Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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The Born rule is a fundamental postulate of quantum mechanics and therefore it cannot be derived from other postulates --precisely your first link emphasizes this--. In particular the Born rule cannot be derived from unitary evolution because the rule is not unitary $$A \rightarrow B_1$$ $$A \rightarrow B_2$$ $$A \rightarrow B_3$$ $$A \rightarrow \cdots$$ The Born rule can be obtained from non-unitary evolutions. |
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Strictly speaking, the Born rule cannot be derived from unitary evolution, furthermore, in some sense the Born rule and unitary evolution are mutually contradictory, as, in general, a definite outcome of measurement is impossible under unitary evolution - no measurement is ever final, as unitary evolution cannot produce irreversibility or turn a pure state into a mixture. However, in some cases, the Born rule can be derived from unitary evolution as an approximate result - see, e.g., the following outstanding work: http://arxiv.org/abs/1107.2138 (accepted for publication in Physics Reports). The authors show (based on a rigorously solvable model of measurements) that irreversibility of measurement process can emerge in the same way as irreversibility in statistical physics - the recurrence times become very long, infinite for all practical purposes, when the apparatus contains a very large number of particles. However, for a finite number of particles there are some violations of the Born rule (see, e.g., the above-mentioned work, p. 115). |
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It is independent, but it is not fundamental, as it applies only to highly idealized kinds of measurements. (Realistic measurements are governed by POVMs instead.) In fact, the role of Born's rule in quantum mechanics is marginal (after the standard introduction and the derivation of the notion of expectation). It is hardly ever used for the analysis of real problems, except to shed light on problems in the foundations of quantum mechanics. |
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The idea of deriving the Born rule (and in fact the whole measurement postulate) from the usual unitary evolution of quantum systems is at the very heart of a realist interpretation of quantum theory. If the quantum state really describes a the true internal state of a system and measurement is just a certain kind of interaction, then there should be only one single law for the time evolution. Quantum theory however is fundamentally non-local and separating systems is conceptually hard, which makes observer and experiment impossible to describe separately. There should be a system containing both parts however and which follows a simple law of time evolution. Of course, the obvious candidate for such a law is unitary evolution, simply because that is what we observe for systems that we isolate as good as possible. It is usually argued that this route leads to the Everett interpretation of quantum theory, where observations are relative to the observer and realized by entangled states. There have been several attempts to derive the Born rule in this context, but all that seem valid require additional assumptions that are questionable (and may in fact be inconsistent with the realist approach or other fundamental assumptions). The reason why there cannot be a derivation that just uses ordinary unitary evolution and results in the Born rule is not even unitarity but the linearity of the theory. Say there is an evolution that takes out input to the measurement output, and we decide to measure a|A>+b|B> in the basis {|A>,|B>}. Then independently from the environment the Born rule predicts that |A> and |B> are invariant under measurement. A superposition (|A>+B>)/sqrt(2) should end up in either |A> or |B> depending on a possible environment state if the Born rule applies. The linearity of the theory requires that the outcome is a superposition of |A> and |B> however (the phase may change though). Everett's answer to this problem is that the superposition comes out, but with the outcomes entangled with the observer seeing either outcome. But this creates two observers that are unaware of their own amplitude. Because of the linearity their future evolution is independent from the branch amplitude, and it's therefore hard to argue that any aspects of their perceived reality would depend on the branch amplitude. Interestingly approaches to fix this issue, like the use of decision theory, advanced branch counting, etc, in some form introduce a nonlinear element to the theory. Be it a measure of branch amplitude, a cutoff amplitude or amplitude discretization, a stability rule (envariance or quantum darwinism). There are also approaches that don't hide the nonlinearity in additional assumptions that may collide with the linear evolution. Those are explicit nonlinear variations of the Schroedinger equation that can in fact produce an evolution that allows the Born rule to emerge. Of course, this is not something that most theorists embrace, simply because the linearity of quantum theory is such an attractive feature. But there's one more approach that I personally favor. The nonlinearity could be only subjective to an observer, caused by incomplete knowledge about the universe. An observer, i.e. a local mechanism realized within quantum theory, can only gather information by interacting with his environment. Certain information however is inaccessible dynamically, hidden outside the observer's light cone or just not available for direct interaction. Considering this, it can be shown that reconstructing the best possible state description an observer can come up with must follow a dynamic law that is not unitary all the time, but also contains sudden state jumps with random outcomes driven by incoming priorly unknown information from the environment. It can be shown that a photon from the environment with entirely unknown polarization can cause a subjective state jump that corresponds exactly to the Born rule. This is of course a bold claim. But please see http://arxiv.org/abs/1205.0293 for a proper derivation and discussion of the details. If you you would like to look at a more gently introduction to the idea you can also read the (less complete but more intuitive) blog I've set up for this: http://aquantumoftheory.wordpress.com |
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