 # Levy process models for jerk noise in robotic systems

The robot differential equations have the general form

p

dX(t) = F(t, X(t), O)dt + Gdt X(t), 0)dNk(t)

k=i

where №(.), k = 1,2, ...,p are independent Poisson processes with rates A*,, k =

1,2, ...,p respectively. Our measurement model is

dZ(t) = h(t, X(t))dt + crvdV(t)

where V(.) is standard vector valued Brownian motion. Let Zt = {Z(s) : s < t}.

We wish to construct the EKF for estimating X(t).O dynamically. Thus,

d(9(i) = 0

and our extended state vector is

£(t) = [X(tf ,O(tff

This is a Markov process and its generator is given by

E[d^(X(t),0(t))|X(i) = a:,0(t) = 0]

= dt[-p,x(t, Of F(t, x, 0) + 52 + Gk(t, x, 0), 9) — tl>(x, 0)) = dt.Ktil>(x, 0)

k

K is the generator of the process ¿(t) and its kernel is given by

Kt(x,0x',0') = F(t,x,0T)(V6(x-x’))6(0-0')+£, Al!(S(x-x'+Gk(t,x,0))-5(x-x'))S(0-0') k

The Kushner-Kallianpur filter for (x, 0) with

^(0) = E[^(X(t), 0(t))IZt]

is given by

^t(d’) = nt(Kt)dt + <7y2(7ri(hid>) - 7rf(/ii)7rt())(iZZ(t) - 7rt(hf)dt)

We then make the EKF approximations. First observe that

Kt^ = Kt[xT,0T]T = [F(t,x,0f,0T]T + ^Ak[Gk(t,x,0)T,0T]T k

= [F(t, X, of + 52 ^kGk(t, X, of, 0T]T

k

So,

7rf (Â'(e) « [F(t, X(t), 0(i))r + 52 XkGk(t, X(i), 0(i))r, 0T]T

fc

Also,

- 7Tt(^)7rf(/lt) «

= PdWW)T

This gives the approximate EKF equation for the conditional mean:

• 4(t) = [F(t,£(< + E XkGk(t,^t))T,0T]Tdt k
• - ht&t»dt)

For the error covariance

P^t) = Cov(SÇ(t)t),6Ç(t) = e(i) - ê(i)

d(^(t)^(0T) = (^(t)).^(i)T + 6^t).dô^t)T

+dst(t).dswr

dô^t) = d£(t) - d£(t) =

F(t,Ç(t))dt+Gk(t,Ç(t))dNk(t)—F(t,Ç(t))dt-^2XkGk(t,Ç(t))dt-av2Pç(t)HT(dZ—hd£(t))dt) k

« (F(t,£(i)) - F(t,t(t))dt + £ GkMt)dMk(t) k

-aÿ2P^t)H^(HtS^t) + avdV(t))

« F'(t,^t))6^t)dt + E Gk(t,WdMk(t)

-aÿ2P^Hl(HtS^t) + avdV(t))

where

Af,(i) = Nk(t) ~ ^kt

is a discontinuous Martingale. It follows (using dV.dVT = dtl, dMkdMj = SkjdNk,dV.dNk = 0) that

dP^t)/dt «

r(t,4(t))pç(i) + pç(i)F'(i>ê(i))T - HtP^t)

+ E ^kGk(t,£(ty)Gk(t, £(t))T

k

This forms the EKF for Poisson noise driven state with white Gaussian measurement noise. Note the notation used:

Ht = h't(£(t)),F(t,£) = [F(t,£)T,0]T,Gk(t,£) = [Gk(t,£)T,0]T

# Digital systems, classical and quantum gates, design of counters using shift registers and flip-flops

For a long time, a lot of digital systems based on implementing Boolean functions and counters has been taught to electronics students in undergraduate courses. Now the time has come to explain that digital gates are generally irreversible and moreover, only a limited amount of classical information can be transmitted over a classical channel ie a channel having a i/o transition probability matrix The Boolean function that represents the output of a JK flip-flop giving Qn+1 = f(Qn, Jn, Kn) = JnKnQn + JnKnQn + JnKn or that of a D or T flip flop in the form Qn+i = f(Qm,Dn) = Dn,Qn+1 = f(Qn,Tn) = QnTn + QnTn should be perhaps introduced via its truth table. The excitation table for the JK flip-flop should then be introduced giving possible values of (Jn,Kn) as Boolean functions of Qn,Qn+i and then how such an excitation table may be used to design a counter in a shift register. How the figure eight consisting of seven linear bulbs can be used to represent all numbers from zero to nine and which of these lights should be switched on to get a decimal number in this range from a four digit binary number using the theory of Boolean functions should be introduced. After introducing all this, the microprocessor should be taught explaining how to write programs using its standard commands like MOV. MV I, PUSH, JUMP for some simple operations like solving simple difference equations, adding numbers from one to N, calculating the DFT of a sequence etc. The functions of the Data bus and address bus should be taught by simple commands like given a binary data loaded on the data bus through an input port and given a specified register location defined by a binary data string on the address bus, how to transfer the data from the data bus to the register should be explained. Simple binary operations like the half-adder, full-adder, multiplication with floating points should be introduced via Boolean functions and how these functions are built into the microprocessor should be taught.

Having taught all these, the irreversibility of gates such as not and, nand, or, exor etc. which have just one binary output and two binary inputs should be explained. In this context, the notion of a 2 x 2 unitary quantum gate acting on single qubit state and its reversibility should be explained. The notion of an r qubit gate as a unitary 2r x 2r matrix acting on an r-qubit quantum gate may be touched upon. Specific examples like the NOT gate which is a two qubit quantum gate, one qubit being a controlling qubit and the other qubit being the controlled qubit may be dealt with. The SWAP gate, phase gate, Hadamard gates, the Fredkin gates and how these gates may be used to build the quantum Fourier transform gate of size 2r x 2r acting on r qubits using just O(2r) operations in contrast to O(r.2r) operations required by a classical FFT should be explained. How to use entanglement as a resource for transmitting a single qubit quantum state using just transmission of two classical bits should be explained (teleportation). More generally how to use entanglement to transmit r-qubits by transmitting just 2r classical bits should be explained. The notion of a qubit consisting of a 2 x 1 complex vector having unit norm expressible as a superposition of a |0 > state and a |1 > state may be explained. How a qubit carries far more information than a classical bit may be touched upon. The interpretation of an r-qubit quantum state in terms of quantum probabilities must be explained thoroughly. In this context, the notion of mixed quantum states and how the state collapses after a measurement is made in the quantum theory unlike the classical theory should be explained.

Some applications of quantum computation like phase finding, order finding, factorizing a number etc. are very important. These ideas revolve around applying a control unitary operator to a state, taking a measurement folllwing this application and obtaining the required result with a high probability on making a measurement of some component of the state.

Other applications revolve around quantum commmunication which is essentially of two kinds, one Cq channel communication in which a classical source alphabet with some probability distribution is encoded into a quantum state. The Cq-channel consists of the transformation x —> p(x) of each source alphabet into a quantum state on the same Hilbert space and then to recover the sent alphabet with high probability by making a measurement on the received state. Just as in Shannon’s classical information theory, the error probabilities will be large, however, if we consider a string u = (xiX2...xn) of source alphabets, and encode this string into the tensor product state p(u) = p(xk) then if n is sufficiently large, we can form Mn such strings with Mn greater than 2n(maxpI(p,p)-6) tliat tlie error probability is arbitrarily small and 6 is also arbitrarily small. Here

Z(p,p) = S(^p(a;)p(a;)) - p(x)S(p(x))

X X

is the mutual information between the source signal and the received state, {/>(»■)} is the source probability distribution and S is the Von-Neumann entropy function of a state. Moreover, the converse of this coding theorem also holds, ie, if we choose any set of Mn string of source alphabets such that log(Mn)/n > maxplfj), p) + <5 for some S > 0, then no matter what decision operators we choose, the error probability will converge to unity as n —> oo. There are many ways to prove this fundamental result based on actual construction of the decision POVM operators and random coding arguments. These latest developments which generalize Shannon’s coding theorem are due to Holevo and Winter. The notion of detection of the source string with small error error probability means that for any e, 6 > 0 and all sufficiently large n, we can construct M = Mn sequences u>n, ..■■>un,Mn of source alphabets, each of length n, positive operators Dnl,DnM, M = Mn such that Dnk < I and Tr(p(wrafc)Z)„fc) > 1 — e,k = 1,2, such that log(Mn)/n > C — ô where C = maxpl(p, p) and further C is the largest such number ie if log(Mn)/n > C + 6 for all sufficiently large n, then minkTr(j)(unk) Dnk) —> 0. C is called the Cq capacity of the channel. My feeling is that it is more advantageous to prove the quantum noiseless Schumacher compression theorem and the quantum Cq noisy Shannon coding theorem and then show the classical theorems of Shannon are special cases of this.