Waiting for a Quantum Bus

Waiting for the quantum bus: The flow of negative probability

A.J. Bracken n, G.F. MelloySchool of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia

a r t i c l e i n f o

Article history:Received 21 July 2014Received in revised form30 August 2014Accepted 2 September 2014Available online 28 September 2014

Keywords:Quantum probabilityProbability backflowNegative probabilityQuantum paradox

a b s t r a c t

It is 45 years since the discovery of the peculiar quantum effect known as probability backflow, and it is20 years since the greatest possible size of the effect was characterized. Recently an experiment has beenproposed to observe it directly, for the first time, by manipulating ultra-cold atoms. Here a non-technicaldescription is given of the effect and its interpretation in terms of the flow of negative probability.

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When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics

1. Introduction

Einstein's conviction that God does not play at dice with theuniverse is now generally seen as misguided. The quantumworld,unlike our everyday classical world, is inherently probabilitistic innature. But it may yet surprise the reader to learn that even inregard to probability itself, the quantum and classical worlds donot always behave in the same way.

Forty-five years ago, British physicist G. R. Allcock publishedthree papers (Allcock, 1969), since widely-cited, on the so-calledarrival-time problem in quantum mechanics. This is the prob-lem of determining the probability distribution over times atwhich a quantum particle may be observed at a given position. Itis converse to a problem that quantum mechanics deals withroutinely, namely to predict the probability distribution overpositions at which a quantum particle may be observed at a giventime. The difference between the two looks slight, but the arrival-time problem is by no means as straightforward as its counterpart,and it has led to continuing argument and research over manyyears (Aharonov, Oppenheim, Popescu, Reznik, & Unruh, 1998;Kijowski, 1974; Muga & Leavens, 2000).

In his studies of the arrival-time problem, Allcock noted a verypeculiar quantum effect that has come to be known as quantumprobability backflow. The effect has provoked continuing interest

in its own right (Bracken & Melloy, 1994; Eveson, Fewster, & Verch,2005; Halliwell, Gillman, Lennon, Patel, & Ramirez, 2013; Melloy& Bracken, 1998a, 1998b; Muga, Palao, & Leavens, 1999; Nielsen,2010; Penz, Grubl, Kreidl, & Wagner, 2006; Strange, 2012; Yearsley,Halliwell, Hartshorn, & Whitby, 2012), leading to a recent proposalto observe it directly, for the first time, by manipulating atoms inthe ultra-cold environment of a so-called BoseEinstein conden-sate (Palmero, Torrontegui, Muga, & Modugno, 2013).

2. Waiting for the classical bus

To appreciate the strangeness of quantum probability backflow,consider a comparable everyday phenomenon. Imagine a longstraight road running past your front door, from far on your left tofar on your rightthink of it as the X-axis, with your house locatedat position x0. You have been reliably informed that there is abus somewhere on the road, possibly to your left, or possiblyalready to your right, but definitely travelling in the left-to-rightdirection. Moreover, the speed of the bus is definitely constant.

Your informants were somewhat uncertain as to just what thatspeed is, but they were able to suggest a distribution of probabil-ities over possible left-to-right velocity values. Fig. 1 provides anexample, where the velocity scale is kilometres per hour. In thisexample, the bus is most likely travelling from left to right at about20 km/h.

Your informants were also not quite sure where the bus waslocated on the road at the time they spoke to you, but they were

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able to suggest another distribution of probabilities, over possiblepositions of the bus along the X-axis at that initial time. Fig. 2shows an example, where the distance scale is kilometres. In thisexample there are matching probability distributions to your leftand right, and so a 5050 chance that the bus is still approachingat the initial time, and that it is about 1 km away.

You wait by the road in the hope of flagging down the bus ifand when it reaches you from the left. The longer you wait, themore doubtful you become that the bus is still on its way. Commonsense tells you that the probability of the bus being somewhere toyour left is not increasing as time passes, given that it is travellingfrom left to right.

Supporting your intuition, Fig. 3 shows distributions of prob-ability over positions of the classical bus at successive times,starting from the assumed initial distribution as in Fig. 2, andassuming a distribution of velocities as in Fig. 1. The probabilityflows steadily towards the right, in the same direction as thevelocity of the bus. This is confirmed in Fig. 4, which shows theprobability that the bus is still on its way from your left, startingfrom the value 0.5, or 50%, and decreasing steadily towards zero.

3. Waiting for the quantum bus

Waiting for a quantum bus is different. To this end, consider a freeparticlewith a mass near that of an oxygen atom, saytravellingalong the X-axis, and subject to the laws of non-relativistic quantummechanics. Such a quantum bus has a wave function satisfyingSchrdinger's equation. This wave function can be chosen so thatthe distributions of probabilities over possible velocities and initialpositions are again as in Figs. 1 and 2, where now distances are

measured in multiples of 105 m (tens of microns), and velocities intens of microns per second.

More generally, all wave functions can be considered for whichthe quantum bus, like the classical one, has a (constant) velocitythat is definitely directed from left to right, with uncertainmagnitude, and an initial position that is also uncertain.

While these conditions seem to be quite analogous to those ofthe classical bus, something completely counter-intuitive canhappen in the quantum case.

Despite the fact that the bus is definitely travelling from left toright, the probability of finding it to your left on measuring itsposition may increase as the time of measurement increases.

Because the bus is certainly somewhere on the line at all times,the probability of finding that its position lies to your right mustdecrease accordingly during this same time interval. In otherwords, probability must flow backwards across position x0, inthe opposite direction to the velocity of the bus, even though thereis no apparent force acting to push the probability backwards.

Fig. 5 shows possible probability distributions over positions ofthe quantum bus at successive times, arising from a choice ofinitial wave function consistent with the probability distributionsshown in Figs. 1 and 2. The curves again show movement of theprobability distribution from left to right, and are not greatlydissimilar to those shown for the classical bus in Fig. 3. What ismore, the graph in Fig. 6, showing the changing value of theprobability that the quantum bus is still to your left, does notappear much different from its classical counterpart in Fig. 4. It iseasy to overlook in Fig. 6 the rise and then fall of the probabilityvalue at very early times, up to about 0.002 s.

If we look more closely at what happens at these early times,we can see the striking difference between the classical andquantum cases. Fig. 7 shows plots of the probability distribution

0 40 80 120 1600

0.01

0.02

0.03

Velocity

Pro

babi

lity

dens

ity

Fig. 1. Distribution of probability over left-to-right velocities in kilometres perhour/tens of microns per second of the classical/quantum bus.

10 5 0 5 100

0.1

0.2

Position

Pro

babi

lity

dens

ity

Fig. 2. Distribution of probability over initial positions in kilometres/tens ofmicrons of the classical/quantum bus.

6 0 6 120

0.1

0.2

Position (kilometres)

Pro

babi

lity

dens

ity

Fig. 3. Distributions of probability over positions of the classical bus. From left toright, at times 0, 0.125, 0.25 and 0.375 h.

0 0.025 0.05 0.0750

0.1

0.2

0.3

0.4

0.5

Time (hours)

Pro

babi

lity

Fig. 4. Probability that the classical bus is to the left of x0, versus time in hours.

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 131914

over positions of the classical bus, at times 0 (solid curve) and0.002 h (dashed curve). The probability flows from left to rightover this time interval, just as it does at later times, and this isconfirmed in Fig. 8, which shows the changing value of theprobability that the classical bus is still to your left. Again, itdecreases steadily.

Figs. 9 and 10 show the analogous results for the quantum bus.While the successive curves in Fig. 9 are again suggestive ofmovement to the right during the time interval of length0.002 s, Fig. 10 shows that the probability that the position ofthe bus will be found on your left actually increases during thisperiod, from the initial value of 0.5 to a value of about 0.508,before starting to decline. In other words, while there is a 50%

probability initially that the position of the bus will be found onyour left, this has increased to about 50.8% after about 0.002 s.

For each curve like those in Figs. 5 and 9, the area under the curveto the left of position x0 gives the probability that the position ofthe quantum bus will be found on your left at the correspondingtime, while the area under the curve to the right gives the probabilitythat it will be found on your right. And for each curve, the sum ofthese two areas has the value one.

For the solid (t0) curve in Fig. 9, these areas are each equal to 0.5.For the dashed (t0.002 seconds) curve however, the area to the leftequals about 0.508, and so is greater than the area to the right, whichequals about 0.492. This is despite the appearance given by the later-time curve of motion to the right from the initial one. The apparent

6 0 6 120

0.1

0.2

Position (tens of microns)

Pro

babi

lity

dens

ity

Fig. 5. Distribution of probability over positions of the quantum bus. From left toright, at times 0, 0.125, 0.25 and 0.375 s.

0 0.025 0.05 0.0750

0.1

0.2

0.3

0.4

0.5

Time (seconds)

Pro

babi

lity

Fig. 6. Probability that the position of the quantum bus is to the left of x0, versustime in seconds.

2 1 0 1 20.08

0.12

0.16

0.2

0.24

Position (kilometres)

Pro

babi

lity

dens

ity

Fig. 7. Distributions of probability over positions of the classical bus at times 0(solid line) and 0.002 h (dashed line).

0 0.0025 0.005 0.00750.45

0.475

0.5

Time (hours)

Pro

babi

lity

Fig. 8. Probability that the classical bus is to the left of x0, versus time in hours.

2 1 0 1 20.08

0.12

0.16

0.2

0.24

Position (tens of microns)

Pro

babi

lity

dens

ity

Fig. 9. Distribution of probability over positions of the quantum bus at times 0(solid line) and 0.002 s (dashed line).

0 0.0025 0.005 0.00750.48

0.49

0.5

0.51

Time (seconds)

Pro

babi

lity

Fig. 10. Probability that the quantum bus is to the left of x0, versus time inseconds.

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 1319 15

motion to the right is more than compensated for by a broadening ofthe left hand peak, and a narrowing of the right hand one.

From Allcock's work it is clear that probability backflow is aninterference effect, made possible because a quantum particle,unlike a classical bus, also has wave-like properties. The appear-ance of mathematically similar retro-propagation effects is notuncommon in classical wave theories (Berry, 2010), so the surprisein the quantum context comes not so much from unusual mathe-matics, as from the fact that it is the probability that flowsbackwards in this case, counter to our classical intuition.

The probability distributions over initial positions and velocitiescan be chosen independently for a classical bus, whereas for aquantum particle, they are tightly related, being determined fromthe wave function and its Fourier transform. This relationshipinvolves Planck's constant h and, as time evolves, also the mass ofthe particle, and it is this same relationship between wave functionand Fourier transform that gives rise to that most profound of allquantum effects, Heisenberg's uncertainty principle.

It is not hard to appreciate the relevance of probability back-flow to the arrival-time problem that was primarily of interest toAllcock 45 years ago. For if the probability of finding a right-moving particle to the left of a given point can increase or decreasewith time, the very notion of the probable time of its arrival at thatpoint, from the left say, may become problematic.

4. Quantifying probability backflow

In an attempt to raise interest in the backflow effect itself, andto quantify it, two aspects were considered by us some 25 yearsafter Allcock's work (Bracken & Melloy, 1994). The first aspectconcerns the possible duration of the effect. It is clear enough, andeasily proved, that the probability of finding the position of thequantum bus to the left of a given point such as x0 mustapproach the value zero for sufficiently long times, while theprobability of finding it to the right of that point must approachthe value one, just as in the case of the classical bus. We are thenled to ask, for how long can the quantum probability flow back-wards past a given point?

The answer to this question is simple, if surprising. Backflowcan persist over any finite time-interval, no matter how long. Foronce a quantum state is found that produces the effect over atime-interval of given duration, another state can be constructedthat produces the same amount of probability backflow over aninterval twice as long, say, by a simple scaling process.

The second and more interesting question is to ask: What is thegreatest amount of probabilitycall this amount P, saythat canflow backwards over a given time-interval? The length of thetime-interval is immaterial, by the scaling argument just men-tioned. This second question leads to a well-defined mathematicalproblem, but one that has, to date, defied exact solution. Ourestimate of P, obtained by treating the problem numerically, wasabout 0.04, or about 4% of the total probability on the line.Subsequent numerical studies (Eveson et al., 2005; Penz et al.,2006; Yearsley et al., 2012) have found the more precise estimate0.0384517. This is about five times greater than the amount ofbackflow in the example treated above.

The number P appears as an eigenvalue in the mathematicalproblem that arises, and the corresponding eigenfunction definesthe wave function of the quantum state that gives rise to the greatestbackflow value. While eigenvalue problems commonly arise forobservable quantities in quantum mechanics, leading to quantumnumbers such as the energy levels of an atom, for example, thequantum number P is quite unusual.

It is a pure (dimensionless) number. Furthermore, its value is notonly independent of the length of the time-interval over which

probability backflow occurs, as already noted, but is also independentof the mass of the quantum particle and, more surprisingly, of thevalue of Planck's constant h, which typically characterizes quantumeffects. In an imaginary world where h had a value many times largeror many times smaller than the one we are familiar with in ourworld, the maximum possible probability backflow over any giventime interval would still be the same for any free quantum particle, ofwhatever mass, namely 0.0384517.

It is often said, rather loosely, that as the value of h is reducedin the mathematical description of a quantum system, the obser-ved properties of the system will gradually approach those of itsclassical counterpart. But probability backflow does not occur at allfor a classical bus, so this dictum fails to apply here.

5. Negative probability interpretation

A striking interpretation of probability backflow can be given ifwe introduce negative probabilities into the quantum description.While such a notion may seem to be absurd, the usefulness ofthis radical concept for conceptualizing other aspects of quantummechanics has been argued long ago in other contexts by Dirac(1942) and Feynman (1987).

The idea of negative probabilities seems to be less absurd whenwe recall how useful a related notion has become in everyday lifethe notion of negative numbers. No-one ever saw a negativeamount of money, for example, but there is a simplifying con-ceptual advantage in regarding withdrawals from your bankaccount as negative deposits when calculating the balance remain-ing after all transactions are considered. Provided that your finalbalance is not negative, your bank manager will not complain thatyou used unphysical negative numbers to help you keep track. Andso it is with negative probabilities.

To see their relevance in the present context, reconsider firstlythe classical bus. Rather than distinct probability distributions overinitial positions and velocities, we can consider a single distribu-tion over both. For the classical state illustrated in Figs. 1 and 2,a three-dimensional plot of such a joint distribution could lookinitially as in Fig. 11.

Note that the distribution is confined to the upper half of thepositionvelocity plane because the bus definitely has a positivevelocity. Fig. 12 shows a contour plot of the initial distribution. Theprobability that the bus is initially to the right of x0 is now givenby the total probability distributed on the right quadrant of Fig. 12,while the probability that the bus is to the left of x0, and so yetto arrive at x0, is given by the total probability distributed on theleft quadrant. Each of these initial probabilities equals 0.5.

As time passes, the distribution on the upper half of the positionvelocity plane evolves by shearing to the right, as indicated by thearrows in Fig. 12, leading after some time to Fig. 13. All the probabilityin the wedge shown in Fig. 12, bounded by the left-hand white lineand the black line, flows across the black line x0 during anappropriate time interval, replacing the probability in the wedgeinitially bounded by the black line and the right-hand white line,which moves further right, as shown in Fig. 13. In this way theprobability in the right quadrant increases while the probability inthe left quadrant decreases by the same amount.

For the quantum bus, the closest analogue of a joint distribu-tion of position and velocity probabilities is given by a constructfrom the wave function due to Wigner (1932). For the quantumstate giving rise to Figs. 1 and 2, the three-dimensional plot ofthis Wigner function, so-called, might look initially as shown inFig. 14.

The Wigner function is not everywhere positive, unlike itsclassical counterpart in Fig. 11. Negative values appear with thelightest shades in the example shown in Fig. 14. For the same

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 131916

example, Figs. 15 and 16 show contour plots of the initial distri-bution and the distribution at a later time, with regions of negativevalues again shown lightest.

Despite the appearance of negative values, the Wigner functiondoes share important properties with the classical distribution.Thus, in the present context, it vanishes for negative velocities.

Fig. 13. Distribution of probability over velocities and positions of the classical bus at a later time: contour plot.

Fig. 12. Distribution of probability over velocities and initial positions of the classical bus: contour plot.

Fig. 11. Distribution of probability over velocities and initial positions of the classical bus: 3D plot.

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 1319 17

Fig. 16. Distribution of quasi-probability over velocities and positions of the quantum bus at a later time: contour plot.

Fig. 15. Distribution of quasi-probability over velocities and initial positions of the quantum bus: contour plot.

Fig. 14. Distribution of quasi-probability over velocities and initial positions of the quantum bus: 3D plot.

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 131918

Furthermore, the total of its values in the right quadrant equals theprobability that the particle will be found to the right of x0 onmeasuring its position at the corresponding time, the total of itsvalues in the left quadrant equals the probability that the particlewill be found to the left of x0, and the sum of these two alwayshas the value one. We may say that the Wigner function definesa quasi-probability distribution, taking both positive and negativeprobability values. Such an interpretation has been much dis-cussed in other contexts. (See for example Feynman, 1987; Scullyet al., 1994; Zachos et al., 2005.)

What is especially important for the discussion of probabilitybackflow is that, as time passes, the Wigner function evolves injust the same way as the classical distribution, by shearing to theright, as indicated by Figs. 15 and 16. In the quantum case,however, the total amount of probability on a wedge like thatbetween the left-hand white line and the black line in Fig. 15 canbe negative. When this negative probability flows to the rightquadrant, arriving between the initial positions of the black lineand right-hand white line after some time, the probability on theright quadrant decreases, and the probability on the left quadrantincreases by the same amount.

We may say that all the probability moves to the right with themotion of the quantum particle, just as in the case of the classicalbus, but now not all that probability is positive. Negative prob-ability moving to the right has the same effect on the totalprobabilities in the left and right quadrants as positive probabilitymoving to the left, thus giving rise to the backflow phenomenon.

While the interpretation of the Wigner function as a quasi-probability distribution has limitations in more general contexts(Feynman, 1987; Scully et al., 1994; Zachos et al., 2005), theusefulness of treating its negative and positive values as prob-abilities when conceptualizing the backflow effect is manifestlyclear. It must be emphasized however that negative probabilityvalues are not directly observable. The total probabilities on theleft and right quadrants, which are observable, are always positive.

Note that the greatest possible negative value of the totalprobability on any wedge like the one on the left in Fig. 15,evaluated over all quantum states with positive left-to-rightvelocities, equals the number P. For the example given here, ithas the value 0.008, only about 20% of the maximum possible.

6. Concluding remarks

The value of P evidently reflects the structure of the quantumdescription of a free particle, in terms of a wave function satisfyingSchrdinger's equation, rather than the values of the only

constants appearing in that description, namely the mass of theparticle and Planck's constant. Experimental observation of thevalue of P would therefore provide some novel verification of thisgeneral mathematical framework per se, unlike more typicalphysical predictions of quantum theory.

Successful observation of probability backflow in experiments,such as that proposed by Palmero et al. (2013), would confirmAllcock's insight from 45 years ago, and highlight importantdifferences in the behaviour of classical and quantum probabilities.

But a further challenge to experimenters would remaintomeasure the mysterious quantity P and verify that it has the value0.0384517 predicted by quantum mechanics.

References

Aharonov, Y., Oppenheim, J., Popescu, S., Reznik, B., & Unruh, W. G. (1998). PhysicalReview A, 57, 4130.

Allcock, G. R. (1969). Annals of Physics, 53 253, 286, 311.Berry, M. V. (2010). Journal of Physics A: Mathematical and Theoretical, 43, 415302.Bracken, A. J., & Melloy, G. F. (1994). Journal of Physics A: Mathematical and General,

27, 2197.Dirac, P. A. M. (1942). Proceedings of Royal Society London A, 180, 1.Eveson, S. P., Fewster, C. J., & Verch, R. (2005). Annales Henri Poincare, 6, 1.Feynman, R. P. (1987). In: F. D. Peat, & B. Hiley (Eds.), Quantum implications: Essays

in honour of David Bohm (pp. 235248). London: Routledge and Kegan Paul.Halliwell, J. J., Gillman, E., Lennon, O., Patel, M., & Ramirez, I. (2013). Journal of

Physics A: Mathematical and Theoretical, 46, 475303.Kijowski, J. (1974). Reports on Mathematical Physics, 6, 361.Melloy, G. F., & Bracken, A. J. (1998a). Annalen der Physik, 7, 726.Melloy, G. F., & Bracken, A. J. (1998b). Foundations of Physics, 28, 505.Muga, J. G., & Leavens, C. R. (2000). Physics Reports, 338, 353.Muga, J. G., Palao, J. P., & Leavens, C. R. (1999). Physics Letters A, 253, 21.Nielsen, M. A. (2010). An underappreciated gem: probability backflow http://

michaelnielsen.org/blog.Palmero, M., Torrontegui, E., Muga, J. G., & Modugno, M. (2013). Physical Review A,

87, 053618.Penz, M., Grubl, G., Kreidl, S., & Wagner, P. (2006). Journal of Physics A: Mathematical

and General, 39, 423.Scully, M. O., Walther, H., & Schleich, W. (1994). Physical Review A, 49, 1552.Strange, P. (2012). European Journal of Physics, 33, 1147.Wigner, E. P. (1932). Physical Review, 40, 749.Yearsley, J. M., Halliwell, J. J., Hartshorn, R., & Whitby, A. (2012). Physical Review A,

86, 042116.Zachos, C. K., Fairlie, D. B., & Curtright, T. L. (2005). Quantum mechanics in phase

space. Singapore: World Scientific.

A.J. Bracken, G.F. Melloy / Studies in History and Philosophy of Modern Physics 48 (2014) 1319 19

http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref1http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref1http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref2http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref3http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref4http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref4http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref5http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref6http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref7http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref7http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref8http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref8http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref9http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref10http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref11http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref12http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref13http://michaelnielsen.org/bloghttp://michaelnielsen.org/bloghttp://refhub.elsevier.com/S1355-2198(14)00099-9/sbref15http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref15http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref16http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref16http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref17http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref18http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref19http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref20http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref20http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref21http://refhub.elsevier.com/S1355-2198(14)00099-9/sbref21

Waiting for the quantum bus: The flow of negative probabilityIntroductionWaiting for the classical busWaiting for the quantum busQuantifying probability backflowNegative probability interpretationConcluding remarksReferences