Simulating Bell violations without quantum computers
Peter D. Drummond, Bogdan Opanchuk, Laura Rosales-Zárate, Margaret D. Reid
SSimulating Bell violations without quantum computers
P D Drummond, B Opanchuk, and L Rosales-Zárate and M DReid
Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University ofTechnology, Melbourne 3122, Australia
Abstract.
We demonstrate that it is possible to simulate Bell violations usingprobabilistic methods. A quantum state corresponding to optical experiments thatviolate the Bell inequality is generated, demonstrating that these quantum paradoxescan indeed be simulated probabilistically. This provides an explicit counter-exampleto Feynman’s claim that such classical simulations could not be carried out.
Submitted to:
Phys. Scr. a r X i v : . [ qu a n t - ph ] O c t imulating Bell violations without quantum computers
1. Introduction
The simulation of quantum dynamics is a hard problem in physics for systems withmany degrees of freedom. While large classical systems can be readily simulated withcomputers, this is difficult with large quantum systems due to the exponentially growingsize of Hilbert space. Since an expansion in eigenstates of the Hamiltonian is virtuallyimpossible for many-body systems, one possible solution is to use probabilistic sampling.This approach was apparently ruled out by a claim of Feynman [1], where he asked
Can quantum systems be probabilistically simulated by a classicalcomputer ? His answer to the question was:
If you take the computer to be the classical kind and there’s nochanges in any laws, and there’s no hocus-pocus, the answer iscertainly, No!
This led Feynman to propose the use of quantum computers for these types ofsimulation, and his argument has motivated extensive research on quantum computing.However, large-scale quantum computers are not yet available. An important questiontherefore is whether probabilistic simulation with classical computers are truly excluded,as this claim would certainly imply. Is there some way that one can in fact useprobabilistic sampling with existing digital computers for these challenging tasks? Andif so, to what degree of practicality?Given the known decoherence problems in constructing quantum computinghardware for quantum simulations, practical alternative strategies using software wouldbe extremely useful. Here we demonstrate, by carrying out a simulation, that Bell’stheorem does not rule out probabilistic simulation methods. Our results treat the caseconsidered by Feynman, which is a four mode photonic state used in experimentaldemonstrations, equivalent to two correlated spin-half particles.This allows us to demonstrate, by means of explicit, direct computer simulation,that the answer to Feynman’s original question is actually “
Yes ”. Our main emphasisis the fundamental question of whether it is possible to probabilistically treat Bellviolations, which was claimed to be impossible in this early literature. In fact, the “not soblack and white” nature of Feynman’s claim was always apparent, given the existence ofpositive phase space methods that were known at the time. It was sometimes interpretedthat these methods would only be useful for semiclassical states, rather than the Bellstate. We show this is not so by carrying out localized probabilistic sampling of a Bell-violating quantum state for the first time, using random sampling to obtain results incomplete agreement with quantum mechanics.One may ask then, in Feynman’s words: what is the “ hocus-pocus ” that allows thesesimulations? Very simply, we only require that average correlations of the simulationoutputs correspond to quantum mechanics. We do not ask the measurement values imulating Bell violations without quantum computers
2. Feynman’s argument
Earlier investigations on the limits to computation had focused on the dissipation ofenergy in standard logic operations. Accordingly, it was rather natural to investigatethe possibility first of dissipationless logic [4], then of quantum logic [1] as alternatives.Feynman’s paper addressed the issue of exponential scaling in quantum dynamics.Was this a problem that classical computers could be used to solve? If not, perhapsquantum computers would be needed. Feynman’s logic here was reasonably clear.Random sampling methods are known as a way to treat many other exponentiallycomplex problems. If one could rule out probabilistic sampling by considering a knownquantum state, this entire strategy could then be eliminated for quantum simulations.To prove this, Feynman turned to Bell’s theorem [5], which shows that all localhidden variable (LHV) theories are inequivalent to quantum mechanics. His argumentused a small quantum system without inherent exponential complexity. However, byassuming that a probabilistic simulation is equivalent to a hidden variable theory,Feynman could argue from this simple case that no general probabilistic method waspossible. It is this claim that we investigate here.To explain this in greater detail, we recall that a Bell inequality is a constraint onobservable correlations of a physical system that obeys a local hidden variable theory.This is a theory that has the property of local realism, as defined by Einstein. In the caseof particles emitted from a common source, measurements of two spatially separatedobservers are obtained by taking random samples of a common parameter λ .Measured values are then functions of some local detector settings and the hidden imulating Bell violations without quantum computers λ , which could be any variable. Mathematically, the correlations in a hiddenvariable theory are obtained from a probabilistic calculation of form: (cid:68) ˆ X ˆ X (cid:69) = ˆ X ( λ ) X ( λ ) P ( λ ) dλ . (1)Here, X j ( λ ) have values that correspond to the experimentally measured eigenvalues.This is just how one might expect to carry out a probabilistic simulation. However, aswe will see, this restriction on the values of X j ( λ ) — which is needed for LHV theories— is not essential for a probabilistic simulation.
3. Optical Bell violations
A popular route to Bell violation experiments is using an atomic cascade, or morerecently using parametric down-conversion, with the resulting state: | Ψ B (cid:105) = 1 √ (cid:16) a † a † + a † − a † − (cid:17) | (cid:105) . (2)Here we suppose that a † ± creates a photon in spatial mode 1, which is detected atsite A with polarization of s = ± respectively, and similarly for operators of mode 2detected at site B . The version of Bell’s inequality given by Clauser, Horne, Shimonyand Holt, (the CHSH form) [6] is especially important, as it gives LHV limits to theexpected correlation for the above experiment conducted by Alice and Bob: C [ A, B ] + C [ A, B (cid:48) ] + C [ A (cid:48) , B ] − C [ A (cid:48) , B (cid:48) ] ≤ , (3)where C [ A, B ] is the correlation, A and A (cid:48) are measurements at location A with twodifferent polarizer angles, while B and B (cid:48) are the corresponding measurements atlocation B. It is usually assumed that assume the observed values are +1 or − . Wenote that a calculation within quantum mechanics shows that, for a singlet quantumstate known as the Bell state, the Bell inequality is predicted to be violated. Quantummechanics predicts that at a relative polarizer angle θ = π/ : ∆( θ ) = 12 (cid:104) (cid:104) ˆ A ˆ B (cid:105) + (cid:104) ˆ A (cid:48) ˆ B (cid:105) + (cid:104) ˆ A ˆ B (cid:48) (cid:105) − (cid:104) ˆ A (cid:48) ˆ B (cid:48) (cid:105) (cid:105) − √ − > . (4)Feynman’s claim was that a classical probabilistic simulation could not replicatethis violation. Hence he argued that classical probabilistic methods using a computercould not be used to simulate quantum dynamics.
4. Positive P-representation
To demonstrate a counter-example - a probabilistic simulation of the bipartite Bellinequalities given above - we use the positive-P representation [3]. This is an expansionof an arbitrary density matrix (cid:98) ρ in coherent state projectors: (cid:98) ρ = ˆ P ( (cid:126)α, (cid:126)β ) (cid:98) Λ( (cid:126)α, (cid:126)β ) d M (cid:126)αd M (cid:126)β. (5) imulating Bell violations without quantum computers (cid:98) Λ( (cid:126)α, (cid:126)β ) = | (cid:126)α (cid:105) (cid:68) (cid:126)β ∗ (cid:12)(cid:12)(cid:12) / (cid:104) (cid:126)β ∗ | (cid:126)α (cid:105) , and | (cid:126)α (cid:105) = | α . . . . α n (cid:105) is a multi-mode coherent state of a bosonic field. The probability function P ( (cid:126)α, (cid:126)β ) is defined onan enlarged, nonclassical phase-space, which allows positive probabilities. This methodleads to an exact mapping between the quantum mechanics of any bosonic field, and aphase-space probability distribution [3]. This was already known by 1982.The correlations of quantum counts ˆ n i = ˆ a † i ˆ a i at different locations are simulatedusing: (cid:104) ˆ n i . . . ˆ n j (cid:105) = ˆ n i . . . n j P ( (cid:126)α, (cid:126)β ) d M (cid:126)αd M (cid:126)β. (6)where n i ≡ α i β i . The effects of a polarizer are simply obtained on taking linearcombinations of mode amplitudes, just as in classical theory.One can obtain the positive-P distribution using a variety of methods, since therepresentation is not unique. Here, we represent the photonic Bell state of Eq (2) usinga generic construction which exists for all quantum states [3, 7]: P ( (cid:126)α, (cid:126)β ) = 1(2 π ) M e − | (cid:126)α − (cid:126)β ∗ | / (cid:42) (cid:126)α + (cid:126)β ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)α + (cid:126)β ∗ (cid:43) . (7)There is a remarkable similarity between the hidden variable theory (1) of Bell,and the positive-P formula (6) for quantum correlations. However, while a hiddenvariable theory obeys Bell’s theorem, the positive-P theory is fully equivalent to quantummechanics, and can violate Bell inequalities. The reason for the difference is due to thedifferent quantities calculated in the correlations. The fundamental observables in Bell’scase, of form X ( λ ) , are equal to observed integer photon counts.The corresponding observables in the positive-P case, of form n (cid:16) (cid:126)α, (cid:126)β (cid:17) , arecomplex numbers whose mean values and correlations correspond to physical meansand correlations. This difference allows the positive-P distribution to be equivalent toquantum mechanics, even though it looks similar to a hidden variable theory. Thispoint was made in an article by Reid and Walls [8], which proposed the modern Bellinequality experiments using parametric down-conversion to generate photon pairs.
5. Simulation results
In our simulations, we choose to expand the Bell state, Eq (2) using the standardmethod of Eq (7). While it is also possible to solve the dynamical stochastic equationsfor the parametric amplifier used in experiments [9], here we are simply interested indemonstrating that probabilistic sampling of the Bell state is possible. In the ideal Bellcase, the actual distribution has M = 4 modes with real dimensions, having the form: P ( (cid:126)α ) = (cid:32) | (cid:126)α A + · (cid:126)α B + | N π ( N + 1) ( N !) (cid:33) e −| (cid:126)α + | −| (cid:126)α − | . (8) imulating Bell violations without quantum computers V i o l a t i on Figure 1.
Probabilistic violation of a Bell inequality with × random samples.The simulated Bell violation ∆( θ ) is graphed as a function of the relative polarizerangle θ . The filled area corresponds to the estimated error range around the mean of ∆( θ ) for the sampled state, while the dashed line is the quantum mechanical prediction. R e σ x −3 −2 −1 0 1 2 3 R e σ x −3−2−10123 p r obab ili t y ( × − ) (a) R e σ x σ x −8 −6 −4 −2 0 2 4 6 R e σ y σ y −6−4−202468 p r obab ili t y ( × − ) (b) Figure 2.
Distribution of variables correspond to spins and correlations.
Here (cid:126)α ± = (cid:16) (cid:126)α ± (cid:126)β ∗ (cid:17) / are the sum and difference coordinates respectively. Clearly, (cid:126)α − can be sampled using Gaussian variates. The notation (cid:126)α A,B + indicates the coherentstate sum vector projected on the A and B observers respectively. The sum distributionis more complex, but can be readily sampled using the von Neumann rejection method,with a standard lambda distribution as the reference distribution.The resulting Bell violation is graphed in Fig (1). This demonstrates completeagreement with quantum predictions, up to a sampling error which can be reduced atwill by taking more samples. However, the results of Fig (2) are more interesting. Thisfigure shows the joint distribution of the Schwinger variables that correspond to thespin projections and their correlations. In a physical measurement, these would all haveeigenvalues of ± . Instead, we see that the variables corresponding to spin measurementsgo outside their quantum bounds. Intriguingly, this is exactly what is predicted for weakmeasurements, which suggests that a close relationship exists between these simulationsand the concept of a weak measurement. imulating Bell violations without quantum computers
6. Conclusions
Our main result is that the earliest argument leading to quantum computers is wrong.There is no impediment to simulating Bell violations probabilistically. Our phase-spacesimulations correctly generate the means and correlations that are predicted by quantummechanics. Such probabilistic simulation methods have already allowed simulationsof the quantum dynamics of quantum solitons [10, 11] and colliding Bose-Einsteincondensates [12], with up to modes and particles. The growth of samplingerror and other scaling issues are important limitations, and will be treated elsewhere.Quantum logic based encryption [13] and atomic clocks [14] are already successful.To develop such technologies in future [15], understanding the consequences of non-idealbehavior is very important. Probabilistic algorithms could therefore have an applicationto the design of these devices. Importantly, fundamental tests of quantum mechanicsrequire simulation methods that do not depend on quantum theory being correct, whichis a strong reason to investigate these issues further. Acknowledgments
L. E. C. R. Z. acknowledges financial support from CONACYT, Mexico. P. D. D. andM. D. R. acknowledge Australian Research Council funding of a Discovery grant.
References [1] Feynman R P 1982
Int. J. Theor. Phys. Proc. Phys. Math. Soc. Jpn. [3] Drummond P D and Gardiner C W 1980 J. Phys. A: Math. Gen. IBM J. Res. Dev. http://dx.doi.org/10.1147/rd.441.0270 [5] Bell J S 1964 Physics http://philoscience.unibe.ch/documents/TexteHS10/bell1964epr.pdf [6] Clauser J F, Horne M, Shimony A and Holt R 1969 Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. A (2) 1260–1276 URL http://link.aps.org/doi/10.1103/PhysRevA.34.1260 [9] McNeil K J and Gardiner C W 1983 Phys. Rev. A (3) 1560–1566 URL http://link.aps.org/doi/10.1103/PhysRevA.28.1560 [10] Carter S J, Drummond P D, Reid M D and Shelby R M 1987 Phys. Rev. Lett. (18) 1841–1844[11] Corney J F, Drummond P D, Heersink J, Josse V, Leuchs G and Andersen U L 2006 Phys. Rev.Lett. (2) 023606[12] Deuar P and Drummond P D 2007 Phys. Rev. Lett. Phys. Rev. Lett. (5) 557–559 URL http://link.aps.org/doi/10.1103/PhysRevLett.68.557 [14] Chou C W, Hume D B, Koelemeij J C J, Wineland D J and Rosenband T 2010 Phys. Rev. Lett. (7) 070802 URL http://link.aps.org/doi/10.1103/PhysRevLett.104.070802 [15] Jones N C, Van Meter R, Fowler A G, McMahon P L, Kim J, Ladd T D and Yamamoto Y 2012
Phys. Rev. X (3) 031007 URL(3) 031007 URL