The measure of existence of a quantum world and the Sleeping Beauty Problem

An attempt to resolve the controversy regarding the solution of the Sleeping Beauty Problem in the framework of the Many-Worlds Interpretation led to a new controversy regarding the Quantum Sleeping Beauty Problem. We apply the concept of a measure of existence of a world and reach the solution known as ‘thirder’ solution which differs from Peter Lewis’s ‘halfer’ assertion. We argue that this method provides a simple and powerful tool for analysing rational decision theory problems. 1. Quantum Sleeping Beauty controversy The Sleeping Beauty Problem (SBP) (Elga 2000, Lewis 2001) inflamed an ongoing controversy amongst researchers in rational Bayesian decision theory. One group claimed that in a certain coin toss scenario, Beauty must have credence one-third for Heads, while another group argued for one-half, with no clear consensus on a solution in sight (Pust 2011, Thorn 2011). Lev Vaidman (2001) proposed to consider the SBP within the framework of the Many-Worlds Interpretation of Quantum Mechanics (MWI). He argued that treating the issue of probability in the MWI using the concept of the measure of existence of a world, makes the SBP simpler, yielding the thirder solution. A few years later, Peter Lewis (2007) also suggested solving the SBP within the MWI framework. He, however, argued for the halfer solution. This prompted a response by David Papineau and Víctor Durà-Vilà (2009a), followed by an exchange of replies (Lewis 2009, Papineau and Durà-Vilà 2009b). Other authors made subsequent contributions to the controversy (Peterson 2011, Bradley 2011, Tappenden 2011, Wilson 2013) proposing some modified scenarios. Peter Lewis introduced the Simplified SBP and put at the heart of his argument, its similarity to the Sleeping Pill Experiment (Vaidman 1998). Papineau and Durà-Vilà, the thirders, responded by questioning this similarity on metaphysical grounds. However, they argued that accepting this similarity and Vaidman’s approach leads to the halfer solution. Yet, we argue that Vaidman’s original suggestion of treating the SBP within the MWI framework using the concept of the measure of existence leads to a straightforward thirder solution, in spite of the strong similarity. 2. From the measure of existence of a world to the illusion of probability In our view, there is no genuine probability in the MWI framework. It is a completely deterministic theory and there is no relevant information that an observer, preparing a quantum experiment, is ignorant about. The quantum state of the Universe at one time determines the quantum state at all times. Vaidman (1998) associated our everyday perception of probability with post-measurement ignorance. In his Sleeping Pill Experiment (SPE), the agent is given a sleeping pill and sleeps through a quantum measurement, which serves as a quantum coin toss. While asleep, she is moved to either room H or to an identical-looking room T, based on the result of the toss – Heads or Tails. Upon awakening, she is asked what her credence in Heads is. It is meaningless to ask the agent before the toss what is the probability to find herself in room H after the toss, since she is the ancestor of both future descendants. However, after the experiment, we can ask her descendant, in which room does she think she is. The fact that there is no direct meaning for the probability of the outcomes of the experiment, does not contradict a genuine uncertainty on the part of the two descendants upon awakening in the propositions ‘I am in the H-world’ and ‘I am in the T-world’. The illusion of probability follows from the identity between the quantum state of the hypothetical Collapse Universe and the quantum state of the corresponding world in the MWI Universe. An observer in a Collapse Universe, performing a sequence of experiments, will have the same memories as an observer in a MWI-Universe in a world with corresponding results. In the MWI, the squared absolute value of the amplitude of a world is called its ‘measure of existence’. The ‘behaviour principle’ teaches us that one should care about one’s descendants according to the measures of existence of their worlds, and thus functions as the Born Rule counterpart in the MWI. Paul Tappenden (2011) has named this the Born-Vaidman Rule. As it is stated in (Vaidman 1998): `The relative measures of existence of the worlds into which the world splits provide the concept of probability' In this paper, we accept the Born-Vaidman rule and show how it helps to analyse the SBP in a straightforward way. Note that an alternative approach which assigns a pre-branching uncertainty (Saunders and Wallace 2008), does not lead to a different conclusion. The analysis in that framework, however, does not benefit from the advantages of our deterministic approach. 3. The Simplified Sleeping Beauty Problem versus the Sleeping Pill Experiment In Peter Lewis’s Simplified SBP, Beauty goes to sleep on Sunday evening. The researchers wake her up twice – once on Monday and once on Tuesday. Upon each awakening, Beauty is asked about her degree of belief in the proposition ‘This is the Monday awakening’. Beauty’s task becomes non-trivial because she forgets the Monday awakening completely, due to a memory erasure pill given to her before she goes back to sleep. Lewis argued that Beauty’s subjective experience is very similar to the experience of the agent awakened in one of the rooms in the SPE. There is a strong case for the similarity advocated by Lewis. What unifies both scenarios is the fact that Beauty’s subjective uncertainty takes place on the background of objective certainty. The researchers possess full information throughout the experiment and do not share Beauty’s ignorance. The effect of the memory erasure pill in the simplified SBP makes Beauty’s experience identical to that of the agent in the SPE. Beauty ought to have the same degree of belief in the propositions ‘This is the Monday awakening’ as in the corresponding proposition ‘I am in the H-world’. Henceforth we identify Beauty with both experiments. In the Simplified SBP, there is only one branch with which we can therefore associate a measure of existence 1. It is the same for Monday and Tuesday, so Beauty should attribute equal credences to both options, which thus have to be 1⁄2. In the SPE, the quantum state of the world upon Beauty’s awakening is and the measure of existence of each branch is 1⁄2. From the equality of the measures of existence it follows that Beauty should attribute credence 1⁄2 again. The measures of existence of the worlds upon Beauty awakenings are equal for the two alternatives in Lewis’s case as well as in the SPE, but different between the two cases, see Figure 1. Like Papineau and Durà-Vilà we think that this is relevant for the solution, however, we disagree when they attribute certainty to the Monday and Tuesday awakenings, but credences of 1⁄2 to the H and T results. They seem to confuse an external observer’s and Beauty’s points of view. An external observer cannot assign credence 1⁄2 for Beauty being in H or T world. Only Beauty possesses credence of 1⁄2 for these events, but she, upon awakening, also has credence 1⁄2 for Monday and for Tuesday. Figure 1: World lines in the Simplified Sleeping Beauty Problem (a) and in the Sleeping Pill Experiment (b). The width of the lines represents the measures of existence of corresponding worlds. The bright areas correspond to awakenings. 4. The Quantum Sleeping Beauty Problem In the full SBP setup, Beauty goes to sleep on Sunday. She knows that a fair coin will be tossed while she is asleep. If the coin lands Tails, then as in the Simplified SBP, she will be awakened once on Monday and once on Tuesday, without having on Tuesday any memory of the previous awakening. If the coin lands Heads, then Beauty is awakened on Monday only. Upon each awakening, she is asked for her credence in the proposition ‘The coin has landed Heads’. Adam Elga (2000) argued that her credence should be 1⁄3, while David Lewis (2001) argued for 1⁄2. Since then, philosophers have been divided between halfers and thirders. In an attempt to resolve the controversy, Vaidman (2001) proposed to use a quantum coin and to consider the problem within the MWI framework. He deduced the thirder solution. Peter Lewis (2007) also addressed the quantum SBP (QSBP) in the MWI framework. However, based on the similarity discussed in the previous section, he replaced the double awakening in the full SBP with a SPE, and deduced the halfer solution. (b) H T