Alhassanain(p) Network for Heritage and Islamic Thought

4. The relational nature of quantum states

There is a sense in which quantum states are relational on the present pragmatist approach. It is important to appreciate their relational character if one is to understand, among other things, the status of von Neumann’s “projection postulate” (wave-packet collapse), why violation of Bell-type inequalities poses no threat of non-local action, why quantum probabilities are not simply Lewisian chances, and how to resolve the “paradox” of Wigner’s friend. But first it is necessary to say what this relational nature amounts to, and to distinguish it from other senses in which quantum states have been taken to be relational.

4.1  Rovelli’s Relationism

Rovelli ([1996], [2005]) proposed a relational view of quantum states. This maintains the tight connection between a quantum state and the values of dynamical variables in that state that has come to be known as the ‘eigenstate-eigenvalue link’: dynamical variableA has valuea ons if and only if the quantum state ofs is an eigenstate, with eigenvaluea , of the self-adjoint operator uniquely corresponding toA . Rovelli assumes that both quantum state and associated values of dynamical variables describe or represent the physical condition of an individual systems 1, but only relative to someother physical systems 2. This second system will in turn have a quantum state and associated values of dynamical variables relative tos 1 (as well as tos 3,s 4,...): even though Rovelli sometimes calls it ‘the observer’, he stresses thats 2 need be neither human, conscious, classically described, macroscopic, nor “special” in any way - it is just some other quantum system. On Rovelli’s relational quantum mechanics, a quantum system has a quantum state and  associated values of dynamical variablesonly relative to (any) other distinct quantum system, and these relative states will in general differ according towhich other system one relativizes them to.

       Rovelli’s relational view of quantum states is quite different from the pragmatist view I am presenting, which begins by dismissing the eigenstate-eigenvalue link as not merely false but arising from confusion between the radically different roles of quantum state ascriptions and claims about the values of dynamical variables in quantum theory. A second basic difference concerns what each view takes quantum states to be relative to. For the pragmatist, a quantum state ascription is not relative to an arbitrary distinct quantum system, but rather to the perspective of an actual or potentialagent - some physically situateduser of quantum theory. While every actual physically situated user of quantum theory may be treated as a quantum system (by some user of quantum theory), not every quantum system is a physically situated user of quantum theory.

4.2 Quantum Bayesian Relationism

       Fuchs ([2010]) advocates what he calls a quantum Bayesian approach to quantum theory (QBism), and seeks to explore its connections to pragmatism, among other philosophies. QBism bears many similarities to the present pragmatist approach. QBism also rejects the eigenstate-eigenvalue link as a misconceived attempt to understand quantum states as yielding descriptions of physical reality. A pragmatist will surely endorse Fuchs’s ([2010]) view that quantum theory as a whole is ‘a users’ manual that any agent can pick up and use to help make wise decisions in this world of inherent uncertainty’. Moreover, QBism agrees that quantum states are relative, to the extent that different agents can consistently assign different quantum states to the same system. But unlike the present pragmatist approach, QBism is committed to a subjective Bayesian view of probability that denies that quantum probabilities derived from the Born rule can ever be authoritative for a rational agent who accepts quantum theory. The key difference is that while, for the QBist, quantum state ascriptions depend on theepistemic state of the agent who ascribes them, on the present pragmatist approach what quantum state is to be ascribed to a system depends only on thephysical circumstances defining the perspective of the agent (actual or merely hypothetical) that ascribes it. I will spell this out in more detail after pointing to a contrast with a third recent account of the relational nature of quantum states.

4.3  Reference-frame Relationism

       Bartlettet al . ([2006], [2007]) take a quantum state ascription to be relative to what they call a reference frame. In ([2006]) they motivate this as a way of resolving disputes that have arisen among physicists in a variety of contexts as to whether it is correct to assign to a system a quantum state that is a superposition or an incoherent mixture of eigenstates of some observable. The example they focus on is a dispute as to whether the quantum state of a laser operating above threshold is a coherent state (with a definite phase) or a mixture (that may be represented either as a uniform integral over projections onto coherent states of every phase, or alternatively as diagonal in a photon number basis). They offer to resolve this and analogous disputes by supposing thateach state ascription may be considered equally correct, but relative to a different reference frame - in this case, relative to a different phase reference frame. They take a reference frame to be embodied in some physical object: in the example, some local oscillator could serve as a phase standard. The laser could be consistently ascribed a coherent state relative to a correlated (‘implicated’) oscillator (such as the main beam in a homodyne detection experiment) and at the same time an incoherent state relative to an uncorrelated  (‘non-implicated’) reference frame (like the beam from an independent laser). In their view

...the whole debate presumes that quantum states only contain information about theintrinsic properties of a system. We submit that this presumption is mistaken; quantum states also contain information about theextrinsic properties of a system, that is, therelation of the system to other systems external to it, and whether or not coherences are applicable depends on the external system to which one is comparing. ([2006], p.28)

A similar analysis would apply to any analogous dispute involving an observable (such as spin-component) which may itself be thought of as relational.

       That quantum states are relational in this sense is interesting, especially because of the way it helps resolve disputes about quantum state ascription that may otherwise have been held up as counterexamples to section 2's claim that these are rare and short-lived. But while endorsing the resolution of these disputes offered by Bartlettet al . ([2006], [2007]), I take quantum states to be relational in a way distinct from relativity to reference frames in their sense.

4.4 Agent-situation Relationism and Wave-Collapse

Specifically, quantum states are relational because any ascription of a quantum state to a system relates that system to a physically characterized situation that may (but need not) be occupied by a physically situated agent. A system’s quantum state is a state appropriate for any agent bearing the relevant kind of physical relation to that system, so the same system may be ascribed different states for different physical agent situations. Note that an agent situation need not actually be occupied by any agent, just as no observer need actually occupy an inertial reference frame, and recall that the term ‘agent’ is being used very broadly so as to apply to any physically instantiated user of quantum theory, whether human, merely conscious, or neither.

       One important aspect of an agent situation is its temporal relation to the time for which a system’s quantum state is to be specified. It is by taking careful note of this relation that one can appreciate the significance of discontinuous changes in quantum states on measurement.

       Consider the following example of a so-called negative result measurement. Suppose a source produces photons linearly polarized at 45° to the vertical: each such photon is heralded by detection, after passing through a 45° oriented polarizer, of a second photon of an entangled pair produced in parametric down conversion in a suitable nonlinear crystal.[19]One such heralded photon is incident on a polarizing beam-splitter, in the vertical channel of which is located a high-efficiency photon detector. If nothing is detected, the photon is ascribed the horizontal polarization state |H ,: the measurement has projected its superposed polarization state as follows

1/√2 ( |H , + |V ,) →|H ,                    (3)

How can such projection be reconciled with the unitary evolution of the combined state of the photon and detector?

       The answer to this question is that the quantum state of the photon’s polarization is a superposition of horizontal and vertical relative to the situation of an agent prior to the decohering interaction with the detector and its environment, but horizontal relative to the situation of an agent after that interaction. Decoherence involves no violation of unitarity. Instead, it warrants an agent in using the latter quantum state rather than the former to guide its expectations after judging that the detector has failed to detect the photon - a judgment that is licensed by the form of the unitarily evolved joint quantum state (which correlates the preferred “pointer basis” of detector states to horizontally/vertically polarized basis states of the photon) and warranted by its record of the detector’s failure to detect the photon. As Zurek ([2009]) explains, the post-interaction state of the detector and its environment acts as a “witness” of the horizontal polarization quantum state that may be consulted in many independent ways by an agent. Quantum theory cannot explain that such an agent records the detector’s failure rather than success - that remains outside its purview. But it can account for the “intrasubjective” concordance of the agent’s records as it performs such multiple independent checks on the detector and the photon itself.[20]

       A standard objection to the claim that decoherence can account for definite outcomes of quantum measurements may seem to apply also to this answer.[21]Even if interactions rapidly and robustly entangle the joint state of photon and detector with the state of their environment so that their joint quantum state is a mixture of product states with no off-diagonal terms, this remains an improper mixture that cannot be understood to represent an agent’s state of ignorance as to their actual joint (pure) product state. There will be some complex collective dynamical variable on the combined photon, detector and environment system whose measurement will almost certainly display statistics distinguishable from those predicted by such a mixture of product states each correlated with a corresponding environment state. By repeatedly measuring the values of this dynamical variable on identically prepared photon+ detector+ initial environment systems, some “super-agent” could verify that their total state evolves unitarily in a way that is inconsistent with the assumption that the photon+detector system is in some (unknown) pure product state.

       This objection cannot be turned into a good argument against the reconciliation of unitary evolution and effective “collapse” offered two paragraphs earlier: That reconciliation nowhere assumed that the quantum state of photon+detector was some (unknown) pure product state. Instead, it simply assumed unitary evolution of the total state, including the environment, to show that, conditional on recording the failure of the detector to detect a heralded photon, the Born probability for a recording of any measurement of the polarization of that photon (even including joint measurements with its environment) is exactly as predicted by assignment to it of quantum state |H ,. This is whythe photon’s quantum state is |H ,, relative to the situation of an agent in a position to access records of the detector’s failure to fire.

       So an agent with a record that the detector has not fired after the decohering interaction between photon and detector (together with its environment) is warranted in ascribing polarization state |H ,to the photon. But the agent isnot thereby warranted in inferring that this photon is horizontally polarized - an inference in conformity to the eigenstate-eigenvalue link. All the Born rule authorizes such an agent to do is to adopt a maximal degree of belief (1) that, following a second (ideal) interaction involving that photon that correlates its horizontal/vertical polarization state with the decohered state of some other polarization detector, the agent would record that detector as recording horizontal. At this point the distinction between quantum state and actual polarization may seem unmotivated. Its significance will become clear in section 5, which analyzes descriptions of a system by more than one agent. It is essential to maintain a distinction between relational quantum state ascriptions and non-relational dynamical variable ascriptions in order to ensure that applications of quantum theory carry objective import.  Consider an experiment set up to investigate violations of Bell inequalities in entangled photon pairs in which Bob measures polarization of photonR along axisb while Alice measures polarization of photonL along axisa . Suppose Alice, Bob and everyone else in their expert community is warranted in agreeing that the experiments are performed on photon pairs in the entangled polarization state

|Φ+ ,=  1/√2 ( |HH , + |VV ,)            (4)

Suppose Bob’s measurement is concluded at a timet b before Alice performs her measurement. Bob then records his detector as indicating polarization ofR along (rather than orthogonal to) theb axis, and ascribes polarization state |b ,to photonL subsequent tot b. This ascription is warranted by considerations parallel to those that warranted ascribing polarization state |H ,to the photon discussed in the immediately preceding paragraphs. Before she performs her measurement onL , no such considerations warrant Alice in ascribing state |b ,to photonL . Instead, Alice is warranted in ascribing toL the same maximally-mixed polarization state given by tracing |Φ+,over the polarization Hilbert space ofR as before Bob’s measurement. The quantum state ofL relative to Alice’s agent situation is different from the quantum state ofL relative to Bob’s agent situation. This is true whether or not thereare any agents Alice and/or Bob actually occupying those situations: all that matters is that for a period of time aftert b the R detector has suitably interacted withR and its environment but theL detector has not.

       In an experiment like this, very little time will elapse between theR andL detection events. Indeed, in certain experiments the interval separating them is space-like rather than time-like. These provide further illustrations of the relational nature of quantum state ascriptions. Suppose that Alice and Bob are moving so that while Bob represents the interaction ofR with his detector to have concluded before the interaction ofL with Alice’s detector, Alice takes these events to have occurred in the opposite time-order. Alice will then be warranted in ascribing quantum state |a ,, say, toR subsequent (for her) to the timet′ a she represents as the conclusion of her measurement ofL . But ifa≠b Bob will never be warranted in ascribing state |a ,(or its Lorentz transform) toR , and nor will Alice ever be warranted in ascribing state |b ,(or its Lorentz transform) toL . Again, there need be no actual agents Alice and Bob moving in these ways. But note that specification of an agent situation here involves not only specification of a time interval, but also of a frame (inertial or otherwise) with respect to which that interval is defined.       

4.5 Why quantum probabilities are not Lewisian chances

   We can now begin to see why quantum probabilities are not simply chances, as Lewis describes them. Section 5.1 will show how this also helps reconcile quantum theory’s violation of Bell inequalities with a physically motivated locality requirement.

       Consider the following space-time diagram (in the laboratory frame), in which the diagonal lines mark boundaries of the causal past or future of the space-like separated eventsM A,M B at whichL ,R respectively interact appropriately with a polarization detector and its environment:

INSERT FIGURE 1 HERE

Suppose one asks: What is the probability att 1 that Alice’s detector will recordL with polarizationa ? Since all agree that the quantum state of the pair att 1 is |Φ+,, the answer ½ follows uniquely by the Born rule. Assume for the moment thata=b , and consider this question: What is the probability att 2 that Alice’s detector will recordL with polarizationa ? Applying the Born rule to her quantum state forL att 2, Alice will give the same answer as before, namely ½. Bob will apply the Born rule tohis quantum state forL att 2 and give the different answer 1. If quantum probabilities obey Lewis’s Principal Principle that he takes to capture all we know about chance, then at least one of these answers must be wrong, since that principle presupposes that, while the chance of an event may change as time passes, it is uniquely defined at any given time. But just as quantum state ascriptions are relational, so also are Born probabilities. Relative to Alice’s agent situation the probability is ½, relative to Bob’s agent situation the probability is 1. Neither probability is subjective, but both are correct. Each is authoritative for any agent that happens to be in the relevant agent situation, and accepting that this is so is a requirement on any agent that accepts quantum theory, whatever the actual situation of that agent.

       Notice that adapting the framework of Lewisian chance to relativistic space-time structure by allowing the chance of an event to depend not on some absolute time but rather on the time in any reference frame will not effect a reconciliation between apparently conflicting quantum probabilities Alice and Bob should assign to the same event here. For Bob in his reference frame, there will be a time interval afterM B and beforeM A during which he should assign probability 1 to Alice’s recording polarizationa on photonL . But this probability assignment could play no role in the decision-making of any agent at rest in Bob’s reference frame but within the back light-cone ofM A, since the outcome of Bob’s measurement lies outside her back light-cone and should be counted by Lewis as inadmissible information for her (assuming no superluminal signaling): the objective probability for her remains ½, as specified by the Born rule applied to her quantum state. The relativization of chance to an arbitrary foliation by space-like Cauchy surfaces would face essentially the same objection: the value of the chance at some point on a Cauchy surface would be irrelevant to the decision-making of an agent at that point. One could try to understand quantum probabilities as Lewisian chances of an event by specifying them only on space-like hypersurfaces restricted to the causal past of that event - the closure of the event’s back light-cone. But this proves problematic for a different reason. To take the value of the chance of an outcome ofM A on a space-like hypersurface within the back light-cone ofM A to be given by Alice’s Born probabilities is to ignore the relevant information provided by the record of Bob’s measurement that is available to Bob att 2: On the other hand taking the value of this chance to be given by Bob’s Born probabilities both makes it arbitrary on which space-like hypersurface withinM A’s causal past they change and raises the specter of non-local action.

       The relational nature of quantum state ascriptions and the consequence that Born probabilities are not simply time-relativized Lewisian chances may be brought home even more forcefully by examining an experiment first proposed by Peres ([2000]): a version of the experiment has recently been successfully performed. Since the experiment combines two independently interesting quantum phenomena, I begin by discussing the first - entanglement swapping - before introducing the second - delayed choice.

         Two agents, Alice and Bob, simultaneously but independently prepare pairs of polarization-entangled photons in (universally agreed) quantum state |Ψ −,= 1/√2 ( |HV ,−  |VH ,):  call Alice’s photons 1,2 and Bob’s 3,4. Alice measures the polarization of photon 1 along axisa , while Bob measures the polarization of photon 4 along axisb . Photons 2 and 3 are passed through optical fiber delays before each is incident on a switchable Bell-state analyzer incorporating a beam splitter, as indicated in the accompanying space-time sketch. A third agent, Victor, then measures the polarization along axisH of any photons emerging to the left of the beam splitter, and the polarization along axisH of any photons emerging to the right of the beam splitter. After allowing for the delay, careful timing permits recording of fourfold coincidence counts from detection of all four photons from two pairs simultaneously prepared by Alice and Bob.

INSERT FIGURE 2 HERE

Consider any such four-fold detection. The initial quantum state has the form of a product of two EPR states |Ψ −,12|Ψ −,34. This may be expanded in terms of the four Bell states

       |Ψ −,= 1/√2 (  |HV ,−  |VH ,)

|Ψ + ,= 1/√2 ( |HV ,+ |VH ,)

|Φ− ,= 1/√2 ( |HH ,− |VV ,)

|Φ+ ,= 1/√2 ( |HH ,+ |VV ,)

as follows

|Ψ −,12|Ψ −,34 = 1/2(|Ψ +,14|Ψ +,23 − |Ψ −,14|Ψ −,23 − |Φ+,14|Φ+,23 + |Φ−,14|Φ−,23)     (5)   

Analysis of the actual experimental setup shows that cases (i ) in which Victor records one photon as detected to each side of the beam splitter (with the same polarization) have non-zero Born probability only from the fourth term in (5), while cases (ii ) in which Victor records both photons as detected to the same side of the beam splitter (with opposite polarizations) have non zero Born probability only from the third term in (5).

       What polarization quantum state should Victor ascribe to a pair 1+4? In a case (i ) he should ascribe the corresponding 1+4 pair the quantum state |Φ−,14, while in a case (ii ) he should ascribe the corresponding pair 1+4 the state |Φ+,14. In either case, Victor should ascribe an entangled state to systems that have never interacted, directly or indirectly. Moreover, in either case Victor ascribes a quantum state to a pair of systemsafter each system has been detected and no longer has any independent existence. The function of such quantum state ascriptions is perfectly standard. By inserting the relevant quantum state into the Born rule, any agent in Victor’s situation can adjust its expectations concerning matters of which it is currently ignorant, namely what is recorded by Alice and Bob’s detectors. Such expectations can be (and in the actual experiment were) compared to Alice and Bob’s records in many cases of type (i ) and many cases of type (ii ). Those records returned statistics in conformity to the Born rule and in violation of Bell inequalities (as the polarization axesa ,b were suitably varied). What this experiment illustrates in a striking way is that an agent may need to form expectations concerning events that have already happened although his physical situation renders him inevitably ignorant of their outcomes. Until physical interactions have suitably correlated Victor’s records with the records of Alice and Bob’s detectors and their environment, the objective Born probabilities he derives from the quantum state for his agent situation remain his most reliable guide to belief about their records.

       There is a further feature of this experiment that involves delayed choice. Clearly, if the beam splitter were not present there would be no swapping of entanglement: 1+2 would remain entangled, as would 3+4, but there would be no entanglement across these pairs. The experiment is therefore designed to allow a “decision”, effectively as to whether or not to introduce the beam splitter, to be postponed untilafter photons 1 and 4 have been detected. (In the actual experiment, implementing the “decision” was a little more complex, and was carried out by a quantum random number generator rather than any agent, human or otherwise.) With this additional feature, whether an agent in Victor’s situation ascribes an entangled or a separable state to photons 1 and 4 depends on events that occur after these photons have been detected. The delayed-choice entanglement-swapping experiment reinforces the lesson that quantum states are neither descriptions nor representations of physical reality. In particular, it undermines the idea that ascribing an entangled state to quantum systems is a way of representing some new, non-classical, physical relation between them. To hold onto that idea in the context of this experiment would require one to maintain not only thatwhich entanglement relation obtains between a pair of photons at some time, but also whetherany such relation then obtains between them, depends on what happens to other independent systems later, after the pair has been absorbed into the environment.

       Note also that the Born probabilities flowing from these assignments of quantum states cannot here be understood as chances that conform to Lewis’s Principal Principle. That principle is supposed to explicate the role of chance in decision-making at a time by saying how an agent should base his credence of an event on the event’s chance at that time. We saw earlier that to be guided by quantum theory in his decision-making, an agent may need to take account of information to which he is privy but that must remain inaccessible to other agents at that time. We now see a situation in which quantum theory supplies objective probabilities concerning a pair of events to no agent until after those events have occurred. Clearly these probabilities cannot guide any agent in forming credences about these events ahead of time. But they can still be useful to a decision-maker like Victor who is kept in ignorance of Alice and Bob’s results.

       Such ignorance would be irremediable in a modified scenario where the choice and Victor’s  measurements arespace-like separated from Alice’s and Bob’s measurements (assuming no-superluminal signaling). In Lewis’s terminology, that scenario would render information about Alice’s and Bob’s results inadmissible for Victor for a whileafter they had already occurred in his reference frame. Even thought they are objective, Born probabilities are indexed not to a time, but to the physical situation of a potential agent relative to the events they concern. These probabilities sometimes, but not always, depend on the temporal aspect of this relation. But whether or not they do, they may depend also on further physical aspects.