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Outcome dependence is a condition in which the result of a nearby measurement has an impact on the result of a distant measurement. It is the inverse of outcome independence, which is a condition in which nearby measurements do not affect distant measurements. Related to outcome dependence is the concept of parameter dependence, which is the condition in which the settings of a nearby measurement apparatus impacts the result of a distant measurement. Outcome dependence has a number of implications in the concepts of factorizability, superluminal causation, and quantum non-locality.

Background Information
Main Article: A Stronger Bell Argument for (Some Kind of) Parameter Dependence

While general relativity provides a big picture, quantum mechanics relays theories about the characteristics, and dynamics of subatomic particles. Planck’s black-body radiation and Albert Einstein’s theory of the photoelectric effect provided the framework in which the theories of quantum mechanics could be made. With quantum entanglement emerging as an important phenomenon in the, Einstein coined the phrase spooky action at a distance to describe how particles in a pair can act on one another and become entangled with each other. This idea became known in the terms of non-locality, meaning the ability of particles to instantaneously know about each other’s state even when separated by large, space-like distances.

The system described as a single quantum system rather than a pair, raised suspicion in Einstein. This is because the entangled particles must influence the other at superluminal speeds, something Einstein was skeptical about. John G. Cramer came along to demonstrate whether communication exists in an entangled pair of particles. However, skeptics suggested that quantum entanglement does not imply faster than light communication (FTLC). This belief arose from the Einstein-Podolsky-Rosen (EPR) paradox, which indicated that the hidden variable needed for FTLC could not exist.

This idea of an entangled system lead to the Bell’s theorem and his thought experiment, which was based on the assumption of realism, locality, and the freedom to choose between measurement settings. The notion of realism is based on the idea that external reality is assumed to exist and have definite properties, whether or not they are observed by someone. On the other hand, the notion of locality is based on two systems that no longer interact, so no real change can take place in the second system in consequence of anything that may be done to the first system.

The EPR/B Experiment
Main Article: Action at a Distance in Quantum Mechanics

In 1935, Einstein, Podolsky, and Rosen published a paper with a thought experiment that suggested that quantum mechanics was incomplete. This famous experiment became known as the EPR experiment, and it led to the realization that there must be a more complete physical reality with potentially some hidden variables, which must be able to describe the state of affairs in the world with more precision than quantum mechanical states.

In 1951, Bohm published a refined version of the EPR thought experiment, which is known as the EPR/B experiment. This experiment begins with the emission of particles in a single state from a source, which diverge into opposite directions and continue travelling until they reach a measuring device. When each of these particles encounter their respective measuring device, which are located at space-like distances from each other, the device measures the state of the particle’s spin components in many directions. It is important to note that due to the space-like distance between both events, no light signals traveling at relativistic speeds, nor any signals traveling at sub-relativistic speeds, can travel between the events. Interestingly, the experiment demonstrates that although the two events are distant, otherwise referred to as non-local, the measured outcomes appear to be correlated.

Although the measurement outcomes at each measuring device appear to be completely stochastic, both of these measurements are correlated in some way. Another way of thinking about this observation is by stating that the joint probabilities of the distant outcomes are not equivalent to the product of the single probabilities for each outcome. The probabilities of the measurement outcomes can be better understood by examining figure 1. The probability that the particle on the left wing will spin in a clockwise fashion along the z-axis would be one-half. If the particle on the left wing indeed spins in a clockwise fashion about this axis, then the particle on the right wing will automatically spin in a counter clockwise fashion about the same axis, even if the measurements are made simultaneously. Therefore, this example strongly suggests that there is a non-local influence affecting the measurement outcomes in both the left and right wings.

Bohm’s observation of a non-local influence between the two measurement outcomes by the left and right measuring devices within the EPR/B experiment can be further supported by the orthodox theory of quantum mechanical wave function collapse. This theory states that particles traveling in opposite directions from a source to a measuring device do not have a spin state until they undergo their first measurement. In turn, the outcome of the first measurement, whether it be at the left or the right wing, defines the spin of that specific particle in a completely stochastic manner. Consequently, the spin of the particle travelling in the opposite direction will instantaneously become defined in a completely deterministic manner thus, demonstrating the non-local influences leading to a correlation between the events occurring at a space-like distance. In order to illustrate this using figure 1, if the particle travelling towards the left wing undergoes the first measurement, which stochastically assigns it a clockwise spin about the z-axis, then the particle travelling towards the right wing will instantaneously acquire a counter clockwise spin with respect to the same axis in a completely deterministic manner. Therefore, a non-local influence between the two events is strongly apparent in the EPR/B thought experiment.

Consequences of the EPR View
Main Article: Action at a Distance in Quantum Mechanics

As previously stated, the EPR view sheds light to the incompleteness of quantum mechanics with respect to providing precise information about the state of a system. This view claims that quantum mechanics can only provide us with information regarding some of the system’s properties and the probabilities of a particle’s measurement outcome, such as the measurement outcome of a particle’s spin state in many directions. An interesting idea that the EPR view suggested was that the correlation between the distant events are not due to changes in the properties of particles, but instead, the correlation is due to changes in the state of knowledge. For instance, if the particle on the left wing of figure 1 has a measurement outcome that indicates a clockwise spin with respect to the z-axis, then it is known that the particle at the right wing will have a measurement outcome indicating a counter clockwise spin about the same axis. However, if the correlation between both events is not due to changes in the properties of the particles, then it can be concluded that the particle found at the right wing had a counter clockwise spin state prior to the measurement outcome of the particle at the left wing. Therefore, this leads to the question of whether the state of knowledge can change if there is no influence between the events occurring at space-like separation.

Overall, Bohm’s observations led to the question of whether a version of the EPR/B thought experiment, in which there is a local, common-cause, could be developed. If so, this would imply that there is no influence between events at a space-like separation, and that the observed correlation occurs as a result of the particles’ spin state at the source. Although, Einstein, Podolski, and Rosen thought this was plausible, Bell suggests otherwise with the development of his theorem in 1964. Thus, resulting in the partial contribution to the development of the outcome dependence principle in the following decades.

Bell's Theorem and Non-Locality
Main Article: Action at a Distance in Quantum Mechanics

The purpose of Bell’s theorem in 1964 was to show that any local models of the EPR/B experiment have certain inequalities about the probabilities of measurement outcomes given specific assumptions. These inequalities, known as Bell’s inequalities, are incompatible with predictions from quantum mechanics. Since the 1970s, quantum mechanical predictions have been continuously supported by observations. Therefore, this sheds light to the possibility of non-locality in this field. As explained in the previous section, non-local models of the EPR/B experiments suggest that correlations between distant outcomes occur as a result of non-local influences for measurements at space-like separation. In comparison, local models of the EPR/B experiments suggest that these correlations occur as a result of a common cause. This common cause was believed to be the spin state of the particle pair before any measurements had taken place, denoted by λ.

Bell’s theorem, which at the time was a thought experiment, proposed that the probability of joint outcomes for a particle pair is equal to the product of the probabilities of the single outcomes. Thereby, for a local EPR/B model, the pair’s spin state and the settings in the left (L) wing would determine the probability of the L-outcome. Similarly, the pair’s spin state and the settings in the right (R) wing would determine the probability of the R-outcome. Furthermore, the pair’s spin state and both the L and R settings would determine the probability of the joint outcomes.

The main idea behind Bell’s theorem lied upon the fact that local events determine the probability of each L and R outcome. Additionally, he proposed that these local events could be confined to the backward light cones of both outcomes, as can be seen in figure 2. This implies that since there is a local event accounting for the correlation between the outcomes, then the distant outcomes are independent of each other. In turn, the probability of the joint outcomes can factorize into probabilities for the single outcomes, an event known as factorizability.

Bell’s theorem also proposed that for each incomplete quantum mechanical state ψ, the distribution of the complete states of λ are independent of the measuring devices in the L and R wings. Essentially, this suggests that the spin states of the particle pair at the source are not correlated with the settings of the measuring devices, a phenomenon known as λ-independence. Thus, failing to follow the λ-independence assumption provides evidence against locality. This assumption, along with factorizability, form the Bell inequalities- which are not supported by orthodox quantum mechanics. In other words, this means that quantum mechanics supports non-locality.

Factorizability
Main Article: Action at a Distance in Quantum Mechanics

Factorizability is a locality condition involving two concepts: parameter independence and outcome independence. The definition of parameter independence and outcome independence are as follows: Parameter independence: "The probability of a distant measurement outcome in the EPR/B experiment is independent of the setting of the nearby measurement apparatus."

Outcome independence: "The probability of a distant measurement outcome in the EPR/B experiment is independent of the nearby measurement outcome."

Thus, outcome dependence is the inverse condition of outcome independence, in which the outcome of a distant measurement is dependent on the outcome of nearby measurement. If there is λ-independence, then either outcome or parameter independence will be violated. It is not possible to reconcile violations of parameter independence, or parameter dependence, with the theory of relativity, as it relies on superluminal signalling and action at a distance. It is however possible to reconcile violations of outcome independence, or outcome dependence, with relativity as it does not require superluminal signalling, only superluminal causation and passion at a distance.

Superluminal Causation and Signaling
Main Article: Action at a Distance in Quantum Mechanics

According to Quantum Mechanics, the failure of factorizability involves non-separability and action at a distance. Non-factorizability therefore implies the possibility of superluminal causation. This is shown using Reichenbach’s common cause principle (PCC), defined as the joint probability of any distinct, correlated events, x and y, which are not causally connected to each other, and which factorize upon union of their partial and separate causes and their common cause. Let CC(x,y) denote the common causes of x and y, and PC(x) and PC(y) denote respectively their partial causes. Then, the joint probability of x and y factorizes upon the Union of their Causal Pasts (henceforth, FactorUCP), i.e., on the union of PC(x), PC(y) and CC(x,y):

"FactorUCP PPC(x) PC(y) CC(x,y) (x & y) = PPC(x) CC(x,y) (x) · PPC(y) CC(x,y) (y)."

The basic idea of FactorUCP is that the objective probabilities of events that do not cause each other are determined by their causal pasts PC(x) or PC(y), and given these causal pasts they are probabilistically independent of each other. The failure of factorizability therefore implies outcome dependence, which in turn implies superluminal causation between the distant outcomes in the EPR/B experiment. Superluminal signalling entails that a signal be transmitted across great distances at speeds greater than that of light, (i.e. almost instantaneously).

Non-separability and Quantum Non-Locality
Main Article: Action at a Distance in Quantum Mechanics

Non-separability is an important concept in quantum non-locality. There are various different types of non-separability.

State separability:"Each system possesses a separate state that determines its qualitative intrinsic properties, and the state of any composite system is wholly determined by the separate states of its subsystems."

Spatiotemporal separability: "The contents of any two regions of space-time separated by a non-vanishing spatiotemporal interval constitute two separate physical systems. Each separated space-time region possesses its own, distinct state and the joint state of any two separated space-time regions is wholly determined by the separated states of these regions."

Process separability: "Any physical process occupying a spacetime region R supervenes upon an assignment of qualitative intrinsic physical properties at spacetime points in R."

Another important concept to consider is holism. The concept of holism is that objects that are made up of a variety of parts have certain properties that are not determined by the properties of their parts. The concept of locality is related to parameter independence, and the concept of separability is related to outcome independence. It is argued that the non-locality involved in parameter dependence is different than the non-locality involved in outcome dependence. State non-separability and holism are required for outcome dependence to occur, and conversely outcome independence is associated with separability. Parameter dependence on the other hand relies on action at a distance and superluminal signalling, as discussed earlier.