Peres criterion

In plain terms

Correlation isn't entanglement. Two systems can be correlated for boring reasons, for instance because they were stamped from the same template, and their results will then agree without anything quantum being involved. Entanglement is the stronger, specifically quantum kind of connection, and telling boring correlation from genuine entanglement is, in general, a provably hard computational problem, so you can ask whether there's at least a quick test that catches some of it.

Peres found one of startling simplicity. Take the joint description of the pair, a matrix called the density operator. Transposition, flipping a matrix across its main diagonal, is a harmless bookkeeping reflection when applied to the description of any single system, as it turns a legitimate state into a legitimate state. Now apply it to one half of the pair only. A legitimate state has eigenvalues, the characteristic numbers of its matrix, that are all probabilities and so never negative, so if the half-transposed result has a negative eigenvalue it's no longer a legitimate state, and the original was entangled, guaranteed.

For the two smallest kinds of pair the test is perfect in both directions, as it catches every entangled state and never raises a false alarm. Beyond that it can miss, as there exist entangled states that pass the test, and the ones it misses turn out to be strange objects in their own right, bound entangled, meaning correlated in the quantum way yet impossible to distil into the clean entanglement quantum protocols use.

In CSD

CSD's machine-checked corpus carries this criterion beside its Bell machinery because the two detect different things, and conflating them is a standing error the programme can't afford. A Bell violation detects entanglement through statistics an experiment could show, and the Peres test detects it through algebra on the description. They disagree, as entangled mixed states exist that violate no Bell inequality.

For a reconstruction of quantum mechanics the distinction is load-bearing. Claims about non-locality and claims about entanglement have different scopes, and the corpus keeps separate instruments for them precisely so its theorems say what they say and no more.

Mathematically

Separable implies positive partial transpose. The converse holds exactly in dimensions two-by-two and two-by-three (Horodecki) and fails above, where PPT- entangled, that is bound entangled, states exist.

The test itself is one eigenvalue computation, which is its entire charm given that separability in general is NP-hard.

The name

Asher Peres (1934-2005) was born Aristide Pressman in Beaumont-de-Lomagne, France, to Polish-Jewish parents, and the family survived the occupation in hiding, so he took a Hebrew name on emigrating to Israel in 1949. He took his doctorate at the Technion under Nathan Rosen, the R of EPR, and spent his career at the Technion thereafter, pursuing foundations one degree of separation from the founding paradox.

The criterion dates from 1996, quantum teleportation, which he co-authored, from 1993, and his 1993 textbook remains the field's standard for saying exactly what is meant. His epigram that unperformed experiments have no results is quoted by every camp in the interpretation wars, each certain it supports them.

Related entries

• Gisin's theoremEvery entangled pure state, however slightly entangled, violates a Bell inequality, so there's no quiet zone.
• Bell state is not a joinThe arena CSD assigns to two systems together is strictly larger than the two arenas simply paired, and the Bell state is a point of the excess, so entanglement is located exactly.
• Partial traceMathlib has no partial trace for matrices, so the corpus built one, the staple operation of quantum information, with its working properties proved and staged for contribution upstream.
• Kronecker productTwo quantum systems become one by multiplying their descriptions rather than adding them, which is why quantum resources multiply where classical ones merely add.
• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.