MIT 6 453 - Problem Set 1 (7 pages)

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Problem Set 1



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Problem Set 1

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7
School:
Massachusetts Institute of Technology
Course:
6 453 - Quantum Optical Communication
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MIT OpenCourseWare http 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 1 Fall 2008 Issued Thursday September 4 2008 Due Thursday September 11 2008 Reading For probability review Chapter 3 of J H Shapiro Optical Progagation Detection and Communication For linear algebra review Section 2 1 of M A Nielsen and I L Chuang Quantum Computation and Quantum Information Problem 1 1 Here we shall verify the elementary properties of the 1 D Gaussian probability density function pdf 2 2 e X m 2 px X 2 2 a By converting from rectangular to polar coordinates using X m R cos and Y m R sin show that 2 2 2 2 2 X m 2 2 2 dX e dX dY e X m 2 Y m 2 2 2 thus verifying the normalization constant for the Gaussian pdf b By completing the square in the exponent within the integrand ejvX X m dX 2 2 2 2 2 verify that Mx jv ejvm v 2 2 2 is the characteristic function associated with the Gaussian pdf c Di erentiate Mx jv to verify that E x m di erentiate once more to verify that var x 2 Problem 1 2 Here we shall verify the elementary properties of the Poisson probability mass function pmf mn m Px n e for n 0 1 2 and m 0 n 1 a Use the power series z e zn n 0 n to verify that the Poisson pmf is properly normalized b Use the power series for ez to verify that Mx jv exp m ejv 1 is the characteristic function associated with the Poisson pmf c Di erentiate Mx jv to verify that E x m di erentiate once more to verify that var x m Problem 1 3 Let x be a Rayleigh random variable i e x has pdf X e X 2 2 2 for X 0 2 px X 0 otherwise and let y x2 a Find py Y the pdf of y b Find my and y2 the mean and variance of the random variable y Problem 1 4 Let x and y be statistically independent identically distributed zero mean variance 2 Gaussian random variables i e the joint pdf for x and y is px y X Y e X 2 2 2 Y 2 2 2 2 2 Suppose we regard x y as the Cartesian coordinates of a point in the plane and let r be the polar coordinate representation of this point viz x r cos and y r sin for r 0 and 0 2 a Find pr R the joint pdf of r and b Find the marginal pdfs pr R and p of these random variables and prove that r and are statistically independent random variables 2 Problem 1 5 Let N x be joint random variables Suppose that x is exponentially distributed with mean m i e X m e m for x 0 px X 0 otherwise is the pdf of x Also suppose that given x X N is Poisson distributed with mean value x i e the conditional pmf of N is PN x n x X X n X e n a Use the integral formula dZZ n e Z n for n 0 1 2 for n 0 1 2 0 where 0 1 to nd PN n the unconditional pmf of N b Find MN jv the characteristic function associated with your unconditional pmf from a c Find E N and var N the unconditional mean and variance of N by di er entiating your characteristic function from b Problem 1 6 Let x y be jointly Gaussian random variables with zero means mx my 0 identical variances x2 y2 2 and nonzero correlation coe cient Let w z be two new random variables obtained from x y by the following transformation w x cos y sin z x sin y cos for a deterministic angle satisfying 0 2 a Show that this transformation is a rotation in the plane i e w z are obtained from x y by rotation through angle b Find pw z W Z the joint pdf of w and z c Find a value such that w and z are statistically independent 3 Problem 1 7 Here we shall examine some of the eigenvalue eigenvector properties of an Hermitian matrix Let x be an N D column vector of complex numbers whose nth element is xn let A be an N N matrix of complex numbers whose ijth element is aij and let denote conjugate transpose so that x x 1 x2 xN and A is an N N matrix whose ijth element is a ji a Find the adjoint of A i e the matrix B which satis es By x y Ax for all x y C N where C N is the space of N D vectors with complex valued elements If B A for a particular matrix A we say that A is self adjoint or Hermitian Assume that A is Hermitian for parts b d b Let A have eigenvalues n 1 n N and normalized eigenvectors n 1 n N obeying A n n n n n 1 for 1 n N for 1 n N Show that n is real valued for 1 n N c Show that if n m then n m 0 i e eigenvectors associated with distinct eigenvalues are orthogonal d Suppose there are two linearly independent eigenvectors and which have the same eigenvalue Show that two orthogonal vectors and can be constructed satisfying A A 0 e Because of the results of parts c and d we can assume that n 1 n N is a complete orthornormal CON set of vectors on C N i e 1 for n m n m 0 for n m Let IN be the identity matrix on this space Show that IN N n 1 4 n n Show that A N n n n n 1 Problem 1 8 Here we introduce the notion of overcompleteness Consider 2 D real Euclidean space 2 T R i e the space of 2 D column vectors x where x x1 x2 with x1 and x2 being real numbers De ne three vectors as follows 0 3 2 3 2 x1 x2 x3 1 1 2 1 2 a Make a labeled sketch of these three vectors on an x1 x2 plane and nd xTn xm for 1 n m 3 Are these three vectors normalized unit length Are they orthogonal b Show that any two of x1 x2 x3 form a basis for the space R2 i e any y R2 can be expressed as y ax1 a x2 bx1 b x3 cx2 c x3 for appropriate choices of the real valued coe cients a a b b c c c Show that the 2 2 identity matrix I2 …


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