Updating the Born rule

Abstract

Despite the tremendous empirical success of quantum theory there is still widespread disagreement about what it can tell us about the nature of the world. A central question is whether the theory is about our knowledge of reality, or a direct statement about reality itself. Current interpretations of quantum theory, regardless of their stance on this question, regard the Born rule as fundamental and add an independent state update (or 'collapse') rule to describe how quantum states change upon measurement. In this paper we present an alternative perspective and derive a unified probability rule that subsumes both the Born rule and the collapse rule. We show that this more fundamental probability rule can provide a rigorous foundation for informational, or 'knowledge-based', interpretations of quantum theory. Our result requires an assumption of instrument noncontextuality, a key notion that generalises previous approaches to non-contextuality. Therefore, the framework also permits one to consider non-contextuality in scenarios with arbitrary causal structure.

Indefinite Causal Order in a Quantum Switch

“…Using the Choi-Jamiołkowski isomorphism [18,19], we represent a CP map as a positive semidefinite operator M A ∈ A I ⊗ A O . The probability to realize the maps fM A ; M B ; …g in an experiment, corresponding to events A; B; …, is given by the "generalized Born rule" [2,[20][21][22],…”

Gleason-Busch theorem for sequential measurements

“…More recently, Shrapnel et al [13] derived the most general frame function on completely positive maps, with a view to understanding recent work on non-fixed causal order. Our work provides a link between the two, starting from a minimal set of axioms to show that the most general sequential measurement rule consistent with these axioms corresponds to a completely positive map.…”

“…There is by now in the literature a long tradition of axiomatic approaches to both the description of measurement in quantum theory [2,[11][12][13], and indeed to derive the structure of quantum theory from simple principles [14][15][16][17]. We note in particular that previous work has addressed a similar scenario to that of interest here: Cassinelli and Zanghi [12] derived the Lüders rule for state update through consideration of conditional probabilities via Gleason type arguments.…”

“…This is however not readily generalised to more general measurements, those which are not described by projectors. More recently Shrapnel et al [13], starting from an assumption that transformations are described by completely positive maps also used an axiomatic approach similar to Busch and Gleason to derive a probability measure which en-compasses both the Born rule and state update rule. Motivated by recent work on indefinite causal order in quantum mechanics [18][19][20][21], Shrapnel et al's work derives the most general rule resulting in a probability measure on the set of completely positive maps.…”

George Sudarshan and Quantum Dynamics

“…Nevertheless, this way of looking at quantum processes naturally resolves the ambiguity of what makes a quantum processes Markovian [35] and when the memory is quantum [18]. It leads to a unifying framework for spatio-temporal correlation [11,39], where a space-time version of the Born rule appears [62]. Later we also generalised Kuah's idea [37] to fit restricted control process tensor [41].…”