4 Coherent States in Quantum Information: an Example of Experimental Manipulation

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462 Photon Counting Distributions

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Given the alphabet A = ( 0 , 1 ), the feedback amplitude u(t), a transmission coef cient , and some subdivision (t 0 = 0, t 1 , , t n , t n+1 = ) of the measurement time interval (or counting interval ) [0, ], the conditional probability w t k | i , u(t) that a photon will arrive at time t k and that it will be the only click during the halfclosed interval (t k 1 , t k ] [7] is called the exponential waiting time distribution for optical coherent states It is de ned as w t k | i , u(t) = (t k ) exp

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(t ) dt

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t k 1

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(429)

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The corresponding exclusive counting densities for the measurement interval are then given by

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p t 1 , , t n | i , u(t) =

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w t k | i , u(t)

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(430)

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They allow one to evaluate, using the Bayes rule, the conditional arrival time probabilities p i |t 1 , , t n , u(t) = p t 1 , , t n | i , u(t) p 0 ( i ) The latter re ect the likelihood that n photon arrivals occur precisely at the times t 1 , , t n , 3) given that the channel is in the state i , the feedback amplitude is u(t), and the detector quantum ef ciency is

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463 Decision Criterion of the Dolinar Receiver

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The receiver decides between hypotheses H0 and H1 by selecting the one that is more consistent with the record of photon arrival times observed by the detector given the choice of u(t) H1 is selected when the ratio of conditional arrival time probabilities, = p 1 |t 1 , , t n , u(t) p 0 |t 1 , , t n , u(t) , (431)

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is greater than one; otherwise it is assumed that 0 was transmitted By employing the Bayes rule, one can reexpress in terms of the photon counting distributions = p t 1 , , t n | 1 , u(t) p 0 ( 1 ) p t 1 , , t n | 0 , u(t) p 0 ( 0 ) =

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p t 1 , , t n | 1 , u(t) p t 1 , , t n | 0 , u(t)

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(432)

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3) Even though the term arrival time is not appropriate from an experimental point of view Time interval is more appropriate

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46 The Dolinar Receiver

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In terms of error probabilities, the likelihood ratio is given by = p H1 | 1 , u(t) p H1 0 , u(t) = 1 p H0 | 1 , u(t) p H1 | 0 , u(t) , for > 1 (433)

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(ie, the receiver de nitely selects H1 ), and = p H0 | 1 , u(t) p H0 | 0 , u(t) = p H0 | 1 , u(t) 1 p H1 | 0 , u(t) , for < 1 (434)

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(ie, the receiver de nitely selects H0 )

464 Optimal Control

The minimization over u(t) of the Dolinar receiver error probability, P D [u(t)] =

0 p

H 1 | 0 , u(t) +

1 p

H 0 | 1 , u(t) ,

(435)

can be accomplished by employing the technique of dynamical programming [47] The optimal control policy, u (t), is identi ed by solving the Hamilton Jacobi Bellman equation, min

u(t)

J [u(t)] + p J [u(t)]T p(t) = 0 , t t

(436)

where the control cost J [u(t)] = P D [u(t)] = given by the conditional error probabilities, p(t) = p H 1 | 0 , u(t) (t) p H 0 | 1 , u(t) (t)

p in an effective state-space picture

(437)

The partial differential equation for J is based on the requirement that p(t) and u(t) are smooth (continuous and differentiable) throughout the entire receiver operation However, like all quantum point processes, our conditional knowledge of the system state evolves smoothly only between photon arrivals Fortunately, the dynamical programming optimality principle allows us to optimize u(t) in a piecewise manner [47] Performing the piecewise minimization leads to the control policy u (t) = 1 (t) 1 + 1 J [u (t)] 1 1 2J [u (t)] 1 (438)

for > 1 (see [42] for the proof), where p [H0 | 1 , u (t)] = 0 and 1 J [u (t)] = 1

t 1 p [H1 | 0 , u 1 (t)]

1 1 2

0 1e