- The statistical mixture of Bob leads to the same momentum distrib- ution as that measured by Alice: the N/2 oscillators in the state | α all lead to a mean momentum +p 0 , and the N/2 oscillators in the state. - In the X variable, the resolution of the detector satisfies δX 1. - Alice therefore has a sufficient resolution to observe the oscillations of the function cos 2 (Xρ. - 2 − π/4) in the distribution P (X. - The shape of the distri- bution will therefore reproduce the probability law of X drawn on figure 12.1, i.e. - If Bob performs a position measurement on the N/2 systems in the state | α , he will find a Gaussian distribution corresponding to the probability law P (X. - He will find the same distribution for the N/2 systems in the state. - Section 12.4: The Fragility of a Quantum Superposition. - (a) The probability distribution of the position keeps its Gaussian envelope, but the contrast of the oscillations is reduced by a factor η.. - (b) The probability distribution of the momentum is given by P (p. - Since the overlap of the two Gaussians ϕ α 1 (p) and ϕ − α 1 (p) is negligible for. - 12.6 Comments 119 whereas only the Gaussian is observed on a statistical mixture. - For times shorter than γ − 1 , the energy of the first oscillator is E(t. - The energy of the second oscillator is. - 12.6 Comments. - The field stored in the cavity is a quasi-perfect harmonic oscillator. - The preparation of the kitten (Sect. - 4) corresponds to the very weak residual absorption by the walls of the superconducting cavity. - We assume here that Alice (A) wants to send Bob (B) some information which may be coded in the binary system, for instance. - (13.1) We denote the number of bits of this message by n. - 13.1 Preliminaries. - The spin operator is ˆ S. - Consider a particle in the state | σ z = +1 . - One measures the component of the spin along an axis u in the (x, z) plane, defined by the unit vector. - e u = cos θ e z + sin θ e x , (13.2) where e z and e x are the unit vectors along the z and x axes respectively. - Show that the possible results of the measurement are. - Show that the eigenstates of the observable (13.3) are (up to a mul- tiplicative constant):. - h/2 when measuring the projection of the spin along the u axis.. - What are the spin states after measurements that give the results. - Immediately after such a measurement, one measures the z compo- nent of the spin.. - (a) What are the possible results and what are the probabilities of finding these results in terms of the results found previously along the u axis (observable (13.3)).. - (b) Show that the probability to recover the same value S z. - h/2 as in the initial state | σ z = +1 is. - h/2 in the last measurement?. - Alice measures the component of the spin of a along a direction θ a and Bob measures the component of the spin of b along a direction θ b. - 13.2 Correlated Pairs of Spins. - 13.1), prepared in the state | ψ = φ( r a , r b. - Σ where the spin state of the two particles is. - 13.2 Correlated Pairs of Spins 123. - A spy, sitting between the source and Bob, measures the component of the b spin along an axis θ s. - In other words, the spin variables are decoupled from the space variables ( r a , r b. - (specifically u = z) are the eigenstates of the u component of the spin of particle a, and similarly for b.. - The pair of particles (a, b) is prepared in the spin state . - (a) Alice first measures the spin component of a along an axis u a of angle θ a . - What are the possible results and the corresponding probabilities in the two cases θ a = 0, i.e. - the x axis?. - (b) Show that, after Alice’s measurement, the spin state of the two particles depends as follows on the measurement and its result. - After Alice’s measurement, Bob measures the spin of particle b along an axis u b of angle θ b. - Give the possible results of Bob’s measurement and their probabilities in terms of Alice’s results in the four following configurations:. - Alice, who controls the source S, prepares an ordered sequence of N n pairs of spins in the state (13.4) (n is the number of bits of the message). - He communicates openly to Alice (by cell phone, www, etc.) the axis and the result of the measurement for each event of this subset. - Suppose that a “spy” sitting between the source and Bob measures the spin of particle b along an axis u s of angle θ s as sketched in Fig. - (a) What are, in terms of θ s and of Alice’s findings, the results of the spy’s measurements and their probabilities?. - (b) After the spy’s measurement, Bob measures the spin of b along the axis defined by θ b = 0. - What does Bob find, and with what probabilities, in terms of the spy’s results?. - (c) What is the probability P (θ s ) that Alice and Bob find the same results after the spy’s measurement?. - (d) What is the expectation value of P (θ s ) if the spy chooses θ s at random in the interval [0, 2π] with uniform probability?. - 13.3 The Quantum Cryptography Procedure 125. - 13.3 The Quantum Cryptography Procedure. - Comment on the two “experiments” whose results are given in Tables 13.1 and 13.2. - Complete the missing item (number 7 in the above procedure), and indicate how Alice can send her message (13.1) to Bob without using any other spin pairs than the N pairs which Bob and her have already analyzed.. - Using Table 13.3, tell how, in experiment 1, Alice can send to Bob the message. - Choice of axes publicly communicated by Bob in the framework of experiment 1, after Alice has said she is convinced that she is not being spied upon. - 13.4 Solutions. - Section 13.1: Preliminaries. - The spin observable along the u axis is S ˆ · e ˆ u. - The possible results of the measurement are the eigenvalues of ˆ S · e ˆ u , i.e.. - (a) If the measurement along u has given. - If the measurement along u has given. - 13.4 Solutions 127 Altogether, one has. - Section 13.2: Correlated Pairs of Spins. - If we make the substitution in expression (13.4), we obtain 1. - (b) This array of results is a consequence of the reduction of the wave packet.. - σ z b = +1 (Alice’s result. - 1 (Alice’s result. - A similar formula holds for a measurement along the x axis, because of the invariance property, and its consequence, (13.5).. - Any measurement on b (a probability, an expectation value) will imply expectation values of operators of the type ˆ I a ⊗ B ˆ b where ˆ B b is a projector or a spin operator. - Since the states under consideration are factorized, the corresponding expressions for spin measurements on b will be of the type
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