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MIT OpenCourseWarehttp://ocw.mit.edu 6.453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Problem Set 8 Fall 2008 Issued: Tuesday, October 28, 2008 Due: Tuesday, November 4, 2008 Reading: For entanglement and measures of entanglement: • L. Mandel and E. Wolf, Optical Coherence and Quan tum Optics (Cambridge University Press, Cambridge, 1995), Sect. 12.14. • D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation (Springer Verlag, Berlin, 200 0), Sect s. 3.4 and 3.5. For qubit telep ortation: • C.C. Gerry and P.L. Knight, Introductory Quantum Optics (Cambridge Uni-versity Press, Cambridge, 2005) Sect. 11.3. • D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation (Springer Verlag, Berlin, 200 0), Sect s. 3.3 and 3.7. For quadrature teleportation: • D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-mation (Springer Verlag, Berlin, 200 0), Sect . 3.9. For optimum binary hypothesis testing: • C.W. Helstrom, Quantum Detection an d Estimation Th eory (Academic Press, New York, 1976) Sects. 4.2 and 6.1. Problem 8.1 Here we shall begin a treatment of optimum binary hypothesis t esting. Suppose that a quantum system is known to be in either state |ψ−1� or |ψ1�, where |ψ−1� =�|ψ1�. Let hypothesis H−1 denote “state = |ψ−1�” and hypothesis H1 denote “ state = |ψ1�.” Assume that these two hypotheses are equally likely, i.e., before we make any measurement on the quantum system, it has probability 1/2 of being in state |ψ−1�and probability 1/2 of being in state |ψ1�. Our task is to make a measurement on this system to determine—wit h the lowest probability of being wrong—whether the system’s state was |ψ−1� or |ψ1� before we make our measurement. (The projection postulate implies that the system’s state will likely be changed by our having made a measurement.) 1Because we know the system can only be in |ψ−1� or |ψ1� we can—and we will— limit all our analysis in the reduced Hilbert space, H ≡ span(|ψ−1�, |ψ1�), i.e., to the Hilbert space of kets of the form |ψ� = α|ψ−1� + β|ψ1�, where α and β are complex numbers. Define a decision operator, ,Dˆ≡ |d1��d1| − |d−1��d−1|where { d−1�, d1�} are a pair of orthonormal kets on the reduced Hilbert space H.| |ˆClearly, Dˆis an observable on H. Suppose that we measure D on the quantum system under study. If the outcome of this measurement is −1, we will say that the state before the measur ement was |ψ−1�. If the outcome of this measurement in 1, we will say that the state before the measurement was |ψ1�. (a) Find the conditional probabilities, Pr( say “state was |ψ−1�” | state was |ψ1�) = Pr(Dˆ= −1 | |ψ1�), Pr( say “state was |ψ1�” | state was |ψ−1�) = Pr(Dˆ= 1 | |ψ−1�). and the unconditional error probability, Pr(e) ≡ Pr(state was |ψ−1�) Pr(Dˆ= 1 | |ψ−1�) + Pr(state was |ψ1�) Pr(Dˆ= −1 | |ψ1�). (b) Suppose that �ψ−1|ψ1� = 0, so that {|ψ−1�, |ψ1�} is an orthonormal basis for H. Find the measurement eigenkets {|d−1�, |d1�} that minimize your error prob-ability expression from (a). [The error probability of your optimum decision operator for this case shows why orthonormal kets are said to be “distinguish-able.”] (c) Suppose that |ψ−1� and |ψ1� are normalized (unit length), but not orthog onal. In particular, let {|x�, |y�} be an orthonormal basis for H, and assume that, |ψ−1� = cos(θ)|x� − sin(θ)|y� and | ψ1� = cos(θ)|x� + sin(θ)|y�, where 0 < θ < π/4. Using the expansions, |d−1� = cos(φ)|x� − sin(φ)|y� and | d1� = sin(φ)|x� + cos(φ)|y�, 22where 0 ≤ φ < 2π, and your Pr(e) result from (a) find the φ value—hence the {|d−1�, |d1�}—that minimizes the error probability fo r this case. [Hint: By assiduous use of trig identities, you should be able to reduce the error probability expression to the following form: 1 Pr(e) = [1 − sin(2φ) sin(2θ)], which is easily minimized over φ.] Problem 8.2 Here we shall continue our treatment of optimum binary hypothesis testing. Suppose that the quantum system considered in Problem 8 .1 is a single-mode optical field with annihilation operator ˆa. (a) Let |ψ−1� = |n−1� and |ψ1� = |n1� be photon number states with n−1 =� n1. Show that making the number operator measurement, Nˆ≡ aˆ†aˆ, on the single-mode field allows a zero-error-probability decision to be made as to whether the state before the measurement was |n−1� or |n1�. (b) Let |ψ−1� = |α−1� and |ψ1� = |α1� be coherent states with � α−1|α1� = cos(2θ) for a θ va lue satisfying 0 < θ < π/4. Find the error probability achieved by the minimum-error-probability decision operator for deciding whether the state before the measurement was |α−1� or |α1�. (c) Evaluate your error probability from (b) when o n-off keying (OO K) is used: |α−1� = |0� and |α1� = |√N�, i.e., when the two coherent states we are trying to distinguish are the vacuum state, and a coherent state with averag e photon number N. Compare this error probability with what is achieved when we make the Nˆmeasurement and say “state was |0� ” when this measurement yields outcome 0 and say “state wa s |√N�” when this measurement yields a non-zero outcome. [Hint: First find the conditional error probabilities, Pr( say “state was |0�” | state was |√N�), and Pr( say “state was |√N�” | state wa s |√0�). and then find t he


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