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Tuesday, June 23, 2026

Mermin’s Inequality: How to Catch a Computer Being Quantum

A single number that tells you whether a machine is genuinely using quantum mechanics, or just doing a convincing classical impression of it.

What the inequality measures

The Mermin inequalityA test of whether a system is truly quantum: you score a fixed set of measurements on several entangled qubits, and a result above its classical limit proves the qubits are using real quantum entanglement, not hidden classical instructions. is a test. You take a group of particles, measure each one, combine the answers in a fixed way, and read off a single number. If that number stays below a set limit, everything you saw could have come from an ordinary classical system: particles carrying hidden instructions, agreed in advance, like actors following a script. If it climbs above the limit, no such script can exist, and the machine is doing something only quantum mechanics allows.1

Below the limit, a classical script explains everything you saw. Above it, nothing classical can.

It is a sharpened version of an older result. Bell’s theorem showed, for two particles, that quantum correlations can break what classical physics permits. Mermin took the same idea to many particles at once, and found that the gap grows quickly as you add more.1

What is really on trial is local realismThe classical assumption that things have definite properties before measurement and are influenced only by their immediate surroundings.. Two everyday assumptions sit underneath it: realism, that a particle has definite properties before anyone looks, and locality, that nothing you do in one place can instantly change a result somewhere far away.2 Common sense says both must hold. For entangled particles, quantum mechanics says you cannot keep both, and the Mermin inequality is how you catch the contradiction in the act.

Why more particles make it sharper

The test runs on a GHZ stateA maximally entangled state of three or more qubits: an all-or-nothing superposition of all-zeros and all-ones. If one qubit is lost or ignored, the remaining GHZ entanglement is destroyed., a group of qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. prepared so they are all 0 and all 1 at the same time, a single superpositionThe quantum ability of a qubit to be in a combination of 0 and 1 simultaneously, which gives quantum computers their power. written (|000…0⟩ + |111…1⟩)/√2.3 You measure each qubit along one of two directions and add the results into the particular sum Mermin built for the job.

Here is the fact that makes it useful: the classical ceiling and the quantum ceiling pull apart as the system grows. For three particles, classical physics caps the score at 2 while quantum mechanics reaches 4. For five particles, the classical limit sits near 4 and quantum mechanics reaches 16.4 The more qubits you genuinely entangle, the harder the result is to fake.

A line chart of Mermin score against the number of qubits, 3 to 5. The quantum maximum climbs steeply (4, 11, 16) far above a flat dashed classical limit, while the real-chip measurements clear the limit at 3 and 4 qubits but fall back below it at 5, where the lost violation is circled in red.
Figure 1: In theory the quantum ceiling races away from the classical limit as qubits are added. On real hardware the chip clears the limit at three and four qubits, then falls back below it at five, where decoherence wins.

What happens on real hardware

That is the theory. Real machines are messier. In 2016 two physicists ran the test on IBM’s five-qubit chip, cooled to a hair above absolute zeroThe coldest temperature physically possible (about -273.15 Celsius), where atomic motion almost stops.. With three qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. they measured 2.85, comfortably past the classical limit of 2. With four, 4.81, just over the line. With five, 4.05, a number you cannot tell apart from an ordinary classical result.4 The quantum-ness had drained away.

Three qubits beat the classical bound with room to spare. Five qubits could not beat it at all.

The culprit is decoherenceThe loss of a qubit's quantum information as it interacts with the outside world. The main reason qubits are fragile., the slow leak of quantum information into the surroundings. Every extra qubit demands more operations, and every operation costs time the fragile state does not have.4

The benchmark you cannot bluff

This is what makes the inequality more than a laboratory curiosity. It is a figure of merit: a way to ask whether a quantum computer is genuinely using entanglementA quantum link where two qubits' states become tied together, so acting on or measuring one affects the other. It is a key resource for quantum computing., or merely burning electricity. A machine that cannot beat the Mermin bound is, for this purpose, an expensive classical simulator.2

So as companies add qubits, the honest question is not how many they have, but whether those qubits still break the inequality together. Counting qubits is easy. Keeping them quantum is the whole game.

Sources

  1. Mermin, N. D. “Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States.” Physical Review Letters 65, 1838–1840 (1990). DOI: 10.1103/PhysRevLett.65.1838.
  2. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S. “Bell Nonlocality.” Reviews of Modern Physics 86, 419–478 (2014). DOI: 10.1103/RevModPhys.86.419. Preprint: arXiv:1303.2849.
  3. Greenberger, D. M., Horne, M. A., Shimony, A. & Zeilinger, A. “Bell’s Theorem without Inequalities.” American Journal of Physics 58, 1131–1143 (1990). DOI: 10.1119/1.16243.
  4. Alsina, D. & Latorre, J. I. “Experimental Test of Mermin Inequalities on a Five-Qubit Quantum Computer.” Physical Review A 94, 012314 (2016). DOI: 10.1103/PhysRevA.94.012314. Preprint: arXiv:1605.04220.