* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Entanglement and Bell theorem
Delayed choice quantum eraser wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Scalar field theory wikipedia , lookup
Particle in a box wikipedia , lookup
De Broglie–Bohm theory wikipedia , lookup
Atomic theory wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum key distribution wikipedia , lookup
Renormalization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Spin (physics) wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Matter wave wikipedia , lookup
Elementary particle wikipedia , lookup
History of quantum field theory wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Identical particles wikipedia , lookup
Canonical quantization wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum state wikipedia , lookup
Quantum teleportation wikipedia , lookup
EPR paradox wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum entanglement wikipedia , lookup
Bell inequality & entanglement The EPR argument (1935) based on three premises: 1. Some QM predictions concerning observations on a certain type of system, consisting of two spatially separated particles, are correct. 2. 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’ 3. There is no action-at-a-distance in nature. EPR paradox 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: 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: 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! 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. Deterministic local hidden variables and Bell’s theorem • Bohm’s theorem: spatially separated spin ½ particles produced in singlet state: 1 2 [unˆ (1) unˆ (2) unˆ (1) unˆ (2)] • 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 nonlocality 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. LHV and Bell’s theorem •Let Aaˆ be the result of a measurement of the spin component of particle 1 of the pair along the direction â and Bbˆ the result of a measurement of the spin component of particle 2 of the pair along the direction bˆ •We denote a unit spin as h / 2 hence B ˆ , Aaˆ = +1 b •The expectation value of this observable is: [ E (aˆ, bˆ)] 1 aˆ 2 bˆ aˆ bˆ •When the analyzers are parallel we have: [ E (aˆ , aˆ )] 1 •The EPR premise 2 assures us that if we measure A we know B 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 d 1 Bell’s definition: • A deterministic hidden variable theory is local if for all â and b̂ and all we have: ( Aaˆ Bbˆ )( ) Aaˆ ( ) Bbˆ ( ) • The meaning of this is that once the state is specified and the particles have separated measurements of A can depend on and â but not b̂ • The expectation value is taken to be: E a, b Aaˆ ( ) Bbˆ ( )d Proof of Bell’s inequality [ E (aˆ , aˆ )] 1 Holds if and only if Aaˆ ( ) Baˆ ( ) Hence: E (aˆ , bˆ) E (aˆ , cˆ) Aaˆ ( ) Abˆ ( ) Aaˆ ( ) Acˆ ( ) d Aaˆ ( ) Abˆ ( )[1 Abˆ ( ) Acˆ ( )]d Since A,B=+1 E (aˆ , bˆ) E (aˆ , cˆ) [1 Abˆ ( ) Acˆ ( )]d Proof of Bell’s inequality Using: d 1 E a, b Aaˆ ( ) Bbˆ ( )d Aaˆ ( ) Baˆ ( ) We have: E (aˆ , bˆ) E (aˆ, cˆ) 1 E (bˆ, aˆ ) Violation of Bell inequality • Taking aˆ, bˆ, cˆ to be coplanar with ĉ making an angle of 2 / 3 with â , and b̂ making an angle of / 3 with both â and ĉ then: aˆ bˆ bˆ cˆ 1 / 2 aˆ cˆ 1 / 2 Which gives: E(aˆ, bˆ) E(aˆ, cˆ) 1 and 1 E (bˆ, aˆ ) 1 / 2 What is the meaning of violating the Bell inequality? • No deterministic hidden variables theory satisfying the locality condition and [ E (aˆ, aˆ )] 1 can agree with all of the predictions by quantum mechanics concerning spins of a pair of spin1/2 particles in the singlet case. In other words: once Bell’s inequality is violated we must abandon either locality or reality! Requirements for a general experiment test • Let us consider the following apparatus: Experiment requirements • The QM predictions take the following form: 1 [ p12 ( )]QM 1 2 f1 g[ 1 2 1 2 F cos( n )] 4 1 [ p1 ]QM 1 f1 1 2 1 [ p1 ]QM 2 f 2 2 2 i Mi mi i Effective quantum efficiency of the detector Mi mi max & min transmission of the analyzers f i Collimator efficiency (probability that appropriate emission enters apparatus 1 or 2 g Conditional probability that if emission 1 enters apparatus 1 then emission 2 enters apparatus 2 F Measure of the initial state purity n=1 for fermions and n=2 for bosons Experiment requirements • Taking the following assumptions: 1 2 , f1 f 2 , 1 2 / 4n g [ 2 ( / ) 2 F 1] 2 If the experimental values are within the domain of the above inequality, then we can distinguish between QM prediction and inequalities. 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. CHSH • Since no idealized system exists, one can abandon the requirement: [ E (aˆ , aˆ )] 1 • CHSH arrived at the following inequality: E(a, b) E(a' , b) E(a, b' ) E(a' , b' ) 2 Which was violated by Alain’s experiment => Proof of non-local correlations occur on a time scale faster than the speed of light. experiments experiments • The third Alain Aspect experiment: faster than light correlation. • Using the following setup: Crash course in information theory… • Ensamble of quantum states with probability p i i We can define the density operator as: pi i i i 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 tr( ) 1 • A mixed state satisfies tr( ) 1 2 2 Schmidt de-composition 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: pi ..... pn then the Shannon Entropy associated with it is: Crash course in information theory… • Definition of typical variable X obeys: Typical sources xi are sources which are highly likely to occur. Von Neumann entropy 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. 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 Setting the limits • Suppose an entangled state has a Schmidt decomposition Setting the limits • An m-fold tensor product can be defined 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 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.