\text{GHZ}\rangle$ ), grant every qubit a pre-existing local value for both $X$ and $Y$ as EPR's locality argument demands, and then show the three " $+1$ " equations alone force a prediction for the all- $X$ product that quantum mechanics flatly denies. Watch the one subtle move: each $Y$ -value must appear an even number of times so it squares to unity — that is the hinge, and it is exact arithmetic over $\{+1,-1\}$ , with no probabilities anywhere.

Step 1. Establish the quantum eigenvalues. Acting with $X$ (bit-flip) and $Y$ ( $Y|0\rangle=i|1\rangle$ , $Y|1\rangle=-i|0\rangle$ ) on $|\text{GHZ}\rangle=\tfrac1{\sqrt2}(|000\rangle-|111\rangle)$ gives, for the all- $X$ product, $X_1X_2X_3|\text{GHZ}\rangle=\tfrac1{\sqrt2}(|111\rangle-|000\rangle)=-|\text{GHZ}\rangle$ , so $X_1X_2X_3=-1$ . For $Y_1Y_2X_3$ : the two $Y$ 's on $|000\rangle$ give $i^2|111\rangle=-|111\rangle$ and on $|111\rangle$ give $(-i)^2|000\rangle=-|000\rangle$ , so $Y_1Y_2X_3|\text{GHZ}\rangle=\tfrac1{\sqrt2}(-|111\rangle+|000\rangle)=+|\text{GHZ}\rangle$ , i.e. $+1$ ; the two cyclic partners follow by symmetry. Why the minus sign in the state matters here: it is precisely what turns the two- $Y$ products into $+1$ while keeping the all- $X$ product at $-1$ . The plus-state $(|000\rangle+|111\rangle)/\sqrt2$ gives the mirror pattern $XXX=+1$ , $XYY=YXY=YYX=-1$ — the same paradox, opposite labels.

Step 2. Grant elements of reality. Because each observer predicts their outcome with certainty from the two distant partners, EPR-locality assigns each qubit fixed values $m_x^{(i)},m_y^{(i)}\in\{+1,-1\}$ for $X_i$ and $Y_i$ , existing before and independent of which meter is actually used.

Step 3. Impose the three certain "two- $Y$ " results on these values:

m_x^{(1)}m_y^{(2)}m_y^{(3)}=+1,\quad m_y^{(1)}m_x^{(2)}m_y^{(3)}=+1,\quad m_y^{(1)}m_y^{(2)}m_x^{(3)}=+1 .

Step 4. Multiply all three equations together:

\big[m_y^{(1)}\big]^2\big[m_y^{(2)}\big]^2\big[m_y^{(3)}\big]^2\; m_x^{(1)}m_x^{(2)}m_x^{(3)}=+1 .

Why this is the whole game: across the three equations each $Y$ -value appears exactly twice — qubit 1 in equations 2 and 3, qubit 2 in equations 1 and 3, qubit 3 in equations 1 and 2 — so every $m_y$ is squared.

Step 5. Use $(\pm1)^2=1$ . Every $Y$ -factor collapses to unity, leaving the local-realist prediction

m_x^{(1)}m_x^{(2)}m_x^{(3)}=+1 .

Step 6. Compare with quantum mechanics. But equation $(3)$ says $X_1X_2X_3=-1$ with certainty. So local realism predicts $+1$ and quantum mechanics guarantees $-1$ for the same all- $X$ experiment. No table $\{m_x^{(i)},m_y^{(i)}\}$ can satisfy all four relations; an exhaustive search over the $2^6=64$ assignments finds zero solutions. Local hidden variables are refuted by a single ideal run. Verified: all four eigenvalues, the parity collapse, and the exhaustive "no assignment exists" search are machine-checked in `verify_aq-qm-ghz-nonlocality.py` (0 of 64 assignments satisfy the four relations; the forced LHV value is $+1$ in every consistent case).

Go deeper · Derivation B · 1 minGHZ entanglement is genuinely tripartite but fragile

Derivation BGHZ entanglement is genuinely tripartite but fragile

We now show the flip side of GHZ's power: it is a purely three-body resource. Lose any single qubit and the two survivors share nothing quantum — only classical correlation. The plan is to compute the reduced two-qubit density operator by tracing out one party, watch the coherences vanish by orthogonality, and read off that the leftover is a separable mixture (concurrence zero). This is what "genuinely tripartite" means operationally, and it sharply distinguishes GHZ from the W state, whose pairwise entanglement survives particle loss.

Step 1. Form the pure-state projector $\rho=|\text{GHZ}\rangle\langle\text{GHZ}|=\tfrac12(|000\rangle\langle000|-|000\rangle\langle111|-|111\rangle\langle000|+|111\rangle\langle111|)$ . The cross terms carry the entanglement.

Step 2. Trace out qubit 3: $\rho_{12}=\sum_{k=0,1}\langle k|_3\,\rho\,|k\rangle_3$ . The cross terms $|000\rangle\langle111|$ have qubit-3 factors $|0\rangle_3\langle1|_3$ , and $\langle k|0\rangle\langle1|k\rangle=0$ for both $k$ , so every coherence dies. Why the coherences vanish: qubit 3's part of $|000\rangle$ and $|111\rangle$ are orthogonal, so tracing it out erases any interference between the two branches — the entanglement was stored in that global coherence.

Step 3. The result is

\rho_{12}=\tfrac12|00\rangle\langle00|+\tfrac12|11\rangle\langle11| ,

a classical mixture of product states. Its concurrence is $C(\rho_{12})=0$ — no entanglement remains.

Step 4. Contrast the W state $|\text{W}\rangle=\tfrac1{\sqrt3}(|001\rangle+|010\rangle+|100\rangle)$ : tracing out one qubit leaves $\rho_{12}$ with concurrence $2/3>0$ . So W entanglement is robust to particle loss but yields no all-or-nothing paradox, while GHZ is fragile but maximally nonlocal. These are the two inequivalent classes of genuine tripartite entanglement — a fact tied to the absence of a Schmidt decomposition for three parties (see The Schmidt Decomposition, which shows tripartite pure states generically have no Schmidt form and GHZ ≠ W are inequivalent classes). Verified: the partial trace, $C(\rho_{12})=0$ for GHZ, and $C(\rho_{12})=2/3$ for W are computed numerically in `verify_aq-qm-ghz-nonlocality.py`.

Limiting cases

GHZ sits at a special point on the ladder of multiparty nonlocality tests.

case	result	meaning
two qubits (Bell/CHSH)	$\lvert S\rvert\le2$ (LHV) vs $2\sqrt2$ (QM)	no deterministic clash exists for two parties — you need the statistical CHSH inequality (see Bell's Theorem and the CHSH Inequality).
three qubits (GHZ, this article)	$X_1X_2X_3=-1$ (QM) vs $+1$ (LHV)	all-or-nothing: one ideal run gives opposite predictions with certainty.
W state (other tripartite class)	$\rho_{12}$ has concurrence $2/3$	genuinely entangled but not GHZ-paradoxical; nonlocality is statistical and pairwise entanglement survives particle loss.
$n$ qubits (Mermin operator)	$\langle M_n\rangle\le 2$ (LHV) vs $4$ for $n=3$ (QM)	the Mermin operator $M=XYY+YXY+YYX-XXX$ turns the deterministic clash into a noise-tolerant statistical test with a wide margin.

Worked examples

Example 1 — Run the paradox by hand. Take $|\text{GHZ}\rangle=(|000\rangle-|111\rangle)/\sqrt2$ and confirm the clash from scratch, then count the impossible assignments.

Step 1. Compute $X_1X_2X_3|\text{GHZ}\rangle$ . Each $X$ flips its bit, so $|000\rangle\to|111\rangle$ and $|111\rangle\to|000\rangle$ , giving $\tfrac1{\sqrt2}(|111\rangle-|000\rangle)=-|\text{GHZ}\rangle$ . Hence $X_1X_2X_3=\mathbf{-1}$ .

Step 2. Compute $Y_1Y_2X_3|\text{GHZ}\rangle$ . On $|000\rangle$ : $(i|1\rangle)(i|1\rangle)(|1\rangle)=i^2|111\rangle=-|111\rangle$ . On $|111\rangle$ : $(-i|0\rangle)(-i|0\rangle)(|0\rangle)=(-i)^2|000\rangle=-|000\rangle$ . So $Y_1Y_2X_3|\text{GHZ}\rangle=\tfrac1{\sqrt2}(-|111\rangle+|000\rangle)=+|\text{GHZ}\rangle$ : eigenvalue $\mathbf{+1}$ . The two cyclic partners give $+1$ by symmetry.

Step 3. Impose the LHV equations $m_x^{(1)}m_y^{(2)}m_y^{(3)}=m_y^{(1)}m_x^{(2)}m_y^{(3)}=m_y^{(1)}m_y^{(2)}m_x^{(3)}=+1$ and multiply. Each $m_y$ squares to $1$ , so $m_x^{(1)}m_x^{(2)}m_x^{(3)}=+1$ .

Step 4. Compare. Local realism forces $X_1X_2X_3=+1$ ; quantum mechanics gives $-1$ . An exhaustive scan of all $2^6=64$ tables $\{m_x^{(i)},m_y^{(i)}\}$ finds exactly $0$ that reproduce all four eigenvalues. The refutation is complete and needs no averaging. Answer: local realism and QM predict $+1$ vs $-1$ ; zero of 64 local assignments work.

Example 2 — The Mermin operator as a noise-tolerant test. Real detectors are imperfect, so experiments measure the Mermin operator $M=X_1Y_2Y_3+Y_1X_2Y_3+Y_1Y_2X_3-X_1X_2X_3$ and compare $\langle M\rangle$ against the local bound.

Step 1. Quantum value. On $|\text{GHZ}\rangle$ each two- $Y$ term contributes $+1$ and the $-XXX$ term contributes $-(-1)=+1$ , so $\langle M\rangle=1+1+1+1=\mathbf{4}$ .

Step 2. Local-realist bound. For any local assignment, $M$ 's value is $m_x^{(1)}m_y^{(2)}m_y^{(3)}+m_y^{(1)}m_x^{(2)}m_y^{(3)}+m_y^{(1)}m_y^{(2)}m_x^{(3)}-m_x^{(1)}m_x^{(2)}m_x^{(3)}$ . Scanning all $64$ sign choices, the maximum is $\mathbf{2}$ . (A quick way to see it: the four terms cannot all be $+1$ simultaneously, exactly the impossibility above.)

Step 3. The gap. QM overshoots the local ceiling by a factor of $2$ : $\langle M\rangle=4$ versus $\le2$ . Because this is now an expectation value, a GHZ experiment with, say, $85\%$ fidelity still reports $\langle M\rangle\approx0.85\times4\approx3.4>2$ — a clean violation. Answer: $\langle M\rangle_{\text{QM}}=4$ against the local bound $2$ , a factor- $2$ margin that survives realistic noise.

Origins · 1 minAnton Zeilinger and how this was discovered

Anton Zeilinger

1945– · Vienna — experimental quantum information

Anton Zeilinger co-invented the GHZ argument in 1989 with Daniel Greenberger and Michael Horne, then spent the following decade turning entanglement from a thought experiment into laboratory hardware — three-photon GHZ states, quantum teleportation, and the entanglement-based tests that pushed Bell violations toward loophole-free. He shared the 2022 Nobel Prize in Physics with Alain Aspect and John Clauser "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science."

History

1935

— Einstein, Podolsky and Rosen state the "elements of reality" criterion and argue quantum mechanics is incomplete

1964

— Bell derives his statistical inequality, converting the EPR debate into an experiment

1989

— Greenberger, Horne and Zeilinger present the all-or-nothing three-particle argument in the Kafatos conference volume

1990

— Mermin popularizes the three-qubit form and introduces the noise-tolerant Mermin operator

2000

— Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger measure a three-photon GHZ state, confirming the quantum predictions

2022

— Aspect, Clauser and Zeilinger share the Nobel Prize for entanglement and Bell-inequality experiments

Going further

The GHZ construction extends the perfect anti-correlation of the singlet (see Quantum Entanglement and Non-Separability and The EPR Paradox and Local Realism) from two particles to three, replacing the statistical Bell/CHSH inequality (Bell's Theorem and the CHSH Inequality) with a deterministic contradiction; the GHZ ≠ W inequivalence follows from the absence of a tripartite Schmidt form (The Schmidt Decomposition). Its state-independent successors live in The Kochen–Specker Theorem and Contextuality. Provenance: upgraded from the AllQFT `qm` base article "The GHZ State and All-or-Nothing Nonlocality" (Sakurai & Napolitano §3.10–3.11; Mermin 1990; Nielsen & Chuang §2.4). Base operator-identity sign corrected ( $XXX=-XYY\cdot YXY\cdot YYX$ ); GHZ-state convention fixed to $(|000\rangle-|111\rangle)/\sqrt2$ throughout so the stated eigenvalue pattern ( $XXX=-1$ , others $+1$ ) is internally consistent. All eigenvalues, the exhaustive no-assignment search, and both concurrences machine-verified. Prerequisites: Quantum Entanglement and Non-Separability · The EPR Paradox and Local Realism · Bell's Theorem and the CHSH Inequality. Leads to: The Kochen–Specker Theorem and Contextuality · The Schmidt Decomposition.

Does the GHZ argument depend on the plus or minus sign in the state?

The sign choice just relabels which product is $-1$ . With $|\text{GHZ}\rangle=(|000\rangle-|111\rangle)/\sqrt2$ you get $XXX=-1$ and $XYY=YXY=YYX=+1$ ; with the plus state you get the mirror pattern $XXX=+1$ and the three two- $Y$ products $=-1$ . Either way, the three "same-sign" equations multiply to force a local-realist value for the fourth product that is the exact opposite of the quantum value. The paradox is identical; only the labels swap. What you must not do is mix conventions within one calculation — that is the internal inconsistency the base article fell into.

Why do we need three particles — what stops a two-particle GHZ paradox?

With two qubits there simply is no set of commuting local products whose eigenvalue equations force a contradiction with a local value assignment; the best you can do is the statistical CHSH inequality, violated by the finite margin $2\sqrt2$ versus $2$ . The third particle is what lets you write down enough certain product relations (three "two- $Y$ " equations) that, multiplied together, over-determine the all- $X$ product. Three is the minimum for the deterministic, single-run refutation.

If each observer can predict their outcome perfectly, why isn't that a hidden variable proving the value was real?

That is precisely EPR's argument — and GHZ shows it leads to contradiction. Each observer can predict their outcome from the two distant results, so by EPR's criterion the value seems real and pre-existing. GHZ's punchline is that these "real" values, though each individually well-defined, cannot be assembled into a globally consistent table: the four eigenvalue equations have no simultaneous $\pm1$ solution. So the elements of reality do not exist, and the flaw is the assumption of local, non-contextual predetermined values.

Is the GHZ paradox the same as contextuality?

They are close cousins. GHZ is a nonlocality proof: the impossible value assignment is ruled out by locality (values fixed independently on spacelike-separated qubits). Contextuality (the Kochen–Specker theorem and the Peres–Mermin square) drops the spatial separation and shows a single system cannot carry non-contextual predetermined values either. GHZ is historically the bridge — it is an "all-or-nothing" clash like the Peres–Mermin square, but with the extra structure of spacelike separation that makes the failure specifically about locality.

Why does tracing out one qubit destroy all the entanglement?

Because GHZ entanglement lives entirely in the global coherence between $|000\rangle$ and $|111\rangle$ . When you trace out one qubit, its two branch-states $|0\rangle$ and $|1\rangle$ are orthogonal, so the interference term $|000\rangle\langle111|$ has a factor $\langle k|0\rangle\langle1|k\rangle=0$ and vanishes for every $k$ . What is left, $\tfrac12(|00\rangle\langle00|+|11\rangle\langle11|)$ , is a classical mixture — perfectly correlated but not entangled (concurrence $0$ ). This "fragility" is the signature of GHZ-type entanglement, in contrast to the W state.

How is the GHZ state actually measured in an experiment?

Prepare many copies (three-photon polarization GHZ states, or ion/superconducting qubits), and for each copy have the three parties measure $X$ or $Y$ in the four combinations $XXX$ , $XYY$ , $YXY$ , $YYX$ . Histogram the $\pm1$ product for each setting. Ideal quantum mechanics predicts the products $-1,+1,+1,+1$ with certainty; real data give these values with high probability, and the noise-tolerant figure of merit is $\langle M\rangle$ for the Mermin operator, which must exceed $2$ to beat local realism. Pan et al. (2000) reported agreement with the quantum predictions and violation of the local-realist bound.

Why does an n-qubit GHZ state help in metrology?

Phase sensitivity improves because the GHZ superposition accumulates phase $n$ times faster than a single qubit: the relative phase between the $|0\cdots0\rangle$ and $|1\cdots1\rangle$ branches is $n\phi$ . This pushes the smallest resolvable phase from the standard quantum limit $\Delta\phi\sim1/\sqrt n$ (independent probes) down to the Heisenberg limit $\Delta\phi\sim1/n$ . The same rigid all-parties-correlated structure that makes GHZ maximally nonlocal is what makes it a maximally sensitive interferometric probe — at the cost of fragility to the loss of even one qubit.