Why aren't quantum correlations maximally nonlocal? Biased local randomness as essential feature of quantum mechanics
aa r X i v : . [ qu a n t - ph ] F e b Why aren’t quantum correlations maximally nonlocal?Biased local randomness as essential feature of quantum mechanics.
Antoine Suarez (Dated: February 14, 2009)It is argued that the quantum correlations are not maximally nonlocal to make it possible tocontrol local outcomes from outside spacetime, and quantum mechanics emerges from timeless non-locality and biased local randomness. This rules out a world described by NL (nonlocal) boxes. Anew type of experiments is suggested.
Bell type experiments demonstrate (within the limitsof a few rather eccentric loopholes) correlations, whichcannot be explained by means of local relativistic influ-ences propagating at velocity v ≤ c [1]. This means thatone has to give up the view that the outcomes at eachpart of the setup result from properties preexisting inthe particles before measurement: Alice’s (respectivelyBob’s) outcomes cannot be explained by the informationthe photon carries when leaving the source and the set-tings in Alice’s (respectively Bob’s) lab.The Suarez-Scarani or before-before experimentdemonstrates that these nonlocal correlations cannot beexplained by time-ordered nonlocal influences [2, 3, 4].Giving up the concept of locality is not sufficient to beconsistent with quantum experiments, one has to giveup the view that one event occurring before in time canbe considered the cause, and the other occurring later intime the effect (nonlocal determinism). The correlationscannot be explained by any history in spacetime, in en-tanglement experiments local random events experienceinfluences from outside spacetime to produce nonlocalorder. [5, 6]However, the orthodox interpretation of quantum me-chanics also claims that in entanglement experiments thelocal outcomes happen in a “full random” way, i.e., ac-cording to a uniform (non-biased) random distribution.In this sense, the orthodox interpretation is at variancewith any model assuming that local outcomes can hap-pen according to a biased random distribution. BothBell’s and Suarez-Scarani’s experiments are compatiblewith such models.The violation of Leggett inequalities was first inter-preted as an experimental falsification of “nonlocal real-ism”, where “realism” refers to the view that the singleparticles carry well defined properties when they leavethe source [7]. Such an interpretation is misleading: Bytesting models fulfilling Leggett inequalities one does nottest “nonlocal realism”, but rather models assuming bothnonlocal randomness and outcomes that depend on bi-ased random local variables [8]. Nevertheless, it is theColbeck-Renner theorem [9] which clearly shows the re-lationship between nonlocality and biased local random-ness in entanglement experiments [8].In this letter I argue that the quantum correlations are )eDiscordancPr( D A ( - ) BS A1 BS B BS A0 BS B1 D B ( - )Source l A s A l B s B D A (+) D B (+) l , l … l =l B S crew Total phase-shift because path-length-difference: c sl ω c sl ωΦ BBBAAA -+-= l A = l … l , l )eConcordancPr( 2 )Vcos1(1 F+= = Pr(a=b) = Pr(a „ b)2 )Vcos1(1 F-=
Alice‘s interferometer Bob‘s interferometerMobile mirrors
FIG. 1: Diagram of a chained Bell experiment using interfer-ometers. The setup makes it possible to perform at the sametime a before-before experiment using acousto-optic waves asmoving beam-splitters. (See text for details). not maximally nonlocal to make it possible biasing localoutcomes from outside spacetime, and propose to con-sider timeless nonlocality and biased local randomnessas primitives to axiomatize quantum theory. This rulesout a world described by NL boxes. I also propose a newtype of entanglement experiments demonstrating theseideas.Consider the experiment sketched in Figure 1: Thesource emits photon pairs. Photon A (frequency ω A ) en-ters Alice’s interferometer to the left through the beam-splitter BS A and gets detected after leaving the beam-splitter BS A , and photon B (frequency ω B ) enters Bob’sinterferometer to the right through the beam-splitterBS B and gets detected after leaving the beam-splitterBS B . The detectors are denoted D A ( a ) and D B ( b )( a, b ∈ { + , −} ). Each interferometer consists in a longarm of length l i , and a short one of length s i , i ∈ { A, B } .Frequency bandwidths and path alignments are chosenso that only the coincidence detections corresponding tothe path pairs: ( s A , s B ) and ( l A , l B ) contribute construc-tively to the correlated outcomes in regions A and B,where ( s A , s B ) denotes the pair of the two short arms,and ( l A , l B ) the pair of the two long arms.Suppose one of the measurements produces the value a ( a ∈ { + , −} ), and the other the value b ( b ∈ { + , −} ). Ac-cording to quantum mechanics the probability P r ( a, b ) ofgetting the joint outcome ( a, b ) depends on the choice ofthe phase parameter Φ characterizing the paths or chan-nels uniting the source and the detectors: P r ( a = b ) = 12 (1 + V cos Φ) P r ( a = b ) = 12 (1 − V cos Φ) (1)where Φ is the phase parameter given by: Φ = ω A l A − s A c + ω B l B − s B c .Bell experiments, using two different values of l A andtwo different values of l B , demonstrate that the correla-tions violate locality criteria, the well known Bell’s in-equalities (see [4] and references therein).In the before-before experiment the beam-splittersBS A and BS B are in motion in such a way that eachof them, in its own reference frame, is first to select theoutput of the photons (before-before timing). Then, eachoutcome should become independent of the other, and(according to a time-ordered causal model) the nonlocalcorrelations should disappear. The result was that thecorrelations don’t disappear, and therefore are indepen-dent of any time-order [4], that is, they come from outsidespacetime [5, 6].Consider now chained Bell experiments using N dif-ferent values of l A ( l , l , ..., l N − ) and N values of l B ( l , l , ..., l N − ), with N ≥
2. We define the function I ( N ) as: I ( N ) = P r ( a = b | Φ( l , l N − ))+ P r ( a = b | Φ( l , l ))+ P r ( a = b | Φ( l , l ))+ ....... + P r ( a = b | Φ( l N − , l N − )) (2)where P r ( a = b | Φ( l , l N − )) means the conditional prob-ability that Alice and Bob get the same outcome if thephase’s value results from long interferometers’ arms setto l , l N − , and P r ( a = b | Φ( l i , l i +1 )) the conditionalprobability that Alice and Bob get different outcomes ifthe phase’s value results from long interferometers’ armsset to l i , l i +1 ; depending on i , l i denotes the arm of Alice’sor Bob’s interferometer.We assume that any two values l i , l i +1 , with i ∈{ , N − } , define the same phase parameter, resultingfrom the equipartition of a value Θ:Φ( l i , l i +1 ) = Θ / N (3)Then, equation (2) can be rewritten as follows: I ( N ) = P r (cid:18) a = b | (2 N −
1) Θ2 N (cid:19) + (2 N − P r (cid:18) a = b | Θ2 N (cid:19) (4) Q I ( N ) I(2)I(3)I(4)I(5)I(6)I(50)I(5000)
I(N)=1/2 (1 + cos( (2N−1) Q /2N ) + (2N−1)/2 (1− cos( Q /2N) ) FIG. 2: I ( N ) as function of Θ, for different values of N (seeEquation (6) in text). For each N , I ( N ) ≥ I (2) ≥ I ( N ) < I ( N ) can be considered an indicator of in-creasing nonlocality.We denote D ( N ) the statistical distance between thedistribution of the local outcomes and the uniform ran-dom distribution in the corresponding chained Bell ex-periment with 2 N measurements.The Colbeck-Renner [9] theorem establishes that: D ( N ) ≤ I ( N ) / I ( N ) = 12 (cid:18) − cos (cid:18) (2 N −
1) Θ2 N (cid:19)(cid:19) + 2 N − (cid:18) − cos Θ2 N (cid:19) (6)Figure 2 represents I ( N ) in function of the phase pa-rameter Θ, for different N . As it appears, the plottedquantum mechanical prediction is not consistent with the“orthodox interpretation” that the local outcomes areuniformly distributed for any entanglement experiment.The fact that I (2) = 2 − √ > π clearly sug-gests that Nature is keen to permit biased local randomoutcomes for any value of the phase parameter Θ.Now I prove theorems showing that essential features ofquantum mechanics emerge from nonlocality and biasedlocal randomness. Basic conditions:
For reasons of scaling and symmetrywe impose:
P r ( a = b | Φ = 0) = 1 (7)
P r ( a = b | Φ) =
P r ( a = b | π − Φ) (8)Additionally, we take account of the fact that naturelikes “smoothness” for fashioning distributions and as-sume: I ( N, Θ ) > I ( N, Θ ) = 0 , Θ < Θ < Θ ⇒ I ( N, Θ ) > I ( N, Θ) > > I (2 , π ) > I ( ∞ , π ) ⇒ I (2 , π ) > I ( N, π ) > I ( N + 1 , π ) > I ( ∞ , π ) , ∀ N > I ( N, Θ) denotes the value of the function I ( N )for the phase parameter Θ. Theorem 1: I (2 , π ) = 0 ⇒ > I (2 , Θ) > < Θ < π .That is, maximal nonlocality for Θ = π necessarilyimplies non-maximal nonlocality for 0 < Θ < π . Proof:
From Equations (4) and (7) one is led to: I ( N, Θ = 0) =
P r ( a = b | Φ = 0)+ 2
N P r ( a = b | Φ = 0) − P r ( a = b | Φ = 0) = 1 (11)Then, from the “smoothness” condition (9) the
Theorem 1 follows.
Theorem 2: > I (2 , π ) > ⇒ ∀ N > , I (2 , π ) >I ( N, π ) > I ( N + 1 , π ) > I ( ∞ , π ) = 0. That is, non-maximal nonlocality for Θ = π and N = 2 is a sufficientcondition for decreasing I ( N, π ): Proof:
Taking account of (8), Equation (4) implies for N = ∞ : I ( ∞ , Θ = π ) = P r ( a = b | Φ = π )+ 2 N (1 − P r ( a = b | Φ = 0) − P r ( a = b | Φ = 0) = 0 (12)Then, from the “smoothness” condition (10) the
The-orem 2 follows.
Theorems 1 and 2 mean that making it possible to biaslocal random outcomes is a necessary and sufficient con-dition to get nonlocal distributions fashioned like thoserepresented in Figure 2. In particular, nonlocal naturecannot be maximally nonlocal for all phases.This rules out a world described by NL (nonlocal)boxes. Actually, one would need a whole spectrum ofNL boxes with biases ranging from I ( ∞ , π ) = 0 to I (2 ,
0) = 1.However a question remains open: Why the particu-lar Bell value I (2 , π ) = 2 − √
2, instead of for instance I (2 , π ) = 2 − √
3? Is it simply motivated by the wishof choosing “nice enough” functions like in (1), in orderto make the work more enjoyable to the physicists, or isthere a deeper reason behind? Q I ( N ) I(2)=0.63I(3)=0.48I(4)=0.42I(5)=0.387I(6)=0.378 MinimumI(7)=0.380I(8)=0.389i(9)=0.402
I(N)=1/2 (1 + 0.97 cos( (2N−1) Q /2N ) + (2N−1)/2 (1− 0.97 cos( Q /2N) ) Values I(N) for Q = p FIG. 3: Functions I ( N ) with Visibility V = 0 .
97. For Θ = π ,the values I ( N ) exhibit a minimum at I(6) In conclusion, the preceding analysis shows that thequantum correlations are not maximally nonlocal tomake it possible to bias local outcomes from outsidespacetime. Thus, nonlocality without signaling and bi-ased local randomness have strong primitive-appeal toexplaining why the laws of nature are quantum. Thismeans that entanglement experiments demonstratingnonlocality alone are basically incomplete, and shouldbe expanded to experiments demonstrating nonlocality,timelessness and increasingly uniform bias altogether.Assuming a visibility factor V (0 ≤ V ≤
1) dependingmainly on the efficiency of the detectors, Equation (6)becomes: I ( N ) = 12 (cid:18) − V cos (cid:18) (2 N −
1) Θ2 N (cid:19)(cid:19) + 2 N − (cid:18) − V cos Θ2 N (cid:19) (13)Experiments with visibility V = 0 .
99 are possible us-ing resting beam-splitters [10], and with V = 0 .
97 usingbeam-splitters in motion (i.e. acousto-optic modulators)[4]. This means, according to (5) and the values of I ( N )in (13) represented in Figure 3, that a before-before ex-periment demonstrating a bias bound decreasing from D = I (2) / .
315 to D = I (2) / .
189 is feasible.I would like to finish by stressing that the possibilityof controlling outcomes from outside spacetime has avery natural correlate in the way the brain functions.When I am typewriting this article, I assume that theauthor is the same who typewrote the article proposingthe before-before experiment in 1997. In this sense myidentity has roots beyond spacetime. I am controllingthe outcomes of my brain, i.e. biasing the random firingsof my neurons, from outside spacetime. The resultpresented in this paper upholds the view that quantumrandomness does not exclude the possibility of orderand control and, therefore, remains susceptible of beinginfluenced by free will [11].
Acknowledgments : I am grateful to Roger Colbeck for insightful discussions, and acknowledge informationin view of experiments by Nicolas Gisin, Renato Renner,Harald Weinfurter, and Hugo Zbinden. [1] Bell J. S.,
Speakable and unspeakable in quantummechanics
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