REFERENCE FRAME

WHAT'S WRONG WITHTHESE ELEMENTS OF REALITY?N. Dovid Mermin

The subject of Einstein-Podolsky-Rosen correlations—those strongquantum correlations that seem toimply "spooky actions at a dis-tance"—has just been given a newand beautiful twist. Daniel Green-berger, Michael Home and AntonZeilinger have found a clever andpowerful extension of the two-particleEPR experiment to gedanken decaysthat produce more than two parti-cles.1 In the GHZ experiment thespookiness assumes an even morevivid form than it acquired in JohnBell's celebrated analysis of the EPRexperiment, given over 25 years ago.2

The argument that follows is myattempt to simplify a refinement ofthe GHZ argument given by thephilosophers Robert Clifton, MichaelRedhead and Jeremy Butterfield.3

Consider three spin-1/, particles,named 1, 2 and 3. They have originat-ed in a spin-conserving gedanken de-cay and are now gedanken flyingapart along three different straightlines in the horizontal plane. (It's notessential for the gedanken trajector-ies to be coplanar, but it makes it eas-ier to describe the rest of the geome-try.) I specify the spin state f of thethree particles in a time-honoredmanner, giving you a complete set ofcommuting Hermitian spin-space op-erators of which * is an eigenstate.

Those operators are assembled outof the following pieces (measuring allspins in units of %fi): a[, the operatorfor the spin of particle i along itsdirection of motion; a'x, the spin alongthe vertical direction; and a'y, the spinalong the horizontal direction orthog-onal to the trajectory. (Any threeorthogonal directions independentlychosen for each particle would do.

David Mermin is a professor atCornell University and director of theLaboratory of Atomic and Solid-StatePhysics. His thoughts about physicsand physicists have just been collectedtogether and published under thepreposterous title Boojums All The WayThrough (Cambridge U. P., New York,1990).

But we're going to be gedanken mea-suring x and y components of eachparticle's spin, so it's nice to think ofthe x and y directions as orthogonalto the direction of motion, since thecomponents can then be straightfor-wardly measured by passage througha conventional Stern-Gerlach mag-net.) The complete set of commutingHermitian operators consists of

1 2 3 1 2 3 1 12 3 / 1 1CTx CT G , £7V O'x (Jv , (Tv (J y (Jx . \ i.)

Even though the x and y compo-nents of a given particle's spin anti-commute—a fact of paramount im-portance in what follows—all three ofthe operators in (1) do indeed com-mute with one another, because theproduct of any two of them differsfrom the product in the reverse orderby an even number of such anti-commutations. Because they all com-mute, the three operators can beprovided with simultaneous eigen-states. Since the square of each of thethree is unity, the eigenvalues of eachare + 1 or — 1, and the 2 s possiblechoices are indeed just what we needto span the eight-dimensional space ofthree spins-V2.

For simplicity of exposition let'sfocus our attention on the sym-metric eigenstate in which each ofthe operators (1) has the eigen-value + 1. (Its state vector is* = (1/V2)(|1,1,1> - | - 1, - 1, - 1»,where 1 or — 1 specifies spin up ordown along the appropriate z axis, butyou don't need to know this. I'm onlytelling you because discussions of EPRalways write down an explicit form forthe state vector and I wouldn't wantyou to think you were missing any-thing.) Because the spin vectors ofdistinct particles commute compo-nent by component, we can simulta-neously measure the x component ofone particle and the y components ofthe other two (using three Stern-Gerlach magnets in three remoteregions of space). Since the threeparticles are in an eigenstate of allthree operators (1) with eigenvalueunity, the product of the results of thethree spin measurements has to be+ 1, regardless of which particle we

single out for the x-spin measurement.This affords an immediate applica-

tion of the EPR reality criterion4:"If, without in any way disturbing asystem, we can predict with certain-ty the value of a physical quantity,then there exists an element of phys-ical reality corresponding to thisphysical quantity." The "element ofphysical reality" is that predictablevalue, and it ought exist whether ornot we actually carry out the proce-dure necessary for its prediction,since that procedure in no way dis-turbs it. Because the product of theresults of measuring one x componentand two y components is unity in thestate tf, we can predict with certaintythe result of measuring the x compo-nent of the spin of any one of the threeparticles by measuring the y compo-nents of the two other, far awayparticles. For if both y componentsturn out to be the same then the xcomponent, when measured, mustyield the value + 1; if the two ycomponents turn out to be different,the subsequently measured x compo-nent will necessarily yield the value— 1. In the absence of spooky actionsat a distance or the metaphysicalcunning of a Niels Bohr, the twofar away y-component measurementscannot "disturb" the particle whosex component is subsequently to bemeasured. The EPR reality criteriontherefore asserts the existence of ele-ments of reality m1, m2 and mj, eachhaving the value + 1 or — 1, eachwaiting to be revealed by the appro-priate pair of far away ^-componentmeasurements.

In much the same way, we can alsopredict the result of measuring the vcomponent of the spin of any particlewith certainty, by measuring one xcomponent and one v component ofthe spins of the other two. Thereare thus elements of reality m\,, m2

and m3, with values + 1 or — 1, alsowaiting to be revealed by far awaymeasurements. All six of the ele-ments of reality m'x and m\, have tobe there, because we can predict inadvance what any one of the sixvalues will be by measurements made

© 1990 American Insnrure of Physics PHYSICS TODAY JUNE 1990

\

A First...Autotuning Temperature Controlfroml.4Kto800Kwith Control Stabilityto ± 2.5 mK—from Lake Shore.While you're having a cup of coffee, your controllerautomatically determines optimum operating parameters.Lake Shore has eliminated the time consuming set-up process for cryogenictemperature controllers with its new Models 320 and 330 HOT (Hands-Off-Temperature) Autotuning Controllers.

Easy to use. The operator simply pushes the "ON" button, keys in the desiredset point, and the HOT Temperature Controller does the rest—without any fur-ther user adjustments. It's that easy.

Temperature Sensor ———Compatible. Two, user-configurable sensor inputs arestandard with the Model 330Autotuning Temperature Con-troller. With the flip of a switch, aeach input can be independentlyconfigured for use with silicondiodes,GaALAs diodes, and platinum resistancethermometers (optional thermocoupleinput available).

Solid Functionality at an affordable price. Other features of the Model 330Autotuning Temperature Controller include:

Turn it on. Key in your setpoint. The Auto-tuning Temperature Controller automaticaldetermines optimum PID parameters basecon system characteristics.

HOT (autotuning) control ormanual PID controlTemperature range of 1.4K to800KDual sensor inputsDual display with ability to readtemperature in kelvin or CelsiusCapability to store user-definedcurves (up to 20)0.01K resolution

Softcal™IEEE-488 and RS-232C interfaces25W/50W heater output with3 range operationDC current source for heaterpower output to assure quiet,stable controlThe HOT controller is also avail-able in a single input version(Model 320)

Your time is too valuable for manual controller set-up. Let the Lake Shore HOT(Hands-Off-Temperature) ^ Controllers do all the work, and youget the results. For more ~ ^ £ " information, call (614) 891-2243.FAX: (614) 891-1392. J

LakeShoreC H O H O N I C S I N C

64 East Walnut Street, Westerville, Ohio 43081 USA Tel: (614) 891-2243 Telex: 24-5415 CRYOTRON WTVL Fax: (614) 891-

High performance in low temperature technology.

© Lake Shore Cryotronics, Inc. 1990.

Circle

REFERENCE FRAME

so far away that they cannot disturbthe particle that subsequently doesindeed display the predicted value.

This conclusion is, of course, highlyheretical, because al

x does not com-mute with ay—in fact the two anti-commute—and therefore they cannothave simultaneous values. (The oper-ators (1) are nicely chosen to hide thisfailure to commute, since the anti-commutations always occur in pairs.)But heresy or not, since the result ofeither measurement can be predictedwith probability 1 from the results ofother measurements made arbitrarilyfar away, an open-minded personmight be sorely tempted to renouncequantum theology in favor of aninterpretation less hostile to the ele-ments of reality.

In the GHZ experiment, however,as in Bell's version of the EPR, theelements of reality are demolishedby the straightforward quantum me-chanical predictions for some addi-tional experiments, entirely unen-cumbered by accompanying meta-physical baggage.

In the GHZ case the demolitionis spectacularly more efficient. Sup-pose, heretically, that the elements ofreality really do exist in each runof the experiment. While we cannotknow all six of their values, thosevalues are constrained by the factthat the values of al

xa'yay, aya^ay andayaya^, all unity in the state 4*, aregiven by the values of the correspond-ing products mlm2

ymy, m].mlml, andmym

2ym\. But if these latter three

quantities are unity, so is their com-bined product. Since each individualm'y is either + 1 or — 1 and eachoccurs twice in the combined prod-uct, that combined product is justmlmiml. So the existence of the ele-ments of reality implies that shouldwe choose to measure the x compo-nents of all three spins in the state *,the product of the three resultingvalues must once again be + 1.

The value of that product can alsobe determined without invoking dis-reputable elements of reality by asimple quantum mechanical calcula-tion, since it is just the result ofmeasuring the Hermitian operator

ala'ial. (2)

You can easily check that this opera-tor also commutes with all of theoperators (1): Once again the numberof anticommutations is always even.This is encouraging, for if the value ofthe operator (2) in the state W isinvariably to be + 1, it had better alsohave * for an eigenstate, a require-ment that is guaranteed by its com-muting with all three members of thecomplete set of commuting operators

(1) whose eigenvalues define *.However:Not only does (2) commute with

each of the operators (1), but you caneasily check that it is a simple explic-it function of them, namely, minusthe product of all three. The (crucial)minus sign arises because here, atlast, in bringing the pairs of opera-tors ay together to produce unity, oneruns up against an odd number ofanticommutations of ay's with a^'s.Since * is an eigenstate with eigen-value + 1 of each of the operators (1),it is therefore indeed an eigenstate ofthe operator (2), but with the wrongeigenvalue, opposite in sign to theone required by the existence of theelements of reality.

So farewell elements of reality!And farewell in a hurry. The com-pelling hypothesis that they exist canbe refuted by a single measurement ofthe three x components: The ele-ments of reality require the product ofthe three outcomes invariably to be+ 1; but invariably the product of thethree outcomes is — 1.

This is an altogether more power-ful refutation of the existence ofelements of reality than the oneprovided by Bell's theorem for thetwo-particle EPR experiment. Bellshowed that the elements of realityinferred from one group of measure-ments are incompatible with the sta-tistics produced by a second groupof measurements. Such a refutationcannot be accomplished in a singlerun, but is built up with increasingconfidence as the number of runsincreases. Thus in one simple versionof the two-particle EPR experiment(which I described in PHYSICS TODAY,April 1985, page 38) the hypothesis ofelements of reality requires a class ofoutcomes to occur at least 55.5% ofthe time, while quantum mechanicsallows them to occur only 50% of thetime. In the GHZ experiment, on the

other hand, the elements of realityrequire a class of outcomes to occurall of the time, while quantum me-chanics never allows them to occur.

It is also appealing to see the failureof the EPR reality criterion emergequite directly from the one crucialdifference between the elements ofreality (which, being ordinary num-bers, necessarily commute) and theprecisely corresponding quantum me-chanical observables (which some-times anticommute).

I was surprised to learn of thisalways-vs-never refutation of Ein-stein, Podolsky and Rosen. After all,quantum magic generally flows fromthe fact that it is the amplitudes thatcombine like probabilities ratherthan the probabilities themselves.But when the probabilities are zero,so are the amplitudes. Guided bysuch woolly thinking, and the failureof anybody to strengthen Bell's resultin this direction in the ensuing 25years, I recently declared in writing5

that no set of experiments, real orgedanken, was known that could pro-duce such an all-or-nothing demoli-tion of the elements of reality. With abow of admiration to Greenberger,Home and Zeilinger, I hereby recant.

References1. D. M. Greenberger, M. Home, A. Zei-

linger, in Bell's Theorem, QuantumTheory, and Conceptions of the Uni-verse, M. Kafatos, ed., Kluwer, Dor-drecht, The Netherlands (1989), p. 69.

2. J. S. Bell, Physics 1, 195 (1964).3. R. K. Clifton, M. L. G. Redhead, J. M.

Butterfield, "Generalization of theGreenberger-Horne-Zeilinger Algebra-ic Proof of Nonlocality," submitted toFound. Phys.

4. A. Einstein, B. Podolsky, N. Rosen,Phys. Rev. 47, 777 (1935).

5. N. D. Mermin, in Philosophical Conse-quences of Quantum Theory, J. T. Cush-ing, E. McMullin, eds., Notre DameU. P., Notre Dame, Ind. (1989), p. 48. •

PHYSICS TODAY JUNE 1990 1 1