Entanglement & Measurement

A composite state is entangled when it cannot be written as a product across the stated partition. For a pure bipartite state, Schmidt rank greater than one establishes entanglement; a mixed state requires a mixed-state criterion, not the same shortcut. Bell states are maximally entangled two-qubit states: each qubit alone is maximally mixed even while joint measurements have strong basis-dependent correlations. Correlation by itself is not sufficient evidence because separable mixtures can also correlate outcomes.

Measurement converts a quantum state into a classical outcome distribution according to the Born rule. Projective measurements use orthogonal projectors that resolve identity; general-measurement effects are positive semidefinite and sum to identity, while measurement instruments additionally define state update. The selected basis, subsystem ordering, outcome labels, post-selection, and whether the measured system remains available are part of the semantics.

Entanglement does not enable faster-than-light signalling: the local reduced state and its outcome statistics do not change with a remote party's measurement choice when the remote result is unavailable. Bell or CHSH violations instead reject a class of local hidden-variable explanations under explicit experimental assumptions. The standard local-hidden-variable bound for absolute CHSH S is two. A CHSH claim needs independently randomized settings, registered trials including losses, uncertainty analysis, and controls for locality, detection, memory, calibration, and selection loopholes.

Production experiments must retain the preparation circuit, partition, basis rotations, target and layout, shots, raw counts, discarded trials, calibration, randomization, estimator definitions, mitigation, and confidence method. The production invariant is measurement-accountable correlation: every entanglement claim is tied to a valid state criterion or preregistered witness, and every reported statistic can be regenerated from complete trials without confusing post-selection, noise, or classical correlation for nonclassical behavior.

A CHSH claim is a statistics claim, and the arithmetic decides whether it survives. The local bound on the absolute value of S is 2 and the quantum maximum is 2 times the square root of 2, about 2.828, so the entire experimental window is 0.83 wide and every error budget has to be read against that width. Each of the four correlators is an average of plus and minus one outcomes, so with 8,192 trials per setting its standard error is about 0.011, and because S sums four of them the standard error on S is about 0.022. An observed S of 2.05 therefore sits a little over two standard errors above the local bound, which is not a result; an S of 2.12 sits at five, which is. Noise moves the mean rather than the spread: symmetric depolarizing noise with parameter p scales the ideal 2.828 by (1 - p), so 1 percent depolarizing gives 2.80 and still violates comfortably, while readout error of 3 percent on each of two detectors multiplies the correlators by roughly 0.94 squared and drags the same state to about 2.50. Correcting for readout with a calibration matrix is legitimate for a state-characterization result and is not legitimate inside a loophole-free claim, because the correction assumes the noise model the experiment is supposed to be testing.

Post-selection is where an entanglement result most often stops being one, and the failure is quantitative rather than philosophical. If trials in which a detector failed to fire are discarded, the surviving subset is chosen by an event that can itself depend on the measurement setting, and a purely local model with a suitable detection strategy can then reproduce correlations above the classical bound. For a maximally entangled pair the detection efficiency has to exceed about 82.8 percent before the fair-sampling assumption can be dropped, which is why serious experiments report the total number of trials attempted, the number discarded, and the reason for each discard, and why an experiment at 70 percent efficiency reports a violation conditional on fair sampling rather than a loophole-free one. The same discipline applies far from Bell tests. On a QPU, running a Bell circuit and then discarding shots whose ancilla parity check failed converts a noisy state into a cleaner apparent one at the cost of a post-selection rate that has to be reported: keeping 40 percent of shots and quoting a fidelity of 0.97 describes a different object than the 0.88 the device actually prepared. State the partition, the criterion, the raw counts, the discarded counts, and the settings randomization, and an entanglement claim can be recomputed by a reader; omit the discards and it cannot be checked at all.

Choosing the right instrument for the claim keeps the cost proportionate. Full state tomography of an n-qubit register needs 3^n measurement settings — 243 at five qubits, 59,049 at ten — and reconstructs far more than an entanglement claim requires. An entanglement witness is an observable whose expectation is bounded on every separable state, so violating that bound certifies entanglement from a handful of settings: for a GHZ state, the standard fidelity witness is built from two settings, all-Z parity and a rotated-basis parity, and certifies genuine multipartite entanglement whenever the estimated fidelity exceeds one half. The condition is that the witness be fixed before the data is examined, because selecting the witness that happens to violate its bound on the observed counts is the same statistical error as choosing a hypothesis after seeing the sample. Mixed states need the same care in the criterion itself: Schmidt rank above one certifies entanglement for pure bipartite states only, and applying it to the noisy state a device actually prepares certifies nothing, since a separable mixture can carry apparent rank from noise alone. Report which criterion was used, what it assumes about the state, and how many settings and shots produced it.