Entanglement , Information , Causality

The paper is divided in two parts. In the first one a summary of the main issues about quantum non–locality is provided. In the second part, the connections with information and causality are considered. In particular, it is shown that a principle of information causality implies that hyper–correlations among experimental settings are not possible but only correlations among possible outcomes. Since a setting is for measuring a particular observable and the eigenbasis of this observable can be considered a code, this means that information codification is a local procedure.


EPR
According to EPR, the correctness of a theory consists in the degree of agreement between its conclusions and human experience-the objective reality, while its completeness is defined as [10]: A theory is complete if every element of objective reality has a counterpart in it.The aim of the EPR article is to show the incompleteness of quantum mechanics in the sense of its inability to give a satisfactory explanation of entities which are considered fundamental-in a word, it is a 'disproof' and not a positive proof.Indeed, theories can be disproved by experience and (even thought) experiments.
The core of the argument is constituted by the Separability principle, which we can express as follows: Two dynamically independent systems cannot influence each other.The separability principle consists in the assumption that any form of interdependency among physical systems is of dynamical and causal type.Therefore, it is important to carefully distinguish the problem of relativistic locality-i.e., the existence of bounds in the transmission of signals and physical effects-from that of separability, which concerns only the impossibility of a correlation between separated systems in the case in which there are no dynamical and causal connections.Part of the EPR argument is that, in the absence of physical interactions, the systems are also separated.
EPR state a sufficient condition for the reality of observables, which can be formulated as follows: If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then, independently of our measurement procedure, there exists an element of the physical reality corresponding to this physical quantity.
The words "without in any way disturbing a system" tells us that the systems are considered as dynamically independent.The aim of EPR is to show that, assuming separability and the sufficient condition a e-mail: gennaro.auletta@gmail.com of reality, quantum mechanics is not complete: in logical terms, for quantum mechanics the following statement holds: [(Suff.Cond.Reality) ∧ (Separability)] =⇒ ¬Completeness, where ∧, ¬, and the arrow are the logical symbols for conjunction (AND), negation and implication, respectively.
The argument of EPR is structured as follows.From (i) the definition of completeness, (ii) the principles of physical reality and separability, and (iii) the fact that, according to quantum mechanics, two non-commuting observables cannot simultaneously have definite values, it follows that the following two statements are incompatible: • The statement r that the quantum mechanical description of reality given by the wave function is not complete and The statement s that when the operators describing two physical quantities do not commute, the two quantities cannot have simultaneous reality.
In formal terms, r s, ( where the symbol means a XOR.The meaning of the statement (2) is the following: if it is possible to show that two non-commuting observables have in fact simultaneous reality, we can logically conclude that quantum mechanics cannot be a complete description of reality (from the falsity of s we infer the truth of r).
Let us consider a one-dimensional system S made of two subsystems S 1 and S 2 interacting during the time interval between t 1 and t 2 , with momenta in the position representations: with momentum eigenfunctions respectively.The vectors |ϕ and |ψ describe the states of the particles 1 and 2, respectively.The eigenfunctions in the position representation are respectively, where x 0 is a fixed position (constant) and the eigenfunction ϕ p (x 1 ) corresponds to eigenvalue +p whilst ψ p (x 2 ) corresponds to the eigenvalue −p of the second particle's momentum (in other words, the two particles are moving away from each other with the same direction into opposite senses).Therefore, the compound system is described by the wave function Now, I summarize the scheme of the first thought experiment [2, Chap.16] [4, Chap.10]: (a) We locally measure the momentum on particle 1: let us assume that we find an eigenvalue p .

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(b) Therefore, the state (6) reduces to (c) Then, it is evident that particle 2 must be in state ψ −p and this result can be predicted with absolute certainty.
(d) However, we were able to formulate such a prediction without disturbing particle 2 (assumption of separability).
(e) Then, as a consequence of (c) and (d) and of the sufficient condition of reality, p(2) x is an element of reality.
Note that steps (a)-(c) are purely quantum mechanical.Only steps (d)-(e) are connected to the specific EPR argument.
However, if we had chosen to consider another observable of particle 1, say x1 , whose eigenfunctions are ϕ x (x 1 ) (whereas ψ x (x 2 ) are the eigenfunctions of the observable x2 of particle 2), then we would have written the state Ψ of the compound system as Let us now repeat the previous procedure for the position measurement.
(a') We locally measure the position on particle 1 and find the eigenvalue x .
(b') Now it is clear that the state (8) reduces to (c') Then, it is evident that the particle 2 must be in the state ψ x and this result can be predicted with absolute certainty.
(e') Then, as a consequence of (c') and (d') and of the sufficient condition of reality, x(2) is an element of reality.
Conclusions (e) and (e') look incompatible on the basis of the fact that position and momentum observables of particle 2 do not commute: going back to Propositions r and s [Eq.( 2)], EPR have in this way shown that, assuming that r (the quantum mechanical description of reality is not complete) is false, s is proved to be false as well since both p(2) x and x(2) have simultaneous reality.Then, the previous assumption must be rejected, and r must be true.Therefore, according to the EPR argument, quantum mechanics cannot be considered as a complete theory and the wave functions ( 6) and ( 8) cannot be considered as complete descriptions of the state of the particles.

Bohm's reformulation
The argument as formulate in the original EPR paper is difficult tom test.However, a great step was provided by David Bohm.Consider now two particles with spin 1  2 that are in a state in which the total spin is zero, that is, they are in a singlet state [8].They can be produced by a single atom radioactive decay.After a time t 0 the two particles begin to separate and at time t 1 they no longer interact.On the hypothesis that they are not disturbed, the law of angular momentum conservation guarantees that they remain in a singlet state.Considering the projection of the spin along the z-direction, the singlet state may be written in the form where the subscripts 1 and 2 refer to the particles.This implies that, if a measurement of the spin component along the z direction of particle 1 leads to a result +1/2, that of particle 2 along the same direction must give the value −1/2, and vice versa.This means that |Ψ 0 is an eigenket of the z component of the spin observables σ1z σ2z of the two systems.
Entanglement is a property of the state that is independent of the basis used.In order to see this rotational invariance, let us write it in terms of the z-component eigenvectors as Then, |Ψ 0 turns out to be also an eigenvector of σ1x σ2x and σ1y σ2y .For example, let us consider the y orientation.First, let us expand the z-eigenkets into the y-eigenkets: Then, we can write the singlet state (10) in the y expansion: Consequently, we have which, by making use Pauli matrices and of a reformulation of expression (11) in the y basis, implies EPJ Web of Conferences Now, let us back-substitute this expression into the z expansion: where I have dropped the symbol ⊗ of the sake of simplicity.

Bell Theorem
Bell assumed the existence of a hidden parameter λ such that, given λ, the function A a describing the results obtained by measuring with a device A the spin of the first particle along a chosen direction a (i.e., the observable σ1 •a), depends only on λ and on a [6].Similarly, the function B b describing the results when measuring with a device B the spin of the second particle along a chosen direction b (i.e., σ2 •b), depends only on b and λ.The separability principle denies that there can be a form of interdependence between two systems if they do not dynamically interact (factorization rule): where therefore A a and B b represent two deterministic functions of the hidden parameter.Eq. ( 17) expresses the fact that the probability distributions for the two particles are mutually independent.I assume that the result of each measurement can be either +1 (representing spin up) or -1 (representing spin down), that is, Following Eq. ( 17), if ℘(λ) denotes the probability distribution of the hidden parameter λ, then the expectation value of the product of the two components σ1 •a and σ2 •b is where Λ represents the set of all possible values of λ.
In the present context, A a (λ) and B b (λ) are functions defining the possible measurement results or the eigenvalues of the measured observables.Since we do not know the values of the hidden parameters λ, we must integrate over all the possible values λ ∈ Λ.Because ℘(λ) is supposed to be a normalized probability distribution, we have and, given the values (18), we also have ICFP 2012 00001-p.5 where I have rewritten the expression ( σ1 •a) ( σ2 •b) in the simplified form a, b .Our aim is to compare the prediction of a deterministic HV theory as expressed by Eq. ( 19) with the quantum mechanical expectation value, which for the singlet state |Ψ 0 [Eq.(10)] is given by This result can be derived when considering the previous products between observables and vectors as sum of Cartesian components The expectation value on the singlet state (11) of these two products gives 9 terms, of which the first three have the form Indeed, similar calculations show that we also have The remaining six cross terms are instead all zero, so that we may finally conclude that When the two orientations a and b are parallel, quantum mechanical calculations [see Eq. ( 15)] show that a, a as it should be since there is a perfect anticorrelation (spin-up versus spin-down) between the results of the two measurements.Since the value given by Eq. ( 27) for perfect anticorrelation is an experimental fact, also a HV theory must satisfy this requirement.On the other hand, a, a = −1 holds if and only if we also have for any direction a.In this case, Eq. ( 19) reaches the minimum value [see also Eq. ( 21)].Under this assumption, we can drop any reference to the B device and rewrite Eq. ( 19) as Now we consider two alternative orientations, say b and c, of the spin measurement of particle 2: EPJ Web of Conferences because of the property (18) and since, for any orientation n, we have [A n (λ)] 2 = 1, which implies Then, from Eq. (30) we may prove the inequality This result is obtained when one considers that for any integrable function f (x), we have and, given again the property (18), we also have Therefore, given the property (20) we finally obtain A reformulation of the Bell inequality (35) is the so-called CHSH inequality, a widely used form, where a is a setting alternative to a as well as b to b.We may associate to this inequality the following Bell operator:

Experiments and Loopholes
Tests of the Bell theorem already started in the mid of 1970s.However, several loopholes were discovered that affected these early experiments and could be dealt with step by step.The first loophole we consider is the locality loophole.In all experiments, one should consider the possibility that the result of a measurement obtained by using a certain polarizer direction depend on the orientation of the other polarizer.This problem was overcome by Aspect's team [1] as outlined in Fig. 1.Another difficulty (second loophole) concerns the angular correlation: Because of the cosinesquared angular correlation of the directions of the photons emitted in an atomic cascade, an inherent polarization decorrelation is present.Hence the very polarization correlation which could result in a ICFP 2012 00001-p.7 , where down-converted photons of wavelength of about 702 nm are produced.Down-conversion can be tuned in order that linearly polarized signal and idler photons emerge at angles of about ±2 • relative to the ultraviolet (UV) pump beam with the electric vector in the plane of the diagram.The idler (i) photons pass through a 90 • polarization rotator, while the signal (s) photons traverse a compensating glass plate C 1 producing an equal time delay.The two photons are then directed from opposite sides towards a beam splitter (BS).The input to the BS consists of an x-polarized s-photon and of a rotated y-polarized i-photon.The light beams emerging from BS, consisting of a mixing of i-photons and s-photons, pass through linear polarizers set at adjustable angles θ 1 and θ 2 , through similar interference filters (IF) and finally fall on two photodetectors D 1 and D 2 .The photoelectric pulses from D 1 and D 2 are amplified and shaped and fed to the start and stop inputs of a time-to-digital converter (TDC) under computer control which functions as a coincidence counter.
violation of one of the Bell inequalities is reduced for non-collinear photons.The problem can be overcome by using SPDC sources instead of atomic cascade ones [14].Pairs of photons resulting from SPDC can have an angular correlation of better than 1 mrad, although in general they need not be collinear.The set up is shown in Fig. 2. A further issue (third loophole) is represented by the detection loophole.In fact, we may raise the question of how high the detection efficiencies must be for the experimental confirmation of the  (a) An ultraviolet pump photon may be spontaneously down-converted in either of two nonlinear crystals, producing a pair of collinear orthogonally polarized photons at half the frequency (type-II phase matching).The outputs are directed toward a second PBS.When the outputs of both crystals are combined with an appropriately relative phase φ, a true singlet-or triplet-like state may be produced.By using a half-wave plate (HWP) to effectively exchange the polarizations of photons originating in crystal 2, one overcomes several problems arising from nonideal phase matching.An additional mirror is used to direct the photons into opposite direction towards separated analyzers.(b) A typical analyzer, including an HWP to rotate by θ the polarization component selected by the analyzing BS, and precision spatial filters to select only conjugate pairs of photons.In an advanced version of the experiment, the HWP could be replaced by an ultrafast polarization rotator (such as Pockels or Kerr cells) to close also the locality loophole.
quantum theoretical predictions.The problem with SPDC-type experiments is that, even with high detection efficiency, one must discard part of the counts, since we are obliged to discard all events where both photons are in the same channel, and one could rise the question whether this selection might represent a bias.Even though this is a remote possibility, in order to exclude any ambiguity a more refined solution is required [12].A possibility is to directly produce a pair of photons in singlettype state, thus avoiding any post-selection.One of the first proposals for doing this is shown and summarized in Fig. 3.By means of this apparatus it is possible to produce output photons in the state

Non-Locality and Information
I have a general remark.Given any quantum system described by the density matrix ρ, its von Neumann entropy is [11] S ( ρ) = −Tr( ρ ln ρ) .
In fact, the density matrix can be seen as the operator which carries maximal information about the state of the system.If we consider an orthonormal basis {|b k } of eigenvectors of the density operator ICFP 2012 00001-p.9ρ for a system S such that where the r k 's are the eigenvalues of ρ, we may rewrite Eq. (41) as The eigenvectors |b k are the possible outcomes of a measurement when we choose to measure the observable of which they are eigenvectors.Let us define the entanglement between systems [5] 1 and 2 as where S (1, 2) is the joint (total) entropy of systems 1 and 2, and are the entropies calculated on the reduced density matrices of the subsystems 1 and 2, respectively, relative to ρ12 .This reflects the fact that entanglement is a quantum form of mutual information: Two entangled systems are correlated because they share an amount of information that is not foreseen classically: indeed the possible outcomes are interdependent.Are there specific quantum mechanical bounds on information acquisition?Is the bound found with inequality (35) a necessity or are there more rigorous bounds?And if they are, what is their meaning?Let us take advantage of the CHSH inequality (37).Since each of the terms in Eq. (37) lies between −1 and +1 [Eq.( 21)], the natural upper bound for the entire expression is +4.This is precisely the case if we demand that the probabilities satisfy only the causal communication constraint [16], i.e., that they do not violate relativistic locality (what is called non-signaling requirement).In this case, we have Indeed, the non-signaling requirement is that the operations one can perform locally here are not influenced by the operations one performs elsewhere, which implies in particular that the probability to obtain a certain outcome (say 1) when choosing the direction a is independent from the outcomes (either + 1 or -1) when elsewhere one choses a direction b or b , that is, Similar considerations hold for any direction.If we consider only this requirement, we are allowed to build the set of probabilities while all other probabilities are zero and where I remark that only the ℘ a ,b probabilities show anticorrelation.
All the four different expectation values in inequality (46) can be formulated as the following one:

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Let Ô1 and Ô2 be two observables on subsystems S 1 and S 2 of a system S, respectively, and ℘(o a , a; o b , b) be the probability that the results of a measurement of Ô1 on S 1 and Ô2 on S 2 yield o a and o b when certain settings of the measurement apparata are a and b, respectively.Since this assumption is of absolute generality, we not need to consider the specific spin model previously introduced.
According to Eberhard, the probability distribution of Ô1 (or Ô2 ), independently of the measurement operations on Ô2 (or Ô1 ), obtained by integrating or summing the probabilities ℘(o a , a; o b , b) over the possible outcomes o b (or o a ), needs to be independent of the other setting b (or a), that is, the two probabilities must depend on local settings only [9]: According to Eberhard, if this requirement were violated, we would have a causal non-local interdependence between the two subsystems.Actually, a violation of the above requirement does not necessarily imply a non-local causal interconnection because there could still be a form of interdependence but satisfying the non-signaling requirement.
In order to prove the theorem, let For any events A and B, the classical probability calculus tells us that the probability of their joint occurrence can be expressed as Given these assumptions, we can obtain the following result that is in accordance with Eqs.(77): and similarly for the other outcomes.This clearly shows that quantum mechanics requires a full independence of the settings (here expressed by the orientation a), which need to be local operations performed in complete separation from other operations that could be performed elsewhere.Quantum correlations are indeed interdependencies of possible outcomes and not of settings.In other words, a violation of the quantum mechanical bound (and of the information causality principle) would imply that there are correlations between settings.If we consider the abstract forms (65) in which I have written the correlations entering in the CHSH inequality, we see that they are expressed in terms of pure conditional probabilities of the form ℘(11|00) Following the customary approach in quantum mechanics (and our physical experience) we have naturally interpreted probabilities of this form as expressing e.g. the probability that both Alice and Bob get the output 1 given that they have both chosen the setting 0. In fact, if Bob knows which was the setting of Alice (whether 0 or 1) he is able to infer which was her outcome.On this procedure is indeed based quantum cryptography.However, nothing forbids to interpret such a probability as telling us that, in a Bayesian inversion, Bob is able to predict that Alice has chosen the setting 1 once that he knows that he and Alice have obtained the outcome 0. This is still allowed by the non-signaling condition (47).As a matter of fact, given Eqs.(65), Bob is able to predict any setting of Alice if he knows her outcome:

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, which will play a crucial role later on.I recall indeed that e.g.a, b is a shorthand for ( σ1 •a) ( σ2 •b) , which allows us to write B ≤ 2 .(39)

Figure 1 .Figure 2 .
Figure 1.One should consider the possibility that the results obtained using a certain polarization direction could depend on the other polarizer.(a) Friedman-Clauser experiment: The correlated photons γ A , γ B coming from the source S impinge upon the linear polarizers A, B oriented in directions a, b, respectively.(b) Experiment proposed by Aspect: The optical commutator C A directs the photon γ A either towards polarizer A 1 with orientation a 1 or to polarizer A 2 with orientation a 2 .Similarly for C B for B 1 and B 2 .The two commutators work independently (the time intervals between two commutations are taken to be stochastic).The four joint detection rates are monitored and the orientations a 1 , a 2 , b 1 , b 2 are not changed for the whole experiment.l is the separation between the switches.

Figure 3 .
Figure3.The question is how high the detection efficiency must be for the experimental confirmation of the quantum predictions.In Aspect's experiment the required is 83%.With SPDC experiments, we are obliged to discard part of the counts (when both photons are in the same channel).Proposed experiment for solving the detection loophole.A possible solution is to directly produce a pair of photons in a singlet-kind state avoiding in this way any post-selection.(a) An ultraviolet pump photon may be spontaneously down-converted in either of two nonlinear crystals, producing a pair of collinear orthogonally polarized photons at half the frequency (type-II phase matching).The outputs are directed toward a second PBS.When the outputs of both crystals are combined with an appropriately relative phase φ, a true singlet-or triplet-like state may be produced.By using a half-wave plate (HWP) to effectively exchange the polarizations of photons originating in crystal 2, one overcomes several problems arising from nonideal phase matching.An additional mirror is used to direct the photons into opposite direction towards separated analyzers.(b) A typical analyzer, including an HWP to rotate by θ the polarization component selected by the analyzing BS, and precision spatial filters to select only conjugate pairs of photons.In an advanced version of the experiment, the HWP could be replaced by an ultrafast polarization rotator (such as Pockels or Kerr cells) to close also the locality loophole.
Po a ,a = | o a , a o a , a | and Po b ,b = | o b , b o b , b | (78) be the projectors on the state |o a , a of subsystem S 1 when the setting is a and the outcome |o a , and on the state |o b , b of subsystem S 2 when the setting is b and the outcome |o b , respectively, and ρ a density matrix which represents the compound state of S = S 1 + S 2 .The probability ℘(o a , a) that, by measuring the observable Ô1 on S 1 , we obtain the outcome |o a (or the eigenvalue o a ), is ℘(o a , a) = Tr Po a ,a ρ .(79) After a measurement of Ô1 when the setting is a with result o a we obtain the transformation ρ → ρ = Po a ,a ρ Po a ,a ℘(o a , a) .(80) If we perform a second measurement on the second subsystem, the conditional probability of obtaining |o b (or o b ) by measuring Ô2 when the setting is b, is given by ℘ (o b , b|o a , a) = Tr Po b ,b ρ = Tr Po b ,b Po a ,a ρ Po a ,a ℘(o a , a) .
) Therefore, the joint probability of obtaining the two results o a and o b given the settings a and b, is given by combining Eqs.(79) and (81): ℘(o a , a; o b , b) = ℘(o a , a)℘ (o b , b|o a , a) = ℘(o a , a) Tr Po b ,b Po a ,a ρ Po a ,a ℘(o a , a) = Tr Po b ,b Po a ,a ρ Po a ,a .
a , a; o b , b) = Tro a Po b ,b Po a ,a ρ Po a ,a = Tr Po b ,b ρ = ℘(o b , b). (84) To derive this result, first note that o a Tr Po b ,b Po a ,a ρ Po a ,a = Tr o a Po b ,b Po a ,a ρ Po a ,a .(85) Moreover, I have made use of the cyclic properties of the trace, i.e., given any three arbitrary observables, we have Tr Ô1 Ô2 Ô3 = Tr Ô3 Ô1 Ô2 = Tr Ô2 Ô3 Ô1 .Po b ,b Po a ,a ρ Po a ,a = Tr Po a ,a Po b ,b Po a ,a ρ .(87) Moreover, Po a ,a and Po b ,b commute because they pertain to different subsystems, and therefore we have Tr o a Po a ,a Po b ,b Po a ,a ρ = Tr o a Po b ,b Po a ,a Po a ,a ρ .(88) However, any orthogonal set of projectors { Po a ,a } satisfies the two properties P2 o a ,a = Po a ,a and o a Po a ,a = Î, from which we finally obtain Tr o a Po b ,b Po a ,a Po a ,a ρ = Tr Po b ,b ρ .(89)Wemay proceed in a similar way starting from the conditional probability ℘ (o a , a|o b , b) in order to derive the second equality (77).What would happen in a world in which the quantum bound is violated but the locality (nonsignaling) requirement is satisfied[3]?Let us now reformulate the quantum-mechanical Eqs.(77) in analogy with Eq. (47) as ℘ a,b (1, 1) + ℘ a,b (1, −1) = p a (1) and ℘ a,b (1, 1) + ℘ a,b (−1, 1) = ℘ b (1),