The Born Rule in a Mixed-State Everettian Multiverse

Table of Links

Abstract & Introduction

Decoherence and Branching

The Born Rule

Discussion

Conclusion and References

3 The Born Rule

3.1 The Sebens-Carroll Program

Sebens and Carroll (2018) proposes a strategy for justifying the Born rule for Alice above: the probabilities ascribed to each branch by the Born rule are to be interpreted as Alice’s self-locating uncertainty as to which branch they’re located in, postmeasurement but before they observe the measurement outcome. In their words:

If we assume that the experimenter knows the relevant information about the wave function, it’s unclear what the agent might be uncertain of before a measurement is made. They know that every outcome will occur and that they will have a successor who sees each possible result. […] We must answer the question: What can one assign probabilities to? Our answer will be that agents performing measurements pass through a period of self-locating uncertainty, in which they can assign probabilities to being one of several identical copies, each on a different branch of the wave function. (ibid., 33)

To elaborate, their strategy relies on (i) the fact that Alice knows the universal wave function, (ii) has undergone branching due to some measurement having been performed, but (iii) may not be able to discern which branch they’re on prior to observing the measurement outcome due to each copy of Alice, post-branching, having qualitatively identical internal states as each other. For Sebens and Carroll, “two agents are in the same internal qualitative state if they have identical current evidence: the pa‹erns of colors in their visual fields are identical, they recall the same apparent memories, they both feel equally hungry, etc.” (2018, 36)

In this ‘post-measurement pre-observation’ period, as they call it, the universal wave function is:

Sebens and Carroll (2018) then proposes an intuitive epistemic principle with which they justify Alice’s use of the Born rule, where the probabilities are now interpreted in terms of subjective credences:

Epistemic Separability Principle (ESP): Suppose that universe U contains within it a set of subsystems S such that every agent in an internally qualitatively identical state to agent A is located in some subsystem which is an element of S. The probability that A ought to assign to being located in a particular subsystem S given that they’re in U is identical in any possible universe which also contains subsystems S in the same exact states (and does not contain any copies of the agent in an internally qualitatively identical state that are not located in S).

Case 1: Pure states

3.2 The McQueen-Vaidman Program

Similar to the Sebens-Carroll program, Mceen and Vaidman (2018) also proposes an interpretation of the Born rule in terms of self-locating uncertainty. ‘is follows earlier a‹empts initiated by e.g. Vaidman (1998) and Tappenden (2011). Mceen and Vaidman’s setup depends on the fiction of a sleeping pill, which induces the same post-measurement pre-observation uncertainty as the Sebens-Carroll program:

The experimenter performs the experiment without looking at the result; she instead arranges to be put to sleep with a sleeping pill and taken to room A if the result was a, and room B if the result was b. ‘e rooms are identical from the inside. So when each of the experimenter’s descendants [post-branching copies] wakes up, they will be uncertain as to which room they’re in, and therefore uncertain as to which result, a or b, obtains in their own world. The question: What is the probability for result a? makes sense for them. It’s not a question about what happened, it’s a question about their self-location. The descendants might know everything relevant regarding the wavefunction of the universe, but still be ignorant about who they are. The descendants are in states of self-location uncertainty. (2018, 2)

However, instead of relying on ESP, they rely on three physical principles:

• Symmetry: Symmetric situations should be assigned equal probabilities.

• No-FTL: Faster-than-light signaling is impossible; the probability of finding a particle in some location with some state cannot be influenced by actions occurring remotely.

• Locality: The probability of finding a particle somewhere in some state depends only on that particle’s quantum state.

Given this situation, each agent – well aware of the symmetry of the situation – is put in a sleeping pill situation: they’re put to sleep before measurement in a ‘ready’ room, and then moved to a ‘found’ room – stipulated to be internally identical as the ‘ready’ room – if the particle is found by the measurement apparatus in their space-station. ‘e measurement then takes place.

3.3 The Deutsch-Wallace Program

Finally, we turn to the Deutsch-Wallace program. In contrast to the previous two, this program provides a justification of the Born rule in WFRE by appealing not to self-locating uncertainty, but to rational choice theory. A rational agent betting on outcomes of measurements for some wave function ought to bet in such a way that the credences they have over these outcomes are governed by the squared-amplitudes of the wave function. Deutsch (1999) provided one of the earliest proofs for this result. However, the most refined result is due to Wallace (2012), who proves a representation theorem to this effect given certain axioms of rational choice and assumptions about the structure of quantum bets.

The decision problem can be summarized schematically as such: a system’s state space – its Hilbert space – can be decomposed into various macrostates π, with their fineness (i.e. size) determined by decoherence and the environment. Any system in a macrostate π is compatible with a set of unitary transformations, which are understood as acts on the system by an agent (for instance, measurement). These acts lead to outcomes in the form of the system ending up in different macrostates on different branches as a result of the unitary transformations. Agents are then asked how they would place monetary bets on these outcomes, on which they will collect rewards after the act is performed; that is, agents are asked to state their preferences for bets on these outcomes. Now, the question is this: what credences should agents rationally assign to these outcomes?

Wallace assumes a set of four ‘richness’ axioms on the structure of the set of possible bets,[2] as well as a set of six ‘rationality’ axioms on the structure of the agent’s rational preferences on pairs of bets.[3] Thee first two are general axioms of rationality, while the latter four are ‘Everettian’ rationality axioms proposed by Wallace. We won’t go into detail stating the axioms, except for one (which we’ll discuss in the following section). ‘e interested reader is invited to read Wallace (2012, §5). Wallace (2012, 172) proves the following theorem with the above set-up:

Born Rule Theorem: There is a utility function on the set of rewards, unique up to positive affine transformations, such that one act is preferred to another if and only if its expected utility, calculated with respect to this utility function and to the quantum-mechanical weights of each reward, is higher.

[1] We could also have made a measurement for z-spin, in which case completely analogous results follow: the density matrix gives rise to a different branching structure.

[2] They are called Reward Availability, Branching Availability, Erasure, and Problem Continuity respectively.

[3] They are called Ordering, Diachronic Consistency, Macrostate Indifference, Branching Indifference, State Supervenience, Solution-Continuity, respectively.

[4] Many thanks to David Wallace for discussions about this point.