Feedback in the problem of distinguishing between two nonorthogonal coherent states
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MOLECULES, OPTICS
Feedback in the Problem of Distinguishing between Two Nonorthogonal Coherent States V. N. Gorbacheva,* and M. V. Chekhovab a
St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, 190000 Russia b Moscow State University, Moscow, 119991 Russia *email: [email protected] Received May 22, 2010
Abstract—Feedback is proposed for distinguishing between two weak coherent states with phases differing by~π. The mutual nonorthogonality of such states gives rise to a discrimination error, which can be reduced by using feedback. An optical quantum channel is discussed where the input is classical information encoded in two weak coherent states. For a channel with feedback, the discrimination error probability is calculated, and the mutual entropy that quantifies the fidelity between input and output is evaluated. We find that the use of a feedback loop in a quantum communication channel can increase the mutual entropy when canonical position or photon number is measured. DOI: 10.1134/S106377611006110X
1. INTRODUCTION When quantum states are used in information transmission systems, as in quantum cryptography, an important problem is that of optimal discrimination between two mutually nonorthogonal quantum states |ψ 1〉 and |ψ 2〉 . There are at least two approaches to the problem. In one of these, three measurement out comes are possible: |ψ 1〉 is identified with probability one, |ψ 2〉 is identified with probability one, and an inconclusive outcome. In the other approach, only two outcomes may be obtained, but either one must contain an error. In particular, if |ψ 1〉 has been sent, then it is incorrectly identified as |ψ 2〉 with probability P(ψ2 |ψ1), and vice versa. This leads to a total error probability expressed as p e = λ 1 P ( ψ 2 ψ 1 ) + λ 2 P ( ψ 1 ψ 2 ),
guished [2]. A more complicated problem frequently arising in quantum information theory (e.g. see [3]) is that of distinguishing between two nonorthogonal coherent states |α〉 and |– α〉 , where α is the state amplitude ( α ≤ 1, see Fig. 1). In the conventional strategy, this is done by homodyne detection, which is equivalent to measuring canonical position: depend ing on whether the outcome of a position measure ment is positive or negative, the output state is con cluded to be |α〉 or |– α〉 , respectively (to be specific, we assume that Reα > 0). However, this strategy can
(1)
where λk is the probability that |ψ k〉 has been sent (k = 1, 2). Helstrom [1] was the first to demonstrate that the lowest possible error probability is determined by the inner product of ψ1 and ψ2: 2 P H = 1 [ 1 – 1 – 4λ 1 λ 2 〈ψ 1|ψ 2〉 ]. 2
p
(2)
This limiting probability is called the Helstrom bound. Expression (2) is a fundamental one, but it gives no clue as to how PH can be achieved. In each particular case, it should be achieved by a strategy based on the nature of |ψ 1〉 and |ψ 2〉 . For example, the Helstrom bound is relatively easy to achieve when two singlephoton polarization states are to be distin 179
|−α〉
|α〉
q
Fig.
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