Homework #2 (due 1/27/09)

2.1. Cal= culate the number of photons.

(a) Calculate the number of photons hitting an area of 1cm^{2 per sec at a distance of 1m from a 60 W light bulb. Assume that the=
light
is emitted at the wavelength of 500nm.}

(b) Answer the same question for the number of photons per cm^{2<=
/sup>
per second from the sunlight on the surface of Earth, assuming the surface =
is
normal to the direction of the sunlight and that the wavelength is also 500=
nm.}

(c) For a linearly polarized plane wave, if the electric field is one atomic unit, what is average power (per cycle) in units of Watts/cm^{2 <=
/sup>?}

2.2. Identifying the Hilbert space.

(a) = For a particle in an infinite 2D square well potential, i.e., V(x,y)=3D0 for = and V(x,y) is infinite otherwise, define the Hilb= ert space, the base state vectors and examples of observables.

(b) For two particles in an infinite 1D square well potential, i.e., V(x)=3D0 for = and infinite otherwise, define the = Hilbert space if the two particles are distinguishable. If the two particles are identical Fermions, what is the Hilbert space? ( We have not talked about identic= al particles yet but you have heard them before).

2. 3. Measurement of polarized light.

(a) For a linearly pol= arized light with the polarization axis at 45^o from the x-axis on the xy-plane, if you use a right-circularly polarized analyzer, what is the fra= ction of the intensity that can be transmitted? What is the polarization of the l= ight after the measurement?

(b) Following the first measurement, you use anot= her linear polarizer ( 45^o with respect to the x-axis) to analyze, w= hat will be the fraction of the intensity that will be transmitted with respect= to the original linearly polarized light?

Study Material:

Adopted from: http://en.wikipe= dia.org/wiki/Quantum_cryptography

Quantum key exchange—use quantum measu= rement concept; use entanglement means

Will use photon polarization-- BB84 protocol - Charles H. Bennett= and Gilles Brassard (1984)

The sender (traditionally referred to as Alice) and the receiver (Bob) are connected by a quantum communication channel w= hich allows quantum states to be transmitted. In the case = of photons this channel is generally either an optical fibre or simply free = space. In addition they communicate via a public classical channel, for example us= ing broadcast radio or the internet. Neither of these channels need to be secur= e; the protocol is designed with the assumption that an eav= esdropper (referred to as Eve) can interfere in any way with both.

The security of the protocol comes from encodi= ng the information in non-orthogonal states. Quantum indeterminacy means that these = states cannot in general be measured without disturbing the original state (see No cloning theorem). BB84 uses two pairs o= f states, with each pair conjugate to the other pair, and the two = states within a pair orthogonal to each other. Pairs of orthogonal states are refe= rred to as a basis. The usual polarization state pa= irs used are either the rectilinear basis of vertical (0°) and horizontal (90°), the diagonal basis of 45° and 135° or= the circular basis of left- and right-hande= dness. Any two of these bases are conjugate to each other, and so any two can be u= sed in the protocol. Below the rectilinear and diagonal bases are used.

Basis	0	1

The first step in BB84 is quantum transmission. Alice creates a random bit (0 or 1) and then randomly selects one of her two bases (rectilinear or dia= gonal in this case) to transmit it in. She then prepares a photon polarization st= ate depending both on the bit value and basis, as shown in the table to the lef= t. So for example a 0 is encoded in the rectilinear basis (+) as a vertical polarization state, and a 1 is encoded in the diagonal basis (x) as a 135&d= eg; state. Alice then transmits a single photon in the state specified to Bob, using the quantum channel. This process is then repeated from the random bit stage, with Alice recording the state, basis and time of each photon sent.<= o:p>

Quantum mechanics (particularly quantum indeterminacy) says there is no possible measurement that will distinguish between the 4 different polariza= tion states, as they are not all orthogonal. The only measurement possible is between any two orthogonal states (a basis), so for example measuring in the rectilinear basis will give a result of horizontal or vertical. If the phot= on was created as horizontal or vertical (as a rectilinear eigenst= ate) then this will measure the correct state, but if it was created as 45° = or 135° (diagonal eigenstates) then the rectilinear measurement will inste= ad return either horizontal or vertical at random. Furthermore, after this measurement the photon will be polarized in the state it was measured in (horizontal or vertical), with all information about its initial polarizati= on lost.

As Bob does not know the basis the photons were encoded in, all he can do is select a basis at random to measure in, either rectilinear or diagonal. He does this for each photon he receives, recording the time, measurement basis used and measurement result. After Bob has meas= ured all the photons, he communicates with Alice over the public classical chann= el. Alice broadcasts the basis each photon was sent in, and Bob the basis each = was measured in. They both discard photon measurements (bits) where Bob used a different basis, which will be half on average, leaving half the bits as a shared key.

Alice's random bit<= /span>	0	1	1	0	1	0	0	1
Alice's random sending basis
Photon polarization Alice send= s
Bob's random measuring basis
Photon polarization Bob measur= es
PUBLIC DISCUSSION OF BASIS
Shared secret key	0		1			0		1

To check for the presence of eavesdropping Alice and Bob now compare a certain subset of their remaining= bit strings. If a third party (usually referred to as Eve, for 'eavesdropper') = has gained any information about the photons polarization, this will have introduced errors in Bobs' measurements. If more than p bits differ = they abort the key and try again, possibly with a different quantum channel, as = the security of the key cannot be guaranteed. p is chosen so that if the number of bits known to Eve is less than this, privacy amplification can be used to reduce Eve's knowledge of the key to an arbitrarily small amount, by reducing the length of the key.

The simplest type of possible attack is the intercept-resend attack, where Eve measures the quantum states (photons) se= nt by Alice and then sends replacement states to Bob, prepared in the state she measures. The probabili= ty Eve chooses the incorrect basis is 50% (assuming Alice chooses her basis random= ly), and if Bob measures this intercepted photon in the basis Alice sent he will= get a random result, i.e. an incorrect result with probability of 50%. The prob= ability an intercepted photon generates an error in the key string is then 50% x 50= % =3D 25%. If Alice and Bob publicly compare n of their key bits (thus discarding them as key bits, as they are no longer secret) the probability = they find disagreement and identify the presence of Eve is

3D"P_d =

So to detect an eavesdropper with probability P_d =3D 0.999999999 Alice and Bob need to comp= are n =3D 72 key bits.