Some more matrix inequalities related to quantum information theory

3.8.1 Discussion

Consider a square matrix A and consider the block matrix

Y -( 1 A

X ~ A I )

Suppose X > 0. Then, A is a contraction, ie,

A* A < I

To prove this, we choose an arbitrary matrix C of the same size as A and consider the inequality

[x*,x*C*]X ( * ) > 0

This expands to give

|| x ||2 + || Cx ||2 +(< x, A*Cx > + < x, C*Ax >) > 0

This gives

I + C*C + A*C + C*A > 0

Replacing C by — C in this inequality gives

1 + C‘C > A*C + CM

This is true for all C. Now choose

C = 1/2

assuming that A is non-singular. Then,

C*C = I

and we get

I > (A* A)1/2

which proves the claim.

3.8.2 Points to remember

[1] Quantum information theory is about the transmission of classical bits and quantum states over noisy channels and revovering the classical bits/quantum states from the received state with as small error probability as possible. In quantum information theory, we study the maximum rate at which classi-cal/quantum information can be transmitted over the channel after appropriate encoding of the source strings or quantum state with zero limiting error probability of decoding. Classical information of an information source is given by the Shannon entropy while quantum information of a quantum state is given by the Von-Neumann entropy. Classical information theory is a special case of quantum information theory for commuting states and observables. The maximum rate at which information can be transmitted reliably over a quantum channel is the maximum mutual information of the channel over all input source probability distributions while the maximum rate at which classical information can be transmitted reliably over a quantum channel after encoding each classical source alphabet into a quantum state is given by the maximum of the classical-quantum mutual information over all source probability distributions. This is a theorem due to Winter and Holevo. The maximum rate at which quantum information can be transmitted over a quantum noisy channel is as yet an unsolved problem although many conjectures about this have been made.

[2] Matrix inequalities in quantum information theory like the matrix Holder’s inequality enable us to get bounds on the error probability. These inequalities are usually derived starting from Renyi’s entropy and that for the Von-Neumann entropy as a limiting case of the Renyi entropy.

[3] Another place in quantum information theory where inequalities are required is in entangled assisted quantum communication. When two persons share an entangled state, then by transmitting classical bits to each other, they can transmit a quantum state as for example in quantum teleportation. This transmission can be achieved at infinite speed in contradiction with the special theory of relativity that sets the speed of light as the limit at which informa-tion/energy can be transmitted. When complete entanglement is not possible, then errors are involved in the decoding process and the estimation of these error probabilities involves making use if quantum information theoretic inequalities involving things like the Fidelity between two states.

[4] Other kinds of quantum information inequalities are important like the concavity of quantum entropy, joint convexity of quantum relative entropy and the monotonicity of the quantum relative entropy under quantum operations. These inequalities and many more are derived from the fundamental Lieb inequality for operators. These can be found in the book by Rajendra Bhatia, ’’Matrix Analysis”, Springer. Another important set of inequalities in quantum information theory is based on Rayleigh’s variational method for calculating the eigenvalues of a Hermitian matrix or the singular values of any matrix in descending or ascending order.

Fresnel and Fraunhoffer diffraction

3.9.1 Discussion

A wave source from the plane area (x, y) € D having amplitude f(x. y) per unit area produces a radiation field pattern

V>(X,y,Z)= f f(x,y)exp(-jky/X - xp + (Y - y)-’ + Z2)dxdt//(X - xp + (Y - yp + Z-'

D

where k = cü/c. When  is very large as compared to |X|, |Y|, |a?|, |y| as happens in diffraction theory using the Young double-slit experiment, we can make a binomial approximation

- æ)2 + (Y - y)2 + /2 ~ Z(1 + ((X _ x)2 + (y _ 2/)2)/2Z2)

= Z + ((X - x)2 + (Y - y)2)/2Z

and we get the approximation

^(XW,Z)^~1exp(jkZ) [ f{x,y)exp^X - x)2 + (Y - y)2)/2Z)dxdy ■Id

This is the Fresnel diffraction formula. If further, we have that |X|, |Y| >>

|x|, |y|, then we have in addition the approximation

(X - x)2 + (Y - y)2 « X2 + Y2 - 2(Xx + Yy)

and the above diffraction formula approximates to

ÿ(X,Y,Z) « -rexp{jk{Z+(X2+Y2)/2Z)) [ f(x,y)exp(-jk(Xx+Yy)dxdy

■Id

This formula is known as the Fraunhoffer diffraction formula and tells us that

the far field amplitude radiation pattern is the 2 — D spatial Fourier transform of the amplitude field over the source area D.

We wish to generalize this formula for source surfaces of arbitrary shape.

Let D be an open connected set in K3 with boundary surface dD. The wave field V>(r),t = (x, y, z) satisfies Helmholtz equation

(V2 + k2)^(r) = 0,x € D

and on the boundary dD at each point, we specify either i/) or d'ÿ/dh. Let

r') denote the Green’s function corresponding to this boundary condition, ie,

(V2 + k2)Gk(r, r') = ¿3(r - r'), r, r' € D

with Gfc(r, r') = 0 if V’ is specified at r € dD and dGk(r,r')/dn = 0 if dp/dh is specified at r € dD. Then, Green’s theorem gives

[ (Gk(r,r,)X2i!>(r) - ^(r)X2Gk(r,r'))d3r

Jd

= / (Gk(r,r')d4>(r)/dh — 4>(r}dGk(r,r')/dn)dS(r) JdD

This gives us

V’(r/) = [ (—Gk(r,r')d$(r)/dn + il>(r)dGk(r,r')ldh)dS(T),r' € D---(1)

JdD

Now, we make the free space approximation for the Green’s function: r ~ exp(-jk - r'|) 4,|r_r,|

and calculate r' € D in terms of its values or its normal derivative on dD. This will yield the generalized diffraction formula on which we can impose the Fresnel and Fraunhoffer approximations. We have

c>Gfe(r, r'}/dh =

exp(—jk — r')(—jk(n,r — r')/ — r'| — (n,r — r')/i— r'|3)

and substituting this into (1) yields the desired diffraction formula. This situation corresponds to an incident wave falling on the surface dD which then becomes a source and generates a diffracted wave field within D. We can equivalently consider the exterior of dD, ie, D' = R3D as the region where the diffraction pattern is to be computed. The bounding surfaces for this region are dD with its normal being directed inward and the surface of an infinite radius sphere where all the fields vanish. In this case, we can consider the far field and near field approximations and obtain respectively the Fraunhoffer and Fresnel diffraction patterns. In the far field zone, we have |r| >> |r'| and in the near field zone, |r| « |r'|.

Exercise: Compute the Fraunhoffer and Fresnel diffraction pattern approximations stated in the above paragraph.

3.9.2 Points to remember

[1] When a wave field falls on a surface, it reradiates to produce a diffraction pattern in accordance to the spatial Fourier version of retarded potential theory. The near field pattern is called the Fresnel diffraction pattern and is obtained by retaining quadratic terms in the exponential of the Green’s function while the far field pattern is obtained by neglecting the quadratic terms and taking into account only the linear terms in the exponential of the Green’s function. This means that the far field amplitude pattern is the spatial Fourier transform of the amplitude on the screen.

[2] In the case of em waves falling on a surface, we can determine the Fresnel and Fraunhoffer patterns by applying Green’s theorem to the surface or equivalently, by calculating the equivalent surface electric and magnetic current densities on the surface induced by the incident em wave and then applying the retarded potential formula to compute the reradiated (ie, diffracted) field pattern in terms of the induced surface current densities.

 
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