aa r X i v : . [ qu a n t - ph ] S e p Quantum Theory and the Limits of Objectivity
Richard HealeyUniversity of ArizonaSeptember 2018
Abstract
Three recent arguments seek to show that the universal applicability ofunitary quantum theory is inconsistent with the assumption that a well-conducted measurement always has a definite physical outcome. In thispaper I restate and analyze these arguments. The import of the first twois diminished by their dependence on assumptions about the outcomesof counterfactual measurements. But the third argument establishes itsintended conclusion. Even if every well-conducted quantum measurementwe ever make will have a definite physical outcome, this argument shouldmake us reconsider the objectivity of that outcome.
Quantum theory is taken to be fundamental to contemporary physics in largepart because countless measurements have yielded outcomes that conform toits predictions. Experimenters take great care to ensure that each quantummeasurement has an outcome that is not just a subjective impression but anobjective, physical event. However, in the continuing controversy in quantumfoundations QBists ([1], [2]) and others ([3], [4], [5]) have come to questionand even deny the principle that a well-conducted quantum measurement has adefinite, objective, physical outcome. This principle should not be abandonedlightly: objective data provide the platform on which scientific knowledge rests. We should demand a water-tight argument before giving it up.In this paper I analyze three recent arguments that quantum theory, consis-tently applied, entails that not every quantum measurement can have a definite,objective, physical outcome. I say ‘can have’, not ‘has’, because each argument Even if no item of data is so certain as to be immune from rejection in the light of furtherscientific investigation. Recall Popper’s ([6], p. 94) famous metaphor:“Science does not rest upon solid bedrock. The bold structure of its theories rises, as itwere, above a swamp. It is like a building erected on piles. The piles are driven down fromabove into the swamp, but not down to any natural or ‘given’ base; and if we stop drivingthe piles deeper, it is not because we have reached firm ground. We simply stop when we aresatisfied that the piles are firm enough to carry the structure, at least for the time being.”
Gedankenexperiment far more extreme even than that of Schr¨odinger’scat. The first two arguments’ dependence on questionable implicit assumptionsseverely limits their significance. But I think the third argument at least suc-ceeds in deflating a certain ideal of objectivity in the quantum domain. I assumethroughout that an outcome of a quantum measurement is definite only if it isunique—an assumption rejected by Everettians such as Deutsch [7] and Wallace[8]. Assuming the objectivity of a physical outcome, an Everettian may takean argument like these considered here as offered in support of that outcome’snon-uniqueness, as suggested by the title of [12].
Brukner’s argument ([3], [4]) applies Bell’s theorem [9] to an extension of Wigner’s[10] friend scenario. My restatement of the most recent version [4] of his argu-ment renames Brukner’s characters and introduces clarifying notation.Before describing his own
Gedankenexperiment , Brukner considers Deutsch’s[7] twist on Wigner’s original friend scenario. So consider first a scenario inwhich Zeus is contemplating possible measurements on Xena’s otherwise phys-ically isolated lab X , inside which Xena has measured the z -spin of a singlespin-1/2 particle 1 prepared in the superposed state in the z -spin basis | x i = 1 / √ |↑i + |↓i ) (1)Assuming the universal applicability of unitary quantum mechanics, Zeus as-signs to the combined system 1 X after Xena’s measurement the entangled state | Φ i X = 1 / √ |↑i | “up” i X + |↓i | “down” i X ) , (2)where | “up” i X (for example) represents a state in which if Zeus were to observethe contents of Xena’s lab he would certainly (with probability 1) find herreporting the outcome of her measurement of z -spin on particle 1 as + ~ / | Φ i X by performing a mea-surement of a dynamical variable A x represented by the operator ˆ A x on H ⊗ H X ˆ A x = |↑i | “up” i X h↓| h “down” | X + |↓i | “down” i X h↑| h “up” | X . (3)This measurement will (with probability 1) yield the outcome +1 while leavingthe state | Φ i X undisturbed. After this successful verification, Zeus’s apparatusand memory establish the truth of statement A + x : “Zeus’s outcome is A x = +1”and the falsity of A − x : “Zeus’s outcome is A x = − z -spin.As Deutsch [7] pointed out, no violation of unitary quantum theory is in-volved if Xena passes a message out of her lab to Zeus reporting that she has Brukner calls this character Wigner, but I have reserved that name for another characterwith analogous powers. A z represented by the operator ˆ A z on H ⊗ H X ˆ A z = |↑i | “up” i X h↑| h “up” | X − |↓i | “down” i X h↓| h “down” | X . (4)If Zeus’s outcome is A z = +1 he may judge this to verify the statement A + z :“Xena’s outcome is z + ,” and falsify A − z : “Xena’s outcome is z − ”, while outcome A z = − A x leaves the state | Φ i X unchanged, Zeus mayfirst perform that measurement to establish the truth of A + x , then measure A z to verify the truth of A + z (or, alternatively, of A − z ). So in this preliminaryscenario Zeus has some reason to believe that not only his own measurementsbut also Xena’s measurement had a definite, physical outcome. Moreover, if hemeasures only A x he can then pass a message with its outcome to Xena, alsowithout disturbing the state | Φ i X : so Xena, too, will have reason to believethat both A + x and A + z (or A − z ) are true and that Zeus’s measurement of A x aswell as her own measurement of 1’s z -spin had a definite, physical outcome.Now consider the statements c ( A + x ): “ A + x would be true if Zeus were tomeasure A x ”, c ( A − x ): “ A − x would be true if Zeus were to measure A x ”; c ( A + z ):“Zeus’s outcome would be A z = +1 if he were to measure A z ”, c ( A − z ): “Zeus’soutcome would be A z = − A z ”. Zeus has reason tobelieve c ( A + x ) is true and c ( A − x ) is false whether or not he measures A x , since | Φ i X predicts the truth of A + x (with probability 1). Whether or not he measures A z , Zeus has reason to believe that one of c ( A + z ), c ( A − z ) is true while the other isfalse in state | Φ i X . Assuming measurements have definite, objective outcomes,he should take his conditional outcome simply to reflect Xena’s actual outcome:Xena got z + if and only if Zeus would get +1, while Xena got z − if and onlyif Zeus would get −
1. Provided that Xena’s measurement had a definite actualoutcome it follows that exactly one of c ( A + z ) or c ( A − z ) is true.After analyzing this preliminary scenario, Brukner [4] introduces his own,more complex, Gedankenexperiment . Each of Xena and Yvonne is located ina separate laboratory. These laboratories are initially completely physicallyisolated, and this isolation is preserved except for the processes specified below.An entangled pair of spin-1 / X and particle 2 in Yvonne’s lab Y . In [4] the initial state assigned to 123in the z -spin basis) is | ψ i = − sin θ/ (cid:12)(cid:12) φ + (cid:11) + cos θ/ (cid:12)(cid:12) ψ − (cid:11) , where (5) (cid:12)(cid:12) φ + (cid:11) = 1 / √ |↑i |↑i + |↓i |↓i ) (cid:12)(cid:12) ψ − (cid:11) = 1 / √ |↑i |↓i − |↓i |↑i ) . Then Xena measures the z -spin of particle 1 in her lab, while Yvonne measuresthe z -spin of particle 2 in her lab. Assume that the measurement in each labora-tory has a definite, physical outcome, registered by a particle detector, recordedin a computer (or on paper) and experienced by Xena or Yvonne respectively.Each of Zeus and Wigner is also located in a separate laboratory. Xena’s lab-oratory is located wholly within Zeus’s, while Yvonne’s is located wholly withinWigner’s. But to this point each laboratory has remained completely physicallyisolated insofar as there has been no direct physical interaction between any ofthese four laboratories.Assuming (no-collapse) quantum theory is universally applicable, there isa correct quantum state for Zeus and Wigner to assign to the joint physicalsystem consisting of the entire contents of both Xena’s and Yvonne’s labora-tories and this state evolved unitarily throughout the interactions involved ineach of their spin-component measurements. (Note that in assigning this state,Zeus and Wigner are here treating Xena and Yvonne themselves as quantum(sub)systems.) Assuming for simplicity that the spin-component measurementswere non-disturbing, we may write this joint state after Xena’s and Yvonne’smeasurements as | Ψ i XY = − sin θ/ (cid:12)(cid:12) Φ + (cid:11) + cos θ/ (cid:12)(cid:12) Ψ − (cid:11) , where (6) (cid:12)(cid:12) Φ + (cid:11) = 1 / √ | A up i | B up i + | A down i | B down i ) (cid:12)(cid:12) Ψ − (cid:11) = 1 / √ | A up i | B down i − | A down i | B up i ) . Here X represents the entire contents of Xena’s lab (including Xena) and Y represents the entire contents of Yvonne’s lab (including Yvonne), except themeasured particles 1 , | A up i , | A down i are eigenstates of ˆ A z .We may define an analogous pair of self-adjoint operators on H ⊗ H Y asfollows: ˆ B z = |↑i | “up” i Y h↑| h “up” | Y − |↓i | “down” i Y h↓| h “down” | Y ˆ B x = |↑i | “up” i Y h↓| h “down” | Y + |↓i | “down” i Y h↑| h “up” | Y where magnitude B z on 2 Y uniquely corresponds to ˆ B z and B x to ˆ B x . The state(6) predicts that the statistics of the (assumed, definite) outcomes of Zeus’s andWigner’s measurements will violate the associated Clauser-Horne-Shimony-Holtinequality S = h A z B z i + h A z B x i + h A x B z i − h A x B x i ≤ h A x B z i , for example, is the correlation function of a probability distri-bution for the outcomes of measurements of magnitudes A x , B z . The inequalityCHSH is violated, for example, by state | Ψ i XY for which S = 2 √ θ = π/ A x or A z andWigner’s measurement of B x or B z varied randomly and independently fromtrial to trial. Violation of (CHSH) by statistics collected in a large number ofsuch trials is perfectly consistent with the assumption Definite Outcomes : In every such trial each of Xena’s, Yvonne’s,Zeus’s and Wigner’s measurements has a definite, physical outcome.The assumption of
Definite Outcomes does not even make it unlikely thata large number of Zeus’s and Wigner’s outcomes in repeated trials will displaycorrelations in violation of (CHSH). Indeed the Born rule predicts that theoutcomes of Zeus’s and Wigner’s measurements will violate CHSH: if θ = π/ S = 2 √
2. Why might one think otherwise?Brukner [4] takes his argument to disprove the following postulate
Postulate (“Observer-independent facts”)
The truth-values of thepropositions A i of all observers form a Boolean algebra A . Moreover,the algebra is equipped with a (countably additive) positive measure p ( A ) ≧ for all statements A ∈ A , which is the probability for thestatements to be true. To evaluate the bearing of his argument on the assumption of
Definite Out-comes one must specify propositions purporting to describe such outcomes.Brukner’s discussion of the preliminary scenario suggests these include A + z , A − z , A + x , and A − x . The symmetry of the Gedankenexperiment further suggeststhey include propositions B + z , B − z , B + x and B − x , each of which states the out-come of an analogous measurement by Yvonne or by Wigner. Is there anyreason to believe that application of Brukner’s Postulate to the propositions B = { A + z , A − z , B + z , B − z , A + x , A − x , B + x , B − x } yields the promised no-go therem?In no repetition are both A x and A z measured—the experimental arrange-ments are mutually exclusive, as are those for B x and B z . If A x is not measured,then neither A + x nor A − x describes an actual outcome: and if B x is not mea-sured, then neither B + x nor B − x describes an actual outcome. So unless A x , B x are measured in a repetition, the propositions of all observers that describethe actual definite outcomes assumed by Definite Outcomes is not the wholeof B but merely a compatible subset B ∗ forming a Boolean algebra which mayreadily be equipped with a (countably additive) positive measure: just use theBorn probabilities from state (6) and extend this to each proposition describingthe outcome of an actual measurement by Xena or by Yvonne by equating itsoutcome to that of the corresponding measurement by Zeus or by Wigner (so,for example, A + z is true if and only if the outcome of Zeus’s measurement of A z is A z = +1, and both propositions have the same probability).5f A x , B x are measured in a repetition, the propositions of all observersdescribing the actual definite outcomes assumed by Definite Outcomes is thewhole of B . But the propositions { A + x , A − x , B + x , B − x } form a Boolean algebrawhose structure is respected by the obvious truth-assignments, and the Bornprobabilities from (6) define a probability measure on this algebra. In the ab-sence of any further constraints it is easy to extend this truth-assignment andprobability measure to the full algebra B .So the assumed actual outcomes in each trial can certainly be described bypropositions A i of all observers forming a Boolean algebra A . Moreover, thisalgebra may be equipped with a (countably additive) positive measure p ( A ) ≧ A ∈ A , which may be taken as the probability for the state-ments to be true in that trial. A no-go therorem is not derivable through theapplication of Definite Outcomes to propositions A i of all observers that describetheir actual outcomes in any, or all, repetitions of Brukner’s Gedankenexperi-ment .What happens if instead the “propositions of all observers” concern not theiractual but their hypothetical outcomes? Consider the set c ( B ) = { c ( A + z ) , c ( A − z ) , c ( B + z ) , c ( B − z ) , c ( A + x ) , c ( A − x ) , c ( B + x ) , c ( B − x ) } of subjunc-tive conditionals describing the outcomes of hypothetical measurements. As-sume that if such a measurement is actually made in a trial then the correspond-ing conditional has the same truth-value as its consequent (so, for example, if A x is measured with outcome A x = +1 then c ( A + x ) is true as well as A + x ). Unlikethe simpler scenario discussed earlier, when A + x , A − x are replaced by the corre-sponding subjunctive statements c ( A + x ) , c ( A − x ): “If A x were measured then thedefinite outcome would be +1 ( − has a truth-value if Zeus does not measure A x . Nor should c ( B + x ) , c ( B − x ) be expected to have truth-values when Wignerdoes not measure B x .Unless A x , B x are both measured in a trial, replacement of propositions aboutactual definite outcomes of a measurement by such conditionals fails to generatea Boolean algebra of propositions of all observers whose truth-value assignmentrespects that algebra. But quantum theory predicts violation of the inequalityCHSH only for the outcomes of actual measurements. Because of the physicalincompatibility of Zeus’s joint measurement of A x and A z and of Wigner’smeasurement of B x and B z , these predictions must concern four distinct kindsof trials, which is what necessitated variation of measurements by Zeus and byWigner from trial to trial. Definite Outcomes implies that the set c ( B ) forms aBoolean algebra whose structure is respected by a joint truth-assignment andis equipped with a (countably additive) positive measure at most in the caseof a repetition in which Zeus measures A x and Wigner measures B x . So theviolation of that inequality in state (6) does not refute Definite Outcomes. Asit stands,
Brukner’s argument ([3], [4]) provides no good reason to doubt thatevery quantum measurement has a definite, objective, physical outcome.In correspondence, Brukner has proposed a slight modification that avoidsthis objection and promises to strengthen the argument. In the modified sce-nario, Zeus measures A x and Wigner measures B x in every trial. As in the6impler Wigner’s friend scenario, Zeus may appeal to the epistemic objectivityof Xena’s outcome to infer that c ( A + z ) has the same truth-value as A + z , and c ( A − z ) has the same truth-value as A − z . Since
Definite Outcomes implies thatone of A + z , A − z is true and the other false, it then follows that in every repeti-tion one of c ( A + z ) , c ( A − z ) is true and the other false, even though Zeus actuallymeasures A x and not A z in that repetition. Similarly, in every repetition oneof c ( B + z ) , c ( B − z ) is true and the other is false. So in each repetition the setof propositions { c ( A + z ) , c ( A − z ) , c ( B + z ) , c ( B − z ) } always forms a Boolean algebrawhose truth-value assignment and probability distribution follow from those as-signed to the assumed actual outcomes of Xena’s and Yvonne’s measurements.This will be true in every trial of this modified scenario. Definite Outcomes now implies that every proposition in the full algebra c ( B ) has a truth-value and these truth-values respect the algebra’s structure.Moreover, any (countably additive) positive measure p ( A ) ≧ c ( B ) must be constrained by a transformed inequality obtained from CHSH byreplacing each reference to an actual outcome by a reference to the correspondinghypothetical outcome (though for A x , B x the hypothetical outcome is the actualoutcome). If quantum theory were to predict violation of this transformedinequality then it would imply that Definite Outcomes is false.But quantum theory predicts probabilities only for the outcomes of actualmeasurements, and neither A z nor B z is actually measured in this modifiedscenario. Only A x , B x and the z -spins of , are measured in each trial, andquantum theory makes no predictions of the joint probability distribution forZeus’s and Yvonne’s pairs of measurement outcomes, or that for Wigner’s andXena’s pairs of measurement outcomes. This is to be expected, since even if Definite Outcomes is true, these outcome pairs are not epistemically accessibleby any observer (including the four agents named in this scenario), so theirstatistics are of no scientific interest.
Here is a simplified restatement of the argument of Frauchiger and Renner ([12],[13]). The appendix compares its strategy to that of the arguments on which itis based and supplies a translation to the notation of [13].Four physical observers are each located in their own separate laboratories.Every laboratory is initially completed physically isolated, and this isolation ispreserved except for the processes specified below. In one laboratory observerXena has prepared a ”biased quantum coin” c in state | ready i c = 1 √ | heads i c + √ √ | tails i c . (7) Though this inference is now questionable, since in this context the antecedent ”Zeusmeasures A z ” of the counterfactuals c ( A + z ) , c ( A − z ) is not merely false but incompatible withZeus’s actual measurement of A x .
7t time t = 0 Xena ”flips the coin” by implementing a measurement on c of observable f with orthonormal eigenstates | heads i c , | tails i c by means of aunitary interaction with c . | ready i c | ready i X − = ⇒ | ψ i cX − = 1 √ | heads i c | heads i X − + √ √ | tails i c | tails i X − . (8)Here and elsewhere I put a numerical superscript n on a state to mark its unitaryevolution up to just after time t = n . X − is a system representing the entirecontents of Xena’s lab (including Xena herself, but neither c nor a qubit system s whose state she is about to prepare), while X is X − + c . | heads i X − , | tails i X − are orthonormal eigenstates of a binary indicator observable on X whose eigen-value x = 1 represents Xena’s outcome “heads” and whose eigenvalue x = − x | heads i X − = | heads i X − (9)ˆ x | tails i X − = − | tails i X − . Assume Xena’s measurement of f on c has a unique, physical outcome: either“heads” or “tails”.At time t = 1, if the outcome was “heads”, Xena prepares the state of a qubitsystem s in her lab in state |↓i s : if the outcome was “tails”, Xena prepares s in state |→i s = 1 / √ |↓i s + |↑i s ). Xena can do this by means of a unitaryinteraction between s and X , yielding the following state | ψ i cX − s = 1 √ | heads i c | heads i X − |↓i s + √ √ | tails i c | tails i X − |→i s (10)= 1 √ (cid:16) | heads i c | heads i X − |↓i s + | tails i c | tails i X − |↓i s + | tails i c | tails i X − |↑i s (cid:17) . (11)Xena then transfers system s out of her lab and into Yvonne’s lab, keeping c inher own lab.Let Y − be a system consisting of the entire contents of Yvonne’s lab (in-cluding Yvonne but not the system s transferred to her by Xena), while Y is Y − + s . At time t = 2 Yvonne measures observable S z on s with orthonormaleigenstates |↓i s , |↑i s by means of another unitary interaction within her lab,yielding state | ψ i cX − sY − = 1 √ | heads i c | heads i X − |↓i s |− / i Y − + | tails i c | tails i X − |↓i s |− / i Y − + | tails i c | tails i X − |↑i s | +1 / i Y − (12)which we can rewrite as | ψ i XY = 1 √ | heads i X |− / i Y + | tails i X |− / i Y + | tails i X | +1 / i Y ) . (13)8et y be a binary indicator observable on Y whose eigenvalue y = 1 representsYvonne’s outcome “+1 /
2” and whose eigenvalue y = − − / y | +1 / i Y = | +1 / i Y (14)ˆ y |− / i Y = − |− / i Y . Assume Yvonne’s measurement of S z on s has a unique, physical outcome:either “+1 /
2” or “ − / XY just after t = 2 may also be expressed as | ψ i XY = 1 √ √ | f ail i X |− / i Y + | tails i X | +1 / i Y ) (13a)= 1 √ | heads i X |− / i Y + √ | tails i X | f ail i Y ) (13b)= 12 √ | f ail i X | f ail i Y + | f ail i X | OK i Y − | OK i X | f ail i Y + | OK i X | OK i Y )(13c)where the states | f ail i X , | OK i X are defined by | OK i X = 1 √ | heads i X − | tails i X ) (15) | f ail i X = 1 √ | heads i X + | tails i X )and the states | f ail i Y , | OK i Y are defined by | OK i Y = 1 √ |− / i Y − | +1 / i Y ) (16) | f ail i Y = 1 √ |− / i Y + | +1 / i Y ) . At time t = 3 Zeus measures observable z on X with orthonormal eigenstates | f ail i X , | OK i X and records a unique, physical outcome: either “fail”, or “OK”.At time t = 4 Wigner measures observable w on Y with orthonormal eigenstates | f ail i Y , | OK i Y and records a unique, physical outcome: either “fail”, or “OK”.Finally, at t = 5 Wigner consults Zeus and notes the outcome of his measurementof z .In arriving at the quantum state assignment (13) (and its equivalents),Wigner has correctly applied unitary quantum theory to the specified inter-actions. Equation (13c) implies that with probability 1 /
12 (slightly more than8%) the outcomes of Zeus’s and Wigner’s measurements will both be “OK”. Wenow investigate Wigner’s reasoning about the outcomes of Xena’s and Yvonne’smeasurements in such a case.
Step 1 At t = 5 Zeus tells me that the outcome of his measurement of z on X at t = 3 was “OK”, so I infer that the unique outcome of his measurementof z on X at t = 3 was “OK”. 9 tep 2 Yvonne measured observable S z on s at time t = 2. If her outcomehad been “ − /
2” and not “+1 / z on X at t = 3 was“fail” and not “OK”. But I inferred in step 1 that the unique outcome of hismeasurement of z on X at t = 3 was “OK”. So I now infer (with probability1) that the unique outcome of Yvonne’s measurement of observable S z on s attime t = 2 was “+1 / Step 3
Xena measured observable f on c at t = 0. If her outcome had been“heads” and not “tails”, then equation (13) implies (with probability 1) thatthe unique outcome of Yvonne’s measurement of S z on s at time t = 2 was“ − /
2” and not “+1 / S z on s at time t = 2 was “+1 / f on c at t = 0 was “tails”. Step 4*
The unique outcome of my measurement of w on Y at t = 4 was“OK”. But equation (13b) implies (with probability 1) that if the unique out-come of Xena’s measurement of f on c at t = 0 had been “tails”, the uniqueoutcome of my measurement of w on Y at t = 4 would have been “fail”. So Iinfer that the unique outcome of Xena’s measurement of f on c at t = 0 was“heads” and not “tails”.Since the conclusion of step 4* contradicts the conclusion of step 3, Wigner’sreasoning has here led to a contradiction. The reasoning depended on severalassumptions, at least one of which must be rejected to restore consistency. Theseinclude the three assumptions: Universality
Quantum theory may be applied to all systems, including macro-scopic apparatus, observers and laboratories.
No collapse
When an observable is measured on a quantum system in aphysically isolated laboratory, the state vector correctly assigned by an externalobserver to the combined system+laboratory evolves unitarily throughout.
Unique outcome
A measurement of an observable has a unique, physicaloutcome.Unique outcome corresponds to what Frauchiger and Renner [13] call (S).The appendix discusses the relation between these three assumptions and Frauchigerand Renner’s assumptions (C), (Q), and (S). But step 4* depends on an addi-tional assumption that should be questioned and, I argue, rejected:
Intervention Insensitivity
The truth-value of an outcome-counterfactual isinsensitive to the occurrence of a physically isolated intervening event.An outcome-counterfactual is a statement of the form O t (cid:3) → O t where O t states the outcome of a quantum measurement at t , t < t , and A (cid:3) → B means “If A had been the case then B would have been the case”: An event thenintervenes just if it occurs in the interval ( t , t ), and it is physically isolated ifit occurs in a laboratory that is then physically isolated from laboratories where O t , O t occur.To see the problem with step 4* of Wigner’s reasoning, focus on the outcome-counterfactual “If the unique outcome of Xena’s measurement of f on c at t = 0 had been “tails”, the unique outcome of my measurement of w on Y at10 = 4 would have been “fail”.” Zeus’s measurement of z on X at t = 3 wasan intervening event that occurred in Zeus’s laboratory Z ∪ X (taken now toencompass the laboratory X on which he performs his measurement of z ). At t = 3, Z ∪ X is still physically isolated from Y and W . So the assumption of Intervention Insensitivity would license step 4* of Wigner’s reasoning.But
Intervention Insensitivity actually conflicts with the other assumptionsof the argument. To see why, consider how Wigner should apply quantum theoryto Zeus’s measurement of z on X at t = 3, in accordance with Universality and
No collapse . Equation (13a) implies | ψ i XY = 1 √ (cid:20) √ | f ail i X |− / i Y + 1 √ | f ail i X − | OK i X ) | +1 / i Y (cid:21) (17)Assume for simplicity that Zeus’s measurement on X is non-disturbing. Wignerknows that Zeus made a non-disturbing measurement of z on X at t = 3. Sothe state he should assign to XY Z immediately following this measurement is | ψ i XY Z = 1 √ √ | f ail i X | “ f ail ” i Z |− / i Y ++ √ ( | f ail i X | “ f ail ” i Z − | OK i X | “ OK ” i Z ) | +1 / i Y = 1 √ | heads i X (cid:16) | f ail i Y + | OK i Y (cid:17) | “ f ail ” i Z + (cid:16) | OK i Y − | f ail i Y (cid:17) | “ OK ” i Z + | tails i X (cid:16) | f ail i Y + | OK i Y (cid:17) | “ f ail ” i Z − (cid:16) | OK i Y − | f ail i Y (cid:17) | “ OK ” i Z (18)What can Wigner legitimately infer about Xena’s outcome at t = 0? Prior to t = 4 he has yet to perform his own measurement of w on Y , and prior to t = 5he remains unaware of the outcome of Zeus’s measurement of z on X at t = 3.But even before t = 4 Wigner can still use | ψ i XY Z to reason hypotheticallyabout Xena’s outcome, conditional on Zeus’s and his own measurements bothhaving the outcome “OK”: on learning at t = 5 that these antecedents are true,he can then infer the truth of the consequent of this conditional. Wigner shouldtherefore replace the incorrect reasoning of step 4* as follows. Step 4
Assume Zeus’s measurement of z on X at t = 3 had a unique, physicaloutcome and that there are then no interactions among X , Y , Z prior to t = 4.Then the state of XY Z at t = 4 is | ψ i XY Z = 1 √ | heads i X (cid:16) | f ail i Y + | OK i Y (cid:17) | “ f ail ” i Z + (cid:16) | OK i Y − | f ail i Y (cid:17) | “ OK ” i Z + | tails i X (cid:16) | f ail i Y + | OK i Y (cid:17) | “ f ail ” i Z − (cid:16) | OK i Y − | f ail i Y (cid:17) | “ OK ” i Z (19)Suppose that Wigner’s unique physical outcome on measuring Y at t = 4 were“OK”. Now consider the hypothesis that Xena’s outcome at t = 0 was “tails”.11quation (19) then implies that the probability of Wigner’s outcome “OK”would have been 1 /
6. On the alternative hypothesis that Xena’s outcome at t = 0 was “heads”, equation (19) also implies that the probability of Wignergetting outcome “OK” would have been 1 /
6. So if Wigner were to get outcome“OK” for his measurement at t = 4 his knowledge of this outcome would notentitle him to infer the outcome of Xena’s measurement at t = 0. Indeed,application of Bayes’s theorem would lead him to conclude that knowledge ofthe outcome of his measurement at t = 4 should have no effect on his estimate ofthe probabilities of Xena’s possible outcomes: they remain prob (“heads”) = 1 / ,prob (“tails”) = 2 / t = 4, and again after further conditionalizing on either possibleoutcome of Zeus’s measurement at t = 3.Consider, for purposes of contrast, how Wigner should reason if he knewthat Zeus performed no measurement at t = 3. In that case he should assignthe following state to XY at t = 4: | ψ i XY = 1 √ | heads i X |− / i Y + √ | tails i X | f ail i Y ) . (20)Knowledge of the outcome “OK” of his own measurement of w at t = 4 wouldthen entitle him to conclude (with probability 1) that the outcome of Xena’smeasurement of f on c at t = 0 was “heads”. This conclusion follows by aninference that parallels step 4* of the reasoning discussed previously. Unlikestep 4* itself, the parallel inference is valid because of the assumption that Zeusperformed no intervening measurement.But that same assumption invalidates the premise of step 1 of the reasoningdiscussed previously. Failing the conclusion of step 1, Wigner would no longerbe entitled to take steps 2 and 3. So if he knew that Zeus performed no measure-ment at t = 3 then Wigner could no longer validly conclude that the outcomeof Xena’s measurement of f on c at t = 0 was “tails”. In this contrasting case,Wigner should correctly, and consistently, conclude that the unique outcome ofXena’s measurement of f on c at t = 0 was “heads”.The preceding analysis of the two contrasting cases (with, and without,Zeus’s intervening measurement) shows clearly why Intervention Insensitivity must be rejected, as inconsistent with
Universality and
No collapse . But itmay appear to raise a worry about locality. For how can a physically isolatedintervening event like Zeus’s distant measurement on X have such an impacton Wigner’s reasoning about local matters in this scenario?The form of the question suggests an answer to the worry it seeks to ex-press. Zeus’s measurement on X certainly does not influence the outcome ofXena’s measurement: if it did, the influence would not be non-local but time-reversed, since Zeus’s measurement occurred later than Xena’s! Xena’s outcomeis what it is, irrespective of Zeus’s measurement. If Zeus’s measurement wereto influence anything it would be Wigner’s outcome, not Xena’s. But Wigner’soutcome “OK” has the same probability (1 /
6) whether or not Zeus performs hismeasurement. It is only the correlation between Xena’s and Wigner’s outcomesthat differs between the cases where Zeus does and does not measure z on X .12hile Zeus’s measurement modifies this correlation, it does so despitebeing causally unrelated to any of its constituent events. This intervention sen-sitivity is not an instance of non-local causal influence. The suspicion that itis may arise from the view that correlations in non-separable states like | ψ i XY are causal because they specify probabilistic counterfactual dependence betweenthe outcomes of distant measurements in violation of Bell inequalities [9]. Whilecontroversy continues [15] as to whether such counterfactual dependence con-stitutes or evidences non-local influence, there are well-known strategies fordenying that it does. So the failure of
Intervention Insensitivity raises no new worry about non-locality.
I first heard this argument in a talk by Matthew Pusey [16], who there credits itto Luis Masanes. They should not be held responsible for my own restatementand further development of the argument.Once again, the argument is set in the context of a
Gedankenexperiment fea-turing four experimenters. For variety I have changed their names to Alice, Bob,Carol and Dan. But while Carol and Dan perform difficult but technically feasi-ble lab experiments, Alice and Bob are credited with even more extreme abilitiesthan the Zeus and Wigner who figured in the previous arguments (though theirexercise of these powers involves no violation of unitary quantum theory).Each of Alice and Bob are in their own separate laboratories, totally physi-cally isolated except for a shared Bohm-EPR pair of spin-1 / A a corresponding tooperator ˆ A a with eigenvalues { +1 , − } on particle 1, while Bob is to measuremagnitude B b corresponding to operator ˆ B b with eigenvalues { +1 , − } on parti-cle 2. a, b label two directions in space along which Alice and Bob (respectively)set the axes of their spin-measuring devices. Alice will choose setting a andperform measurement of A a at spacelike-separation from Bob’s choice of setting b and measurement of B b .But before performing these measurements, Alice and Bob first delegate asimilar task to their friends, Carol and Dan respectively. Carol occupies herown separate laboratory, initially totally physically isolated from Alice’s: Danoccupies his own separate laboratory, initially totally physically isolated fromBob’s. Carol and Dan perform measurements on the Bohm-EPR pair: Carolmeasures (normalized) spin-component C c on particle 1, while Dan measures(normalized) spin-component D d on particle 2. Carol’s choice of setting c andmeasurement of C c are each spacelike-separated from Dan’s choice of setting d and measurement of D d . Assume Carol’s and Dan’s measurements each have adefinite, physical outcome that is registered, recorded and experienced by themseparately in their labs. My preferred strategy [14] depends on an interventionist approach to causal influence.
13t is important to note that Alice and Carol both perform their measurementson the very same particle 1, and that Bob and Dan perform their measurementson the very same particle 2. To make this possible, after performing Carol’smeasurement particle 1 is transferred out of her lab and into Alice’s lab, andafter performing Dan’s measurement particle 2 is transferred out of his lab andinto Bob’s lab. Assume that measurement causes no physical “collapse” of thequantum state, so that each spin-measurement proceeds in accordance with aunitary interaction between the measured particle and the rest of the exper-imenter’s lab, and that this is consistent with its having a definite, physicaloutcome recorded by the experimenter in that lab. It follows that Carol’s mea-surement entangles the state of her lab C with that of particle 1, while Dan’smeasurement entangles the state of his lab D with that of particle 2.But Alice and Bob use their superpowers to undo these entanglements byapplying very carefully tailored interactions, in the first case between 1 and C , and in the second case between 2 and D . This restores C and D to theiroriginal states, and also restores the original spin-entangled state of 1 + 2. Thatis how it is possible for Alice and Bob to perform spin-measurements on thesame Bohm-EPR pair as Carol and Dan.By assumption, we now have a situation in which successive measurementsof spin-component (in the c and a directions) have been performed on parti-cle 1 of an individual Bohm-EPR pair, while successive measurements of spin-component (in the d and b directions) have been performed on particle 2 ofthat pair. By assumption, each of these measurements has a definite, physicaloutcome registered, recorded and experienced by an experimenter in his or herlaboratory. Finally suppose that this entire situation is repeated very manytimes, each time with a different Bohm-EPR pair, giving rise to a statisticaldistribution of results for the four outcomes in each trial.We may use quantum theory to predict the corresponding probability distri-bution by applying the Born rule to appropriate quantum states. From Alice’sperspective, events in a given trial unfold in the following sequence. At time t the particles are in state | ψ i = 1 √ |↑i |↓i − |↓i |↑i ) (21)while C, D are in states | ready i C , | ready i D respectively. Then Dan measuresthe d -spin of 2 by means of a unitary interaction ˆ U D as followsˆ U D |↓ d i | ready i D = | d -down i D (22)ˆ U D |↑ d i | ready i D = | d -up i D . So at time t when Dan has recorded the definite outcome as either d -down or d -up, Alice assigns the following state to 12 D Ψ = 1 √ |↑ d i | d -down i D − |↓ d i | d -up i D ) . (23)14hortly after t Carol measures the c -spin of particle 1 by a unitary interactionˆ U C ˆ U C |↓ c i | ready i C = | c -down i C (24)ˆ U C |↑ c i | ready i C = | c -up i C . So at time t when Carol has recorded the definite outcome as either c -down or c -up, Alice assigns the following state to 12 DC Ψ = 1 √ U C |↑ d i | ready i C | d -down i D − ˆ U C |↓ d i | ready i C | d -up i D ) (25)Next Alice “undoes” the effects of Carol’s measurement by applying an interac-tion between 1 and C with unitary ˆ U † C , and assigns the state Ψ at time t to12 D (which is no longer entangled with that of C )Ψ = 1 √ |↑ d i | d -down i D − |↓ d i | d -up i D ) . (26)Shortly after t , Alice measures a -spin on 1 and at time t gets a definite, phys-ical outcome of either a -down or a -up. Then Bob “undoes” Dan’s measurementon particle 2 by implementing an interaction in accordance with unitary ˆ U † D ,before measuring the b -spin of 2 and at time t getting a definite, physical out-come of either b -down or b -up.By applying the Born rule to the Bohm-EPR spin-state at t , Alice predictsthe probabilistic correlation function E ( c, d ) for Carol’s and Dan’s measurementoutcomes E ( c, d ) = − cos( c − d ) . (27)To predict the correlation function E ( a, d ) for Alice’s and Dan’s measurementoutcomes, Alice reasons as follows. If Carol had performed no measurement and C and 1 had never interacted, then between t and t Alice and Dan would justhave been recording a correlation between outcomes of an a -spin measurementon 1 and an earlier d -spin measurement on 2—a standard Bell experiment withsettings and measurements performed at timelike separation. For such a case,the Born rule predicts E ( a, d ) = − cos( a − d ) . (28)In the present case, C and 1 interacted twice between t and t , but theseinteractions had no overall effect on the state of the joint system 12 D at thetime when Alice performed her measurement of a -spin: its state was the sameat t as it had been at t (Ψ = Ψ ). It follows that in the present case alsoquantum theory predicts the correlation function E ( a, d ) = − cos( a − d ) . (29)After the effects of Dan’s measurement on 2 have been “undone” by Bob’simplementation of the interaction ˆ U † D , Alice should again recognize that the15utcomes of her measurement of a -spin on 1 and Bob’s spacelike-separatedmeasurement of b -spin on 2 constitute a record of a correlation in a standard(spacelike separated) Bell experiment, with predicted correlation function E ( a, b ) = − cos( a − b ) . (30)So far we have been considering the events involved in a single trial fromAlice’s perspective. But those same events should also be considered from theperspective of Bob. If the labs of Alice, Bob and friends are all in the samestate of motion, then the events we have considered will play out in the samesequence also from Bob’s perspective. But it is well known that the time-orderof spacelike separated events is not invariant under transformations of inertialframe.Suppose that Alice’s lab and Carol’s lab are in one state of motion relative toframe F (moving to the right at speed v , say), while Bob’s lab and Carol’s lab arein a different sate of motion (moving to the left at speed v , say). To make surethat Alice is in position to manipulate 1 and C we may assume that they bothremain inside, and move together with, Alice’s lab A : and to make sure thatBob is in position to manipulate 2 and D we may assume that they both remaininside, and move together with, Bob’s lab B . This arrangement is depicted inFigure 1. Relative to the state of motion of Bob and Dan, the same events playout over a period marked by the sequence of times h t ∗ , t ∗ , t ∗ , t ∗ , t ∗ , t ∗ i .Note that in the ∗ ’d frame Carol’s measurement precedes Dan’s and Bob’sprecedes Alice’s. Most important, note that the state of 12 C is the same at t ∗ asat t ∗ . Paralleling Alice’s reasoning, Bob should therefore conclude that in thissituation the correlation function for his outcome when measuring the b -spin of2 and Carol’s outcome when measuring the c -spin of 1 is E ( b, c ) = − cos( b − c ) . It is a central assumption of this third argument that every spin measure-ment by Alice, Bob, Carol or Dan has a definite, physical outcome—either spinup or spin down with respect to the chosen direction. It follows that in a longsequence of trials of the
Gedankenexperiment just described there will be a sta-tistical distribution of actual outcomes, with a set of outcomes that may belabeled h a, b, c, d i in each trial. Statistical correlations between pairs of actualexperimental outcomes may be represented in the usual way by statistical corre-lation functions corr ( a, b ) , corr ( b, c ) , corr ( c, d ) , corr ( a, d ). It follows that thesestatistical correlations will satisfy the inequality | corr ( a, b ) + corr ( b, c ) + corr ( c, d ) − corr ( a, d ) | ≤ . (31)Note that no locality assumption is required to derive this inequality here, sinceit is mathematically equivalent to the existence of a joint distribution over theactual, physical outcomes whose existence has been assumed [17].But we saw that quantum mechanics predicts probabilistic correlation func-tions E ( a, b ) , E ( b, c ) , E ( c, d ) , E ( a, d ) for these pairs of outcomes that may becompared to the inequality | E ( a, b ) + E ( b, c ) + E ( c, d ) − E ( a, d ) | ≤ . (32)16t is well known that quantum theory predicts violation of inequality (32) for cer-tain choices of directions a, b, c, d . If the particles and labs of the experimentersin the Gedankenexperiment had been at relative rest, then the choice of four di-rections in a plane defined by rotations of a ◦ = 0 ◦ , b ◦ = 45 ◦ , c ◦ = 90 ◦ , d ◦ = 135 ◦ from a fixed axis would yield maximal violation of (32) with predicted value | E ( a, b ) + E ( b, c ) + E ( c, d ) − E ( a, d ) | = 2 √ . (33)The relativistic relative motion of labs and particles makes it necessary to takeaccount of the associated Wigner rotation of vectors, affecting the predictedvalue for this choice of directions. But the inequality is still maximally violatedfor a different choice of directions . [18] specifies the necessary directions in section 4. Rather than being coplanar (withrespect to frame F ) these may be chosen to lie on a cone centered on the direction of motionof the lab in which that spin measurement is performed. Conclusion
Each of the three arguments analyzed in this paper sought to establish a contra-diction between the universal applicability of unitary quantum theory and theassumption that a well-conducted quantum measurement always has a definite,physical outcome. The first argument succeeded in doing so only by implicitlyrelying on assumptions that the work of Bell [9] and Kochen and Specker [19]gives us good reasons to reject—in Einstein’s [20] words, that in the circum-stances described in the associated
Gedankenexperiment the individual system(before the measurement) has a definite value for all variables of the system,and more specifically, that value which is determined by a measurement of thisvariable. Failing some such naive realist assumptions, nothing justifies the ar-gument’s application of quantum theory to predict probabilities for outcomesof hypothetical measurements which would be incompatible with those actuallyperformed.Though it does not rely on such naive realist assumptions, the second argu-ment also depends on a superficially plausible assumption about the outcomesof counterfactual measurements I called intervention insensitivity , according towhich the truth-value of an outcome-counterfactual is insensitive to the occur-rence of a physically isolated intervening event. But in the circumstances of theassociated
Gedankenexperiment , intervention insensitivity is itself incompatiblewith the universal applicability of unitary quantum theory. Since a contradic-tion then follows even if each (well-conducted) quantum measurement does not have a definite, physical outcome in the
Gedankenexperiment , the argumentdoes not establish its intended conclusion.Unlike the first two arguments, the third argument relies on no implicitassumptions about the outcomes of hypothetical measurements, since all theoutcomes it considers are assumed to be actual. I think it succeeds in showingthat, in the circumstances described in the associated
Gedankenexperiment , theuniversal applicability of unitary quantum theory implies (with probability ap-proaching 1) that there is no consistent assignment of values to the (supposedlydefinite, physical) outcomes of the measurements in the sequence of trials thereconsidered.This result prompts further reflection on how to understand quantum the-ory. But the circumstances of the
Gedankenexperiment in the third argumentare so extreme as forever to resist experimental realization. There are no fore-seeable circumstances in which the argument would require us to deny that awell-conducted quantum measurement has a definite, physical outcome. Thearguments considered in this paper give us no reason to doubt the sincerity ortruth of experimenters’ reports of definite, physical outcomes. But I think thethird argument should make us reconsider the extent and nature of their objec-tivity. This paper was intended to both motivate and prepare the way for thepursuit of that project. 18
Appendix
In restating the argument of ([12], [13]) I have changed the notation to try tomake it easier to follow. The following table supplies a translation between mynotation and that used in [13].Agent Lab Measured System Measured observable Other observable¯ F ! Xena L ! X R ! c heads/tails ! fF ! Yvonne L ! Y S ! s up/down ! S z ¯ W ! Zeus • ! Z ¯ L ! X ¯ w ! z • ! xW ! Wigner • ! W L ! Y w ! w • ! y Readers familiar with Wigner’s original “friend” argument [10] will be primedto attribute extraordinary powers to the experimenter I have named Wigner, andI thought it appropriate to name a second character with such almost “God-like”powers Zeus. This naturally suggested also giving the experimenters chargedwith less extraordinary tasks names whose initial letters are also at the end ofthe alphabet, with corresponding labels for their labs and measured observables.While such changes are merely cosmetic, my restatement deliberately lacksone feature emphasized by the authors of the argument of [13] that they call“consistent reasoning”, illustrate in their Figure 1, and formalize in their as-sumption (C). Both in the original and in my restatement it is Wigner ( W )whose reasoning is the ultimate focus of the argument. But the authors of theoriginal argument consider it important that Wigner’s reasoning incorporatesthe reasoning of the other experimenters ( via assumption (C)).It is vital to check whether Wigner’s reasoning is both internally consistentand consistent with the reasoning of the other experimenters in this Gedanken-experiment . My restatement makes it clear how Wigner can consistently applyquantum theory without considering the reasoning of any other experimenters.But are the conclusions of this independent reasoning by Wigner consistent withthose of the other experimenters, based on their own applications of quantumtheory? Indeed they are, provided each experimenter has applied quantum the-ory correctly . The problem with the argument of Frauchiger and Renner is thatone experimenter (Xena/ ¯ F ) has applied quantum theory incorrectly .Recall step 4* of the reasoning in my restatement of this argument (see § z renders it fallacious. Frauchiger and Renner initially attributeparallel reasoning to Xena/ ¯ F and then use assumption (C) to attribute itsconclusion also to W igner. To see where things go wrong if Xena/ ¯ F reasonsthis way, I quote from [13]. 19Specifically, agent ¯ F may start her reasoning with the two state-ments s ¯ FI = “If r = tails at time n : 10 then spin S is in state |→i S at time n : 10” s ¯ FM = “The value w is obtained by a measurement of L w.r.t. { π Hok , π
Hfail } ”.”They conclude that ¯ F can infer from s ¯ FI and s ¯ FM that statement s ¯ FQ holds: s ¯ FQ = “If r = tails at time n : 10 then I am certain that W will observe w = f ail at n : 40”.Starting with s ¯ FQ , they then apply assumption (C) to the reasoning of theother agents successively, eventually to establish that W igner may conclude s W = “If ¯ w = ok at time n : 30 then I am certain that I will observe w = f ail at n : 40”,which (given (S)) is inconsistent with W ’s independent conclusion (based onassumption (Q)) s WQ =“I am certain that there exists a round n ∈ N ≥ in which it isannounced that ¯ w = ok at time n : 30 and w = ok at n : 40.”But this chain of reasoning is based on a mistaken starting point, since ¯ F has applied quantum theory incorrectly in asserting statement s ¯ FQ . Compare s ¯ FQ with the corresponding conclusion of Wigner’s fallacious reasoning in step 4* of §
3: “If the unique outcome of Xena’s measurement of f on c at t = 0had been “tails”, the unique outcome of my measurement of w on Y at t = 4 would have been “fail”.Agent ¯ F ’s reasoning was equally fallacious here. The problem starts withstatement s ¯ FI : ¯ F is correct to assign state |→i S to S at time n : 10 for certainpurposes but not for others. Suppose, for example, that ¯ F had “flipped thequantum coin R ” by passing that system through the poles of a Stern-Gerlachmagnet. By applying unitary quantum theory, ¯ F should conclude that this willinduce no physical collapse of R ’s spin state but entangle it with its translationalstate, and thence with the rest of her lab [21]. So while ¯ F would be correctthen to assign state |→i S to S at time n : 10 for the purpose of predicting theoutcome of a subsequent spin measurement on S alone, she would be incorrectto assign state |→i S to S at time n : 10 for the purpose of predicting correlationsbetween S (or anything with which it subsequently interacts) and her lab ¯ L (oranything with which it subsequently interacts).By using the phrase ‘is in’, statement s ¯ FI ignores the essential relativity of S ’s state assignment at time n : 10 to these different applications. By using s ¯ FI to infer s ¯ FM , agent ¯ F is, in effect, taking ¯ F ’s coin flip to involve the physicalcollapse of R ’s state rather than the unitary evolution represented by equation208). So agent ¯ F is mistaken to assert s ¯ FQ , and W would be wrong to incorporatethis mistake in his own reasoning by applying assumption (C).Frauchiger and Renner [13] justify ¯ F ’s inference from s ¯ FI and s ¯ FM to s ¯ FQ byappeal to assumption (Q). I have argued that ¯ F is not justified in asserting s ¯ FQ ,since ¯ F is justified in using the state assignment licensed by s ¯ FI for the purposeof predicting the outcome of a measurement on S only where S ’s correlationswith other systems (encoded in an entangled state of a supersystem) may beneglected. But the sequence of interactions in the Gedankenexperiment succes-sively entangle the state of S with those of R , ¯ L , L and ¯ W . So in reasoningabout the outcome of W ’s measurement of w , ¯ F must take account of thisprogressive entanglement of the states of S and ¯ W .Specifically, to predict the outcome of W ’s measurement of w , ¯ F must rep-resent that measurement as the second part of W ’s joint measurement on thesystem ¯ W + L . This interaction between W and ¯ W was represented in § W and L that undercuts ¯ F ’s justification for using the stateassignment |→i S in inferrring s ¯ FQ from s ¯ FI and s ¯ FM . Only by neglecting the priorinteraction between ¯ W and L can ¯ F draw the erroneous conclusion s ¯ FQ . W igner can reason consistently about the unique, physical outcomes of allexperiments in the Gedankenexperiment of ([12], [13]) without any appeal tothe reasoning of the other agents involved. Each of these other agents mayreason equally consistently. And their collective reasoning is perfectly in accordwith assumption (C) as well as the universal applicability of unitary quantumtheory and the existence of a unique, physical outcome of every measurementthat figures in the
Gedankenexperiment of ([12], [13]).
Acknowledgement 1
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