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MIT 6 453 - Problem Set 3

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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 3 Fall 2008 Issued: Thursday, September 18, 2008 Due: Thursday, September 25, 2008 Problem 3.1 Here we shall extend the results of problem 2.2 to include classically-random polar-izations. Suppose we have a +z-propagating, frequency-ω photon whose p olarization vector (in problem 2.1 notation) is, αx ,i ≡ αy where αx and αy are a pair of complex-valued c lassical random variables that satisfy | αx| 2+ | αy| 2 = 1, with probability one. (Two joint complex-valued random variables, αx and αy, are really four joint real-valued random variables, viz., the real and imaginary parts of αx and αy.) The Poincar´e sphere representation for the average behavior of this random po-larization vector is, ⎡ ⎤ ⎡ ⎤ ∗ 22Re[�α2Im[�ααy�] αy�] r1 x⎣ r2 ⎦ = ⎣ ⎦ ∗ xr ≡ , 2�|αx� − �|αy| | r3 where—in keeping with the quantum notation for averages—�·� denotes ensemble average. (a) Use the Schwarz inequality to prove that rT r ≡ r 21+ r 22+ r 23≤ 1, i.e., the r vector lies on or inside the unit sphere. (b) Let ia and ib be a pair of deterministic, orthogonal, complex-valued unit vectors, viz., � 1, k = l †ikil = δkl ≡ , k = l0, where k and l can each be either a or b. By means of wave plates, a polarizing beam splitter, and a pair of ideal photon counters, it is possible to measure whether the photon is polarized along ia or along ib. The statistics of this measurement satisfy, Pr(polarized along ia) = 1 + rT a r 2 , (1) Pr(polarized along ib) = 1 + rT b r 2 , (2) 1where ra and rb are the Poincar´e sphere representations of ia and ib, respectively. Show that ra = −rb, so that Eqs. (1) and (2) constitute a proper probability distribution. (c) Suppose that the photon’s random polarization leads to r = 0, i.e., r1 = r2 = r3 = 0. Show that 1 Pr(polarized along ia) = Pr(polarized along ib) = ,2for all pairs of deterministic, orthogonal complex-valued unit vectors {ia, ib}, and thus that r = 0 represents a state of completely random polarization. Contrast the preceding measurement statistics with what will be obtained when ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 r = ⎣ 1 ⎦ ra = ⎣ 0 ⎦ rb = ⎣ 0 ⎦ , 0 1 −1 and when ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 r = ⎣ 1 ⎦ ra = ⎣ 1 ⎦ rb = ⎣ −1 ⎦ , 0 0 0 are the Poincar´e sphere representations of the photon and the pair of orthogonal polarizations being measured. Problem 3.2 Here we introduce the notion of a density operator, i.e., a way to account for classical randomness limiting our knowledge of a quantum system’s state. Consider a quantum mechanical system whose state at time t is not known. Instead, there is a classically-random probability distribution for this state. In particular, suppose that there are M distinct unit-length kets, { |ψm� : 1 ≤ m ≤ M }, and that the system is known to be in one of these states. Moreover the probability that it is in state |ψm� at time t is pm, for 1 ≤ m ≤ M, where { pm : 1 ≤ m ≤ M } is a classical probability distribution: �M pm ≥ 0 and m=1 pm = 1. (a) Suppose that we measure the observable Oˆat time t, where Oˆhas distinct eigenvalues, { on : 1 ≤ n < ∞ }, and a complete orthonormal set of associated eigenkets, { |on� : 1 ≤ n < ∞ }. GIVEN that the state of the system at time t is |ψm�, we know that the Oˆmeasurement will yield outcome on with conditional probability Pr( on | |ψm� ) ≡ |�on|ψm�|2, for 1 ≤ n < ∞ and 1 ≤ m ≤ M. Use this conditional probability distribution to obtain the unconditional probability, Pr(on), of getting the outcome on when we make the Oˆmeasurement at time t. 2� � � � (b) Define a density operator for the system by, Mρˆ ≡ pm|ψm��ψm|. m=1 Show that ˆρ is an Hermitian operator, and verify that your answer to (a) can be reduced to Pr(on) = �on|ρˆ|on�, for 1 ≤ n < ∞. (c) Show that the expected value of the Oˆmeasurement, i.e., ∞�Oˆ� ≡ on Pr(on), n=1 satisfies �Oˆ� = tr(ˆρOˆ), where tr( Aˆ) for any linear Hilbert-space operator, Aˆ, is the trace of that oper-ator, defined as follows. Let { |k� : 1 ≤ k < ∞ } be an arbitrary complete set of orthonormal kets on the quantum system’s state space, so that ∞Iˆ= |k��k|. k=1 Then ∞tr(Aˆ) ≡ �k|Aˆ|k�, k=1 i.e., it is the sum of the operator’s diagonal matrix-elements in the {|k�} rep-resentation. Comment: The trace operation is invariant to the choice of the CON basis used for its calculation. Hence a propitious choice of the basis can be a great aid in simplifying the calculation of averages involving a density operator. Problem 3.3 Here we will explore the difference between a pure state and a mixed state, i.e., the difference between knowing that a quantum system is in a definite state |ψ� as opposed to having a classically-random distribution over a set of such states, namely a density operator ˆρ. Because the density operator is Hermitian, it has eigenvalues and eigenkets. Let us assume that these form a countable set, viz., ˆρ has eigenvalues, { ρn : 1 ≤ n < ∞ }, and associated eigenkets { |ρn� : 1 ≤ n < ∞ }, that satisfy ρˆ|ρn� = ρn|ρn�, for 1 ≤ n < ∞. 3� � � � Without loss of generality, we can assume that these eigenkets form a complete or-thonormal set, i.e., �ρm|ρn� = δnm, ∞Iˆ= |ρn��ρn|. n=1 (a) Show that the eigenvalues {ρn} satisfy 0 ≤ ρn ≤ 1, for 1 ≤ n < ∞, and ∞ρn = 1. n=1 (b) Show that tr(ˆρ) = 1 for any density operator (c) Suppose that the quantum system is in a pure state, i.e., it is known to be in the state |ψ�. Show that this situation can be represented in density-operator form by setting ρ1 = 1 and |ρ1� = |ψ�, viz., a pure state has a de nsity ope rator with only one eigenket whose associated eigenvalue is non-zero. Show that tr(ˆρ2) = 1 for any pure-state density operator. (d) When the density operator has two or more eigenkets with non-zero eigenvalues we s ay that


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MIT 6 453 - Problem Set 3

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