Homework on Quantum Cryptography and Quantum Computing (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words. Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof. A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should justify all of your answers for full credit.

Quantum Cryptography

Alice and Bob are using a Mach-Zehnder interferometer to transmit the key to the code they will be using, as described in your handout from Schroedinger's Machine by Milburn. If neither adjusts a mirror, or if the sum of mirror shifts is 180^o, the light will undergo complete constructive interference at Bob's lower detector and complete destructive interference at Bob's upper detector. If the sum of Bob's mirror shift and Alice's mirror shift is 45^o, 225^o, or 135^o, the light will be equally split between detectors. If the sum is 90^o, the light will all reach Bob's upper detector. If Eve has intercepted the signal, the interference pattern will be different.

Alice randomly chooses from her four possible mirror positions, with position A = no shift, B = 45^o, C = 90^o, and D = 135^o. She ends up with the following sequence of positions:

Bob randomly chooses from his two mirror positions, with position G = 45^o , and H = 90^o.

1. Bob and Alice must discard any signal that results in the light being evenly split between detectors and can only use bits for which all the light hits the lower detector (which they call a 1) or all the light hits the upper detector (which they call a 0). Which possible combination(s) (such as AG, or CH) of Alice and Bob's mirror positions will result in a 1? Which will result in a 0?
AH results in a total shift of 90^o, so it is a 0 (light reaching upper detector only). BG results in a total shift of 90^o, so it is a 0 (light reaching upper detector only). CH results in a total shift of 180^o, so it is a 1 (light reaching lower detector only). DG results in a total shift of 180^o, so it is a 1 (light reaching lower detector only).

2. Upon receiving the transmission, Bob calls Alice and tells her, "In the first string of ten, I got a signal on bit 3, when I used position H, on bit 4, when I used position G, and bit 7, when I used position H. The next string, I got a signal on bits 11 (G), 15 (H), 17 (H), and 18 (G). The third string gave a signal on bits 21 (H), 22 (G), 23 (H), 25 (G), 26 (H), and 28 (G). In the final string, I saw a signal on bits 36 (H), 38 (H), and 39 (G). Based on what Alice now knows, determine the key Bob and Alice have developed. To what decimal number does this correspond?
I have added two rows to the table with Alice's mirror positions. The second row is the information Bob sends her. Combined with our results from the previous question, we arrive at the third row for bits: 0100010000110010, or 17458 in decimal.

3. You have figured out how Alice knows what the key is. Bob, of course, knows which detector the light went into, so he knows the key. Explain why Eve cannot determine the key by overhearing the conversation between Bob and Alice.
The conversation contains no mention of which bit Bob might have detected, nor does it tell Eve what Alice's settings were. Bob knows the code because he measured it; Alice can deduce the code based on her settings and what Bob tells her. Eve cannot do any better than flip a coin for each bit Alice and Bob discuss.

4. What might be some warning signs to Alice and Bob indicating that Eve has intercepted the light signal upon which the key is based?
By intercepting the signal between Alice and Bob, Eve will destroy any interference effects that have built up to that point. This will change the interference pattern Bob detects. In the unlikely event that Eve's effect is only to swap 1s and 0s, Alice and Bob can check the "parity" of certain sequences of data. They share whether the number of 1s in a given sequence is odd or even. The more sequences they use to compare, the more confident they are that the data is uncorrupted and unintercepted.

Quantum Computing

You have read or heard about several quantum phenomena critical to quantum computing: quantum decoherence (measuring/copying changes state), quantum teleportation, quantum entanglement, and quantum Fourier sorting. Pick one of them and describe the phenomenon and how it relates to quantum cryptography and/or computation.

1.	Bob and Alice must discard any signal that results in the light being evenly split between detectors and can only use bits for which all the light hits the lower detector (which they call a 1) or all the light hits the upper detector (which they call a 0). Which possible combination(s) (such as AG, or CH) of Alice and Bob's mirror positions will result in a 1? Which will result in a 0? AH results in a total shift of 90^o, so it is a 0 (light reaching upper detector only). BG results in a total shift of 90^o, so it is a 0 (light reaching upper detector only). CH results in a total shift of 180^o, so it is a 1 (light reaching lower detector only). DG results in a total shift of 180^o, so it is a 1 (light reaching lower detector only).
2.	Upon receiving the transmission, Bob calls Alice and tells her, "In the first string of ten, I got a signal on bit 3, when I used position H, on bit 4, when I used position G, and bit 7, when I used position H. The next string, I got a signal on bits 11 (G), 15 (H), 17 (H), and 18 (G). The third string gave a signal on bits 21 (H), 22 (G), 23 (H), 25 (G), 26 (H), and 28 (G). In the final string, I saw a signal on bits 36 (H), 38 (H), and 39 (G). Based on what Alice now knows, determine the key Bob and Alice have developed. To what decimal number does this correspond? I have added two rows to the table with Alice's mirror positions. The second row is the information Bob sends her. Combined with our results from the previous question, we arrive at the third row for bits: 0100010000110010, or 17458 in decimal.
3.	You have figured out how Alice knows what the key is. Bob, of course, knows which detector the light went into, so he knows the key. Explain why Eve cannot determine the key by overhearing the conversation between Bob and Alice. The conversation contains no mention of which bit Bob might have detected, nor does it tell Eve what Alice's settings were. Bob knows the code because he measured it; Alice can deduce the code based on her settings and what Bob tells her. Eve cannot do any better than flip a coin for each bit Alice and Bob discuss.
4.	What might be some warning signs to Alice and Bob indicating that Eve has intercepted the light signal upon which the key is based? By intercepting the signal between Alice and Bob, Eve will destroy any interference effects that have built up to that point. This will change the interference pattern Bob detects. In the unlikely event that Eve's effect is only to swap 1s and 0s, Alice and Bob can check the "parity" of certain sequences of data. They share whether the number of 1s in a given sequence is odd or even. The more sequences they use to compare, the more confident they are that the data is uncorrupted and unintercepted.