Locally causal and deterministic interpretations of quantum mechanics: parallel lives and cosmic inflation

Abstract

Several locally deterministic interpretations of quantum mechanics are presented and reviewed. The fundamental differences between these interpretations are made transparent by explicitly showing what information is carried locally by each physical system in an idealized experimental test of Bell’s theorem. This also shows how each of these models can be locally causal and deterministic. First, a model is presented which avoids Bell’s arguments through the assumption that space-time inflated from an initial singularity, which encapsulates the entire past light cone of every event in the universe. From this assumption, it is shown how quantum mechanics can produce locally consistent reality by choosing one of many possible futures at the time of the singularity. Secondly, we review and expand the Parallel Lives interpretation of Brassard and Raymond-Robichaud, which maintains local causality and determinism by abandoning the strictest notion of realism. Finally, the two ideas are combined, resulting in a Parallel Lives model in which lives branch apart earlier, under the assumption of a single unified interaction history. The physical content of weak values within each model is discussed, along with related philosophical issues concerning free will.

Appendix

Here we go through several examples intended to clarify the Parallel Lives model. We discuss entangled states, interference effects, memory and quantum erasers, quantum relativity, and the possibility of non-unitary evolution.

1.1 Entangled states

Here we explain the detailed behavior of a simple entangled state in the Parallel Lives model. Consider the entangled state,

$$\begin{aligned} |\psi \rangle = \frac{4}{5} |0\rangle |0\rangle + \frac{3}{5}|1\rangle |1\rangle . \end{aligned}$$

(2)

The two qubits are sent to space-like separated detectors A and B and measured in the computational basis, and then the results are subsequently collected by the experimenter and compared. There are several important details to take note of:

• There is a preferred basis for the two qubits, and it may not be the computational basis in which the state is represented above. For simplicity in this example, suppose it is the computational basis.

• This means that each qubit is experiencing two different lives, $|0\rangle$ and $|1\rangle$, and 16/25 of the total probability fluid is in the channel for $|0\rangle$ and 9/25 of it is in the channel for $|1\rangle$—although the two lives are oblivious to the amount of fluid that flows along with them, and oblivious to one another. Each system carries a record of $|\psi \rangle$ in its local interaction history.

• When the measurement is performed, detector A reports $|0\rangle$ with probability 16/25, and $|1\rangle$ with probability 9/25, as it becomes correlated with one of the two lives of that qubit. The same is true for the qubit at detector B. These are just the usual probabilities predicted by quantum mechanics for the reduced density matrix of each qubit (i.e., oblivious of what happens to the other qubit).

• The physical system that carries the result of measurement A back to the experimenter is now experiencing two different lives, with the same relative proportion of probability fluid, and likewise for the system that carries the result of measurement B.

• Because of the record of $|\psi \rangle$ carried by both systems, the life of system A in which the result of measurement A was $|0\rangle$ only becomes correlated with the life of system B with the result $|0\rangle$, and likewise for the lives with $|1\rangle$. Importantly, the amount of fluid is identical in the channels that become correlated, and now the experimenter has two lives, one in which the result $|0\rangle |0\rangle$ was obtained (with 16/25 of the fluid), and one in which the result $|1\rangle |1\rangle$ was obtained (with 9/25 of the fluid).

• Clearly both lives of the experimenter observe the correct entanglement correlation for $|\psi \rangle$, and the total amount of probability fluid is conserved, which is to say that all lives of all systems are continuous. The entire mechanism is locally causal and deterministic.

It is also worthwhile to consider the less simple case in which the preferred basis is the $\sigma _x$ eigenbasis, $|+\rangle = \frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}}|1\rangle$, and $|-\rangle = \frac{1}{\sqrt{2}} |0\rangle - \frac{1}{\sqrt{2}}|1\rangle$. Then we can rewrite $|\psi \rangle$ as,

$$\begin{aligned} |\psi \rangle = \frac{7}{10} |+\rangle |+\rangle + \frac{1}{10} |+\rangle |-\rangle + \frac{1}{10}|-\rangle |+\rangle + \frac{7}{10}|-\rangle |-\rangle . \end{aligned}$$

(3)

In this basis, each of the two lives of each qubit carries half of the probability fluid, which we see by summing the mod-squared amplitudes of the terms with each single-qubit state $|+\rangle$ or $|-\rangle$,

$$\begin{aligned} \left( \frac{7}{10}\right) ^2 + \left( \frac{1}{10}\right) ^2 = \frac{1}{2}. \end{aligned}$$

(4)

Again each system carries a record of the state $|\psi \rangle$ in its local interaction history (i.e., a record of what happened when the two systems did interact locally in the past).

When the measurements A and B occur, the probability fluid now mixes (interferes) as it flows from the channels of the $\sigma _x$ basis into the channels of the computational ($\sigma _z$) basis of the measurement, and the record of $|\psi \rangle$, written in the computational basis, determines how much fluid flows into each channel, so that again 16/25 flows in the channel for $|0\rangle$ and 9/25 in the channel for $|1\rangle$. Notice that regardless of the preferred basis for the qubits of the original Bell state, after the measurement, the preferred basis is the computational basis.

This case also raises an interesting question. Each system carries 1/2 $|+\rangle$ and 1/2 $|-\rangle$, but then interference at the measurement causes 16/25 to flow into $|0\rangle$ and 9/25 to flow into $|1\rangle$, but what was the probability for a $|+\rangle$ life to find itself in the $|0\rangle$ channel? This detail does not seem to be specified by the current model—it is clear that all of the fluid is conserved, but due to the interference, it is not clear how the two pre-measurement channels flow into the two post-measurement channels. Perhaps a Bohmian-like treatment of the individual trajectories can answer this question.

In the above example, the order of events played no important role, and this description works the same is any relativistic reference frame. All physical systems propagate at typical luminal or subluminal speeds and carry with them all of the information that is needed to correctly correlate the parallel lives when multiple systems interact locally.

1.2 Quantum relativity

In order to conceptualize the universe of many parallel lives, we also introduce the notion of quantum relativity, which formalizes our earlier discussion of the Wigner’s friend thought experiment. Recall that in this model, there is just one universal wavefunction that describes an elaborate superposition state for all systems. The experience of a life within the universal wavefunction is subjective—it is a special projection onto a single eigenstate of a given system in that system’s preferred basis, or one specific channel through which probability fluid flows. The complete life experience of a given life of a given system is exactly the complete record of its local interaction histories with other systems, and systems they had previously locally interacted with, etc., and this experience is unique for every life of every system.

For each life of each interacting system, the experience of an interaction is a collapse onto a product state of the interacting systems. However, because each system experiences splitting into parallel lives independently, whenever two systems interact locally, they can enter a superposition relative to remote systems. Specifically, when systems A and B interact locally, the individual lives of A and B experience a collapse into a product state of the two systems, but relative to system C which did not participate in the interaction, systems A and B have not collapsed, but rather undergone a unitary transformation into an entangled superposition state whose terms are the different product-state lives the two systems are experiencing. This product basis is also the new preferred basis for both systems. If system C interacts with system A or system B in the future, the individual lives of C will also experience a collapse, but relative to system D there is no collapse, this is now just an entangled superposition of all three systems A, B, and C, and so on.

Within a single universal wavefunction, quantum relativity is the fact that an interaction that is experienced as a collapse by the lives of the interacting systems, is instead experienced as a unitary evolution by all other systems in the wavefunction, and both experiences are perfectly consistent. At the level of the universal wavefunction, there is never a collapse—which conventionally leads us to conclude that only unitary evolution occurs.

1.3 The Mach–Zender interferometer, Wheeler’s delayed-choice, and Elitzur–Vaidman interaction-free measurements

Consider a Mach–Zender interferometer, tuned for perfect constructive and destructive interference, respectively. After passing through the first beam splitter, half of the fluid flows down each arm of the beam splitter—the two arms are orthogonal in position and momentum space. At the second beam splitter there is interference between the fluid coming from the two arms. Interference occurs because the fluid coming from each arm is once again in the same place, with the same momentum, and thus no longer orthogonal—interference occurs only when non-orthogonal fluid flows together.

This probability fluid picture, with half of the fluid flowing through each arm of the interferometer, also provides a clear explanation for Wheeler’s delayed-choice experiment [29, 30], and the Elitzur–Vaidman interaction-free measurements [31], as locally causal and deterministic.

In a Wheeler delayed-choice experiment using a Mach–Zender, if the second beam splitter is not present, the 1/2 measure of fluid from the two arms simply remains orthogonal and crosses paths without any interaction, arriving at both detectors.

In interaction-free measurement, the fluid from one arm of the interferometer is simply blocked by the ‘bomb’, which means it cannot participate in the interference at the second beam splitter. For the lives that experience the interaction-free measurement, it is exactly because this probability fluid fails to arrive that the presence of the bomb is revealed. On the other hand, the lives that were blocked most certainly did experience interacting with the ‘bomb’.

1.4 Interference and memory, quantum erasers

Now, if a system is capable of encoding a memory into Hilbert space of which arm it took through the beam splitter, then this information makes the fluid that recombines at the second beam splitter orthogonal, and thus there is no interference. The Hilbert space into which the path-information is encoded can be internal to the system traversing the interferometer, or it can be encoded into the environment by an interaction—which is the explanation due to decoherence.

A general quantum eraser experiment [32, 33] involves an entangled pair of systems, which we call signal and idler, the orthogonal information is encoded into the idler system and the signal system interferes with itself (through an interferometer, diffraction grating, etc.). If the orthogonal information is measured in the idler system, no interference is observed in the signal system. On the other hand, if the orthogonal information is ‘erased’ by a unitary operation before the idler is measured, then the interference pattern is observed in the signal.

In the delayed-choice quantum eraser [34], the decision to erase the information is postponed until well after the signal photon has been recorded. The post-selected signal data corresponding to the cases in which the idler was erased reveal complementary interference patterns, while the data corresponding to cases where the idler was not erased show no interference at all.

This is perfectly consistent when one considers how the parallel lives of entangled systems become correlated as the experimenter collects all of the data. Nevertheless, it is a subtle and nontrivial case that warrants a detailed description here. We will consider the delayed-choice quantum eraser experimental setup of Kim et al, in which a photon passes through a double slit and is then incident on a BBO crystal beyond either slit. At each site, spontaneous parametric down conversion converts an incoming photon into an anticorrelated entangled pair of photons, and we define a signal and idler photon for each site. The two signal paths are directed to a screen where they can interfere, and are then measured. The two idler paths take a longer route, such that they are still in mid-flight when the signal photon is measured. After this, the idler paths pass through beam splitters, which effectively ‘decide’ whether or not the orthogonal information about which slit the photon passed through is erased. In one case the two idler paths interfere at a third beam splitter which erases the orthogonal information, while in the other the idler from each path is directly measured, revealing exactly which path the photon took. The coincident arrival of the signal and idler photons are tracked so that the signal data can later be post-selected into separate sets for each of the four possible detectors the idler arrived at. In the two cases where the orthogonal which-slit information was not erased, the signal data shows no interference pattern. In the two cases where the orthogonal information was erased, the signal data reveals two different interference patterns, complementary in sense that their fringes are $\pi$ out of phase. Because of this complementarity, the interference pattern is completely hidden in the full data set.

Now we discuss the probability fluid picture of this experiment. The fluid for the incident photon arrives at the first beam splitter and divides in half. The 1/2 fluid at each BBO site creates a signal and idler photon which each carry a 1/2 measure of probability fluid. The two signal channels interfere and are measured—all signal fluid arrives at the screen. The 1/2 measure of idler fluid from site A arrives at beam splitter A and divides into 1/4 going to detector A and 1/4 going to beam splitter C. Likewise, the 1/2 measure of idler fluid from site B arrives at beam splitter B and divides into 1/4 going to detector B and 1/4 going to beam splitter C. The idler fluid from each site arrives at beam splitter C and divides into 1/8 each, which interfere to become 1/4 going to each of the two ‘erased’ detectors C$_1$ and C$_2$. Given our earlier discussion of entanglement correlations in the Parallel Lives model, we can see that when the signal and idler data are collected, the experimenter now experiences many parallel lives in which the entanglement correlations are obeyed, meaning that pattern of erased idler photons matched with signal photons will reveal interference, and the pattern of unerased idler photons matched with signal photons will show no interference. It is important to emphasize that every life of each system (signal and idler) exists independently before the experimenter collects the data, and quantum theory tells us the probability (including entanglement correlations) that each possible pairing of the lives of the two systems will be experienced by the experimenter, and furthermore that every signal life finds a match with an idler life.

We should emphasize the difference in this model between the Quantum Eraser and the Wheeler’s delayed choice, which seem more closely related than they are. In the delayed-choice experiment, the presence or absence of the second beam splitter changes the quantity of fluid that flows to each detector—specifically, if the beam splitter is present, no fluid at all flows to one of the detectors due to destructive interference. After the photon passes through the first beam splitter, adding the second beam splitter causes interference which effectively erases the ‘which-path’ information, while leaving out the second beam splitter allows the ‘which-path’ information to be directly measured.

Because the delayed-choice quantum eraser experiments use entangled states to encode and potentially erase the orthogonal information, the same quantity of probability fluid always flows through the parts of the experiment, and no interference patterns are visible at this level. The interference patterns only emerge when the lives of the signal and idler become locally correlated according to the usual quantum probabilities.

As an aside, it is worth mentioning that in all discussions of the Mach–Zender interferometer, or the double-slit experiment, we have made an implicit assumption that when the particle is incident on a beam splitter (double-slit), the particle enters a superposition that is not entangled with the beam splitter (double slit). If it were entangled in this way, this would mean the device itself had encoded orthogonal information, and it would never be possible to see an interference pattern. Indeed, double-slit experiments have been performed in which atoms near the slits can acquire a record of which slit the particle passed through, and the visibility of the interference fringes is measurably reduced. Similar loss of visibility is observed in the large-molecule double-slit experiments of Arndt et al. [35,36,37,38], as various decoherence issues become increasingly difficult to avoid. Interactions can create a superposition for one system, or an entangled superposition of several systems. However, in the entangled case, if the system has decohered and interacted with the environment, then the interference vanishes because each life of the experimenter sees only one term in the superposition of the experimental system.

1.5 Irreversible processes and non-unitary evolution

In this picture, the unitary evolution seems to follow from conservation of probability fluid, but fluid conservation alone is actually a significantly weaker constraint. For example, consider an irreversible logical process like the erasure of an isolated single-qubit system, as an interaction in which all fluid from two distinct channels flows together into a single channel. The fluid is still conserved in this process, and indeed the trajectories of individual particles in the fluid are reversible, but the experience of a particle in the fluid would be oblivious to which channel it had come from. Internally, the particles carry a record of their local interaction history, but isolated single-qubit systems cannot encode any memory about which of the two channels they came from, and thus the present experience of a life is fully defined by the channel in which the fluid now flows—at the level of experience, the information about the past has been erased. There is nothing about local causality, determinism, or our parallel lives guiding principle that would prevent this situation from occurring, but this operation is not, in general, represented by a unitary operator in Hilbert space—it is instead a projection operator. This may allow for a Parallel Lives model that does not strictly adhere to the Church of the Larger Hilbert space view, in which there can be only unitary operations.

It is interesting that this could only occur when non-orthogonal fluid flows together, meaning the condition for interference is apparently also the condition for the projective interactions discussed above.

The black hole information paradox may also be avoided if we only impose the condition that probability fluid is conserved as it crosses the event horizon, rather than insisting on unitary evolution.