Born rule and unitary evolution

Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?

• As the page about postulates you linked to correctly says, the Born-like rules to calculate probabilities from state vectors and operators are among the general postulates of quantum mechanics. It doesn't mean that they can't be derived from some other assumptions. However, the other assumptions clearly have to be connected with the notion of "probability" in one way or another, so they will be either a special or generalized formulation of the Born rule, anyway. Saying that the evolution is unitary doesn't say anything about probabilities - it can't "replace" the Born rule. – Luboš Motl Nov 23 '12 at 17:43
• @LubošMotl I felt that if experimental apparatus must obey the same laws as the system under observation, then Born rule must follow from unitary evolution in all situations. Please can you elaborate on this comment "However, the other assumptions clearly have to be connected with the notion of "probability" in one way or another, so they will be either a special or generalized formulation of the Born rule" – Prathyush Nov 24 '12 at 4:28
• I gave a derivation of the Born rule in the last answer to the question physics.stackexchange.com/q/19500 – Stephen Blake Aug 6 '13 at 20:37
• related or duplicate: physics.stackexchange.com/q/73329 – Ben Crowell Aug 6 '13 at 22:27

7 Answers

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.

• This argument is actually not valid because it does not count in unknown states from the environment which could differ for different outcomes. – A.O.Tell Nov 24 '12 at 13:35
• That is not true. Adding the environment and its equation of evolution gives an isolated system whose exact evolution is non-unitary. – juanrga Nov 26 '12 at 11:15
• You are arguing that the same input state gives different output states, which is not unitary. That argument is false because you don't know that the input state is different for different outcomes, simply because you don't know the state of the unknown environment, by definition, that leads to the different outcomes. I'm not saying that your conclusion is wrong, but your argument certainly is. – A.O.Tell Nov 26 '12 at 12:36
• Either if you assume that the same initial environment state $A\otimes E$ or not $A\otimes E_1,A\otimes E_2,A\otimes E_3\dots$ the evolution of the composite isolated system continues being non-unitary. von Neuman understood this and introduced his non-unitary evolution postulate in orthodox QM. – juanrga Nov 26 '12 at 20:46
• That's not what you wrote in your answer however – A.O.Tell Nov 26 '12 at 21:34

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).

• Unfortunately the article is completely wrong. I know two of the authors and their works on perpetual machines and supposed violations of the second law of thermo. – juanrga Nov 24 '12 at 11:28
• Thank you, I will take a look at the article referred to see if there is any weight in their arguments. Probably they are wrong as juanrga says, as most papers in this field are. – Prathyush Nov 24 '12 at 12:58
• @juanrga: Maybe you're right, and the article is indeed completely wrong, but until you offer some specific arguments, why should I believe you, rather than the authors and the referees of their published articles? You mentioned their articles on other topics, but I am not sure this is relevant. – akhmeteli Nov 24 '12 at 13:34
• @Prathyush: You may wish to start with their article arxiv.org/abs/quant-ph/0702135 , which is much shorter (see references to their journal articles there). – akhmeteli Nov 24 '12 at 13:53
• @akhmeteli Thank you I will look into it, Indeed since I haven't gone deeply into the article, I must not comment on its factual accuracy. May I ask what you thought about the article? – Prathyush Nov 24 '12 at 17:32

The use of the word "postulate" in the question may indicate an unexamined assumption that we must or should discuss this sort of thing using an imitation of the axiomatic approach to mathematics -- a style of physics that can be done well or badly and that dates back to the faux-Euclidean presentation of the Principia. If we make that choice, then in my opinion Luboš Motl's comment says all that needs to be said. (Gleason's theorem and quantum Bayesianism (Caves 2001) might also be worth looking at.) However, the pseudo-axiomatic approach has limitations. For one thing, it's almost always too unwieldy to be usable for more than toy theories. (One of the only exceptions I know of is Fleuriot 2001.) Also, although mathematicians are happy to work with undefined primitive terms (as in Hilbert's saying about tables, chairs, and beer mugs), in physics, terms like "force" or "measurement" can have preexisting informal or operational definitions, so treating them as primitive notions can in fact be a kind of intellectual sloppiness that's masked by the superficial appearance of mathematical rigor.

So what can physical arguments say about the Born rule?

The Born rule refers to measurements and probability, both of which may be impossible to define rigorously. But our notion of probability always involves normalization. This suggests that we should only expect the Born rule to apply in the context of nonrelativistic quantum mechanics, where there is no particle annihilation or creation. Sure enough, the Schrödinger equation, which is nonrelativistic, conserves probability as defined by the Born rule, but the Klein-Gordon equation, which is relativistic, doesn't.

This also gives one justification for why the Born rule can't involve some other even power of the wavefunction -- probability wouldn't be conserved by the Schrödinger equation. Aaronson 2004 gives some other examples of things that go wrong if you try to change the Born rule by using an exponent other than 2.

The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former.

Although photons are inherently relativistic, there are many situations, such as two-source interference, in which there is no photon creation or annihilation, and in such a situation we also expect to have normalized probabilities and to be able to use "particle talk" (Halvorson 2001). This is nice because for photons, unlike electrons, we have a classical field theory to compare with, so we can invoke the correspondence principle. For two-source interference, clearly the only way to recover the classical limit at large particle numbers is if the square of the "wavefunction" ($\mathbf{E}$ and $\mathbf{B}$ fields) is proportional to probability. (There is a huge literature on this topic of the photon "wavefunction". See Birula 2005 for a review. My only point here is to give a physical plausibility argument. Basically, the most naive version of this approach works fine if the wave is monochromatic and if your detector intercepts a part of the wave that's small enough to look like a plane wave.) Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector.

The Born rule says that probability doesn't depend on the phase of an electron's complex wavefunction $\Psi$. We could ask why the Born rule couldn't depend on some real-valued function such as $\operatorname{\arg} \Psi$ or $\mathfrak{Re} \Psi$. There is a good physical reason for this. There is an uncertainty relation between phase $\phi$ and particle number $n$ (Carruthers 1968). For fermions, the uncertainty in $n$ in a given state is always small, so the uncertainty in phase is very large. This means that the phase of the electron wavefunction can't be observable (Peierls 1979).

I've seen the view expressed that the many-worlds interpretation (MWI) is unable to explain the Born rule, and that this is a problem for MWI. I disagree, since none of the arguments above depended in any way on the choice of an interpretation of quantum mechanics. In the Copenhagen interpretation (CI), the Born rule typically appears as a postulate, which refers to the undefined primitive notion of "measurement;" I don't consider this an explanation. We often visualize the MWI in terms of a bifurcation of the universe at the moment when a "measurement" takes place, but this discontinuity is really just a cartoon picture of the smooth process by which quantum-mechanical correlations spread out into the universe. In general, interpretations of quantum mechanics are explanations of the psychological experience of doing quantum-mechanical experiments. Since they're psychological explanations, not physical ones, we shouldn't expect them to explain a physical fact like the Born rule.

Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062

Bialynicki-Birula, "Photon wave function", 2005, http://arxiv.org/abs/quant-ph/0508202

Carruthers and Nieto, "Phase and Angle Variables in Quantum Mechanics", Rev Mod Phys 40 (1968) 411; copy available at http://www.scribd.com/doc/147614679/Phase-and-Angle-Variables-in-Quantum-Mechanics (may be illegal, or may fall under fair use, depending on your interpretation of your country's laws)

Caves, Fuchs, and Schack, "Quantum probabilities as Bayesian probabilities", 2001, http://arxiv.org/abs/quant-ph/0106133; see also Scientific American, June 2013

Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, Springer, 2001

Halvorson and Clifton, "No place for particles in relativistic quantum theories?", 2001, http://philsci-archive.pitt.edu/195/

Peierls, Surprises in Theoretical Physics, section 1.3

• Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector. Could you explain this in more detail? – yippy_yay Aug 8 '13 at 20:52
• @SebastianHenckel: This is not completely thought out and may be wrong. But suppose that the rule for electrons is not the Born rule but a rule saying that probability is $\propto|\Psi|^p$, where $p\ne 2$. If you scatter an EM wave off of an electron, they interact through some wave equation such that the scattered part of $\Psi$ is proportional to the amplitude of the EM wave: amplitude is proportional to amplitude. But then the electron is acting like a detector, and $p\ne 2$ means that the probability of detection isn't proportional to the probability that the photon was there. – Ben Crowell Aug 8 '13 at 21:08
• I like this argument. The interaction between the photon and the electron however is quantum electrodynamics all the way through, and that's something I don't know much about. However, thanks for making a connection between electrons and waves I never thought about. The pure de Broglie argument always seemed very ad hoc, and this makes it somewhat more plausible. – yippy_yay Aug 8 '13 at 21:30
• It took me a while to read this answer. you said "The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former." Isn't Born rule applicable even in Relativistic Quantum mechanics(Any field theory in general), not in the sense of KG equation but the KG field. Also Would you comment on my recent answer on a related topic, physics.stackexchange.com/questions/76132/… – Prathyush Sep 4 '13 at 8:54
• @Prathyush: My relativistic field theory is pretty weak, so if you want a really coherent explanation of why the Born rule doesn't apply to the KG equation, you're probably better off posting that as a question and letting someone more competent answer. But basically I think the concept is that in relativistic QM, we have to give up on the idea of having eigenstates of position, so the whole Copenhagen-ish interpretation of a position measurement as projecting the wavefunction down to a delta function doesn't really work. – Ben Crowell Sep 4 '13 at 15:45

This is clearly a somewhat controversial topic, but Zurek claimed to derive the Born Rule from other postulates.

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

• I don't know if environment is a necessary concept in the measurement problem, For example, will a photographic Plate work in perfect vacuum. Thought I don't have the opportunity of experiment with such a situation, I believe a photographic must work normally in vacuum where there is no environment or extraneous photons. – Prathyush Nov 25 '12 at 17:02
• Even in your perfect vacuum you always have an interacting environment. And of course the environment may not be needed for the resolution of the measurement problem, but it might possibly be necessary, and so you cannot simply exclude it. It is at least a plausible source for randomness due to our lack of information about its state. – A.O.Tell Nov 25 '12 at 17:11
• In some situations where you cannot remove it from the experimental setup you will have to include the environment in the theory. What do you mean even in perfect vacuum you have the interaction environment? The basic process in a photographic plate is a light sensitive chemical reaction right? So an environment wont play a role – Prathyush Nov 25 '12 at 17:16
• The environment always plays a role in quantum theory. You cannot remove the quantum fields from space, no matter how perfect your vacuum is. There will always be interaction on some level, and ignoring that is surely not helpful for understanding the properties of quantum systems. You seem to be thinking is more or less classical terms with your photographic plate example. – A.O.Tell Nov 25 '12 at 17:21
• Also, in order to see if your plate has been affected by light you have to look at it. So at the very latest then you will subject it to massive interaction with an unknown environment – A.O.Tell Nov 25 '12 at 17:22

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.

• One day I will learn about POVM's its been on my list of To Do's for a long time. – Prathyush Nov 24 '12 at 19:23
• POVMs can be regarded as Born type measurements in a larger space, so you're back where you started. – A.O.Tell Nov 24 '12 at 22:45
• @A.O.Tell: On the formal level, yes. But in this larger space, one never does any measurements that would deserve that name. – Arnold Neumaier Nov 26 '12 at 9:37
• That statement would require an exact definition of what a measurement is and how it is applied to a subsystem. Also, it makes no practical difference. If you know how a Born style measurement works you understand how a POVM works. – A.O.Tell Nov 26 '12 at 12:39
• "It is hardly ever used for the analysis of real problems [...]" This is wildly wrong in my experience. – Ben Crowell Aug 6 '13 at 20:19

Yes it can be derived from unitarity of quantum evolution. This was shown by Lesovik in his paper, Derivation of the Born rule from the unitarity of quantum evolution.