Bell inequality & entanglement

1. Bell inequality & entanglement

2. The EPR argument (1935)based on three premises: • Some QM predictions concerning observations on a certain type of system, consisting of two spatially separated particles, are correct. • A very reasonable criterion of the existence of ‘an element of physical reality’ is proposed: ’if, without any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity’ • There is no action-at-a-distance in nature.

3. EPR paradox

4. EPR paradox • Before making the measurement on spin 1 (in z direction) the state vector of the system is: • After measurement on particle 1, (for argument’s sake say we measured spin down), the state of particle 2 is:

5. EPR paradox • Since there is no longer an interaction between particle 1 and 2, and since we haven’t measured anything of particle 2, we can say that it’s state before the measurement is the same as after:

6. EPR paradox • We could apply the same argument if we have measured the spin in the x direction and receive: In other words: it is possible to assign two different state vectors to the same reality!

7. Bell’s theorem • If premise 1 is taken to assert that all quantum mechanical predictions are correct, then Bell’s theorem has shown it to be inconsistent with premises 2 & 3.

8. Deterministic local hidden variables and Bell’s theorem • Bohm’s theorem: spatially separated spin ½ particles produced in singlet state: • All components of spin of each particle are definite, which of course is not so in QM description => HV theory seems to be required. • The question asked by Bell is whether the peculiar non-locality exhibited by HV models is a generic characteristic of HV theories that agree with the statistical predictions by QM. • He proved the answer was: YES.

9. LHV and Bell’s theorem • Let be the result of a measurement of the spin component of particle 1 of the pair along the direction and the result of a measurement of the spin component of particle 2 of the pair along the direction • We denote a unit spin as hence , = +1 • The expectation value of this observable is: • When the analyzers are parallel we have: • The EPR premise 2 assures us that if we measure A we know B

10. Local Hidden Variables defined. • Since QM state does not determine the result of an individual measurement, this fact suggests that there exists a more complete specification of the state in which this determinism is manifest. We denote this state by • Let be the space of these states • We represent the distribution function for these states by

11. Bell’s definition: • A deterministic hidden variable theory is local if for all and and all we have: • The meaning of this is that once the state is specified and the particles have separated measurements of A can depend on and but not • The expectation value is taken to be:

12. Proof of Bell’s inequality Holds if and only if Hence: Since A,B=+1

13. Proof of Bell’s inequality Using: We have:

14. Violation of Bell inequality • Taking to be coplanar with making an angle of with , and making an angle of with both and then: Which gives: and

15. What is the meaning of violating the Bell inequality? • No deterministic hidden variables theory satisfying the locality condition and can agree with all of the predictions by quantum mechanics concerning spins of a pair of spin-1/2 particles in the singlet case. In other words: once Bell’s inequality is violated we must abandon either locality or reality!

16. Requirements for a general experiment test • Let us consider the following apparatus:

17. Experiment requirements • The QM predictions take the following form: Effective quantum efficiency of the detector max & min transmission of the analyzers Collimator efficiency (probability that appropriate emission enters apparatus 1 or 2 Conditional probability that if emission 1 enters apparatus 1 then emission 2 enters apparatus 2 Measure of the initial state purity n=1 for fermions and n=2 for bosons

18. Experiment requirements • Taking the following assumptions: If the experimental values are within the domain of the above inequality, then we can distinguish between QM prediction and inequalities.

19. Summery: for direct test of inequality the requirements are: • A source must emit pairs of discrete-state systems, which can be detected with high efficiency. • QM must predict strong correlations of the relevant observables of each pair, and the pairs must have high QM purity. • Analyzers must have extremely high fidelity to allow transmittance of desired states and rejections of undesired. • The collimators must have high transmittance and not depolarize the emissions. • A source must produce the systems via 2-body decay, or else g becomes g<<1. • For locality’s sake: Analyzer parameters must be changed while particles are in flight. (no information exchange between detectors.

20. CHSH • Since no idealized system exists, one can abandon the requirement: • CHSH arrived at the following inequality: Which was violated by Alain’s experiment => Proof of non-local correlations occur on a time scale faster than the speed of light.

21. experiments

22. experiments • The third Alain Aspect experiment: faster than light correlation. • Using the following setup:

23. Crash course in information theory… • Ensamble of quantum states with probability We can define the density operator as:

24. Crash course in information theory… • A quantum system whose state is known exactly is said to be in pure state, otherwise it is said to be in mixed state. • A pure state satisfies • A mixed state satisfies

25. Schmidt de-composition

26. Crash course in information theory… • Shannon Entropy: quantifies how much information we gain, on average, on a random variable X, or the amount of uncertainty before measuring the value of X. • If we know the probability distribution of X: then the Shannon Entropy associated with it is:

27. Crash course in information theory… • Definition of variable X obeys: Typical sources are sources which are highly likely to occur.

28. Von Neumann entropy

29. Entanglement distillation and dilution • Suppose we are supplied not with one copy of a state , but with a large number of it. Entanglement distillation is how many copies of a pure state we can convert into entangled Bell state. Entanglement dilution is the reverse process.

30. Entanglement distillation and dilution • Defining a specific bell state as a ‘standard unit’ of entanglement, we can quantify entanglement. • Defining an integer n which represents the number of Bell states, and an integer m representing the number of pure states that can be produced, then the limiting ratio n/m is the entanglement of formation of the state

31. Setting the limits • Suppose an entangled state has a Schmidt decomposition

32. Setting the limits • An m-fold tensor product can be defined

33. Alice and Bob live by the limits => We have an upper limit for entanglement formation! In a similar manner it was shown that there is a lower limit for entanglement distillation which is also

34. bibliography • Quantum optics- an introduction/ M. Fox p.304-323 • Quantum optics/ M.Scully & M.Tsubery p.528-550. • Bell’s theorem: experimental tests and implications, J. Clauser& A. Shimony, Rep. Prog. Phys, Vol.41, 1978. • Experimetal tests of realistic local theories via Bell’s theorem, PRL vol.47, nu.7, 1981 A.Aspect et al. • Quantum computation and quantum information, M.Nielasen and I.Chuang, p.137,580, 607.