43 Binary Coherent State Communication

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(iii) nonzero dead-time, or nite detector recovery time after registering the arrival of a photon, (iv) nite bandwidth of any signal processing necessary to implement the detector, (v) uctuations in the phase of the incoming optical signal

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43 Binary Coherent State Communication 431 Binary Logic with Two Coherent States

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Let us consider an alphabet consisting of two pure coherent states, 0 = | 0 0 | , 1 = | 1 1 | ,

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corresponding to the logic states 0 and 1, respectively Without loss of generality, 0 (t) can be chosen as the vacuum, 0 (t) = 0, that is, | 0 = |0 , while 1 (t) = 1 (t) exp i( t + ) + cc , where is the frequency of the optical carrier and is (ideally) a xed phase The envelope function, 1 (t), is normalized such that

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(41)

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| 1 (t)|2 dt = n ,

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(42)

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where n is the mean number of photons arriving at the receiver during the measurement interval, 0 u t u That is, | 1 (t)|2 is the instantaneous average power of the optical signal for logic 1 By combining the incoming signal with an appropriate local oscillator, one can always transform the amplitude keying with the alphabet of two coherent states A = {|0 0|, | |}, with | = | 1 , to the phase-shift keyed alphabet, 1 2 1 1 , 2 2 1 2 ,

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via the unitary displacement, D 1 = exp( 1 ( a a)) Similarly, if | 0 = |0 , / 2 2 a simple displacement can be used to restore | 0 to the vacuum state

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432 Uncertainties on POVMs

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In the case of nonorthogonal quantum states as codewords, the receiver attempts to ascertain which state was transmitted by performing a quantum measurement,

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

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say, , on the channel The operator is described by an appropriate POVM represented by a complete (here countable) set of positive operators [37] resolving the identity, i = I d ,

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i v 0 ,

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(43)

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where n indexes the possible measurement outcomes We already gave a simple example of a continuous POVM in Section 323 In the same vein, an example of a nite POVM in the Euclidean plane is given by the following cyclotomic polygonal resolution of the unity: 2 n

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2 q = I d ,

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= | | =

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cos2 cos sin

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cos sin sin2

For binary communication, for which the POVM resolution of the unity reads 0 + 1 = I d , the measurement by the receiver amounts to a decision between two hypotheses: H0 , that the transmitted state is 0 , selected when the measurement outcome corresponds to 0 , and H1 , that the transmitted state is 1 , selected when the measurement outcome corresponds to 1

433 The Quantum Error Probability or Helstrom Bound

Now, possibilities of errors mean that there is some chance that the receiver will select the null hypothesis, H0 (or H1 ), when 1 (or 0 ) is actually present Thus, we have in terms of conditional probabilities p (H0 | 1 ) = tr[ 0 1 ] = tr[(I d 1 ) 1 ] , p (H1 | 0 ) = tr[ 1 0 ] (44)

The total receiver error probability, say, p[ 0 , 1 ], is then given by p[ 0 , 1 ] =

(H1 | 0 ) +

(H0 | 1 ) ,

= 1,

(45)

where 0 = p 0 ( 0 ) and 1 = p 0 ( 1 ) are the probabilities that the sender will transmit 0 and 1 , respectively; they re ect the prior knowledge that enters into the hypothesis testing process implemented by the receiver, and, in many cases 0 = 1 = 1/2 Minimizing the error in receiver measurement over all possible POVMs ( 0 , 1 ) leads to the so-called quantum error probability or Helstrom bound, P H = min p[ 0 , 1 ] ,

0 , 1

(46)