wk7/8/9ish

Question 1

What is super dense coding

Answer 1

Encoding two classical bits into one part of a pair of entangled qubits. This technically means two bits of information were encoded into one quantum state

Question 2

what is the protocol for super dense coding

Answer 2

-Alice and Bob starts with bell state: 1/root(2) ( |00> + |11> ) where Alice has the first |0> + |1>
-Call two bits to encode x and y.
-If y = 1 then we run an X gate on Alices qubits (flips the bits)
-then if x = 1 we run a Z gate on Alices qubits (flips the state from positive to negative)
-Bob then receives Alice’s qubits and performs a cnot. Alice’s qubits are control and his pair are the target.
-Finally Bob runs a hadamard on the qubits received from Alice
-Final result gives 4 distinct states on measurement, one corresponding to each of the possible encodings

Question 3

What is quantum teleportation

Answer 3

The process of teleporting a quantum state. The state start off in arbitrary state |𝛙> and is teleported via an entangled Bell state to another location. Both the original state that is sent and the Bell state are destroyed in the process

Question 4

Explain the quantum teleportation protocol

Answer 4

-Alice and Bob share bell state |B_0> and Alice starts with arbitrary quantum state |𝛙>

-Alice couples the quantum state |𝛙> with her half of the Bell state by performing a controlled not operation with |𝛙> as the control and her half of the Bell as the target

-Alice then applies a Hadamard gate the state |𝛙>

• Then, a measurement is performed by Alice on both |𝛙> and her half of Bell state. By algebraic manipulation, we can group Alice’s different measurement outcomes with the corresponding outcomes on Bob’s side of the Bell state
• If we want to send Bell state |𝛙> = a|0> + b|1> and Alice receives the measurement corresponding to this state, we need do nothing else, Bob has the state

-Since measurement are inherently random though, we may need to tell Bob to make adjustments to his side of the bell state via a classical channel e.g.

If we measure |Y> = a|1> + b|0>, we must tell Bob to apply an x gate to flip the qubits.

Question 5

Why does quantum teleportation not imply faster than light communication

Answer 5

We must still send over a classical message to inform Bob whether he must amend his quantum state as measurement outcomes are random

Question 6

Does Alice and Bob know the state |𝛙> and what happens to Alice’s original state |𝛙> and her half of the Bell state

Answer 6

• Neither Alice or Bob have to know the state |𝛙>
  -Alice’s half of Bell state is collapsed after measurement as is bob’s
  -Alices original state |𝛙> is also destroyed

Question 7

Describe the basic BB84 quantum key distribution protocol

Answer 7

1) Alice sends random qubit to Bob, this has a 50/50 chance of being a random qubit in x basis or z basis (4 possibilities with equal probability)

2) Bob recivies the qubits and measures them in random basis

3) Alice announces which qubits were measured in which basis. Both discard the qubits which were not measured in the same basis

4) If measuring in the wrong basis (e.g. evesdropper):
- 50/50 chance of measuring in same basis
-if same basis, no change to qubit
-if different basis, random outcome which is has 50% chance of giving the correct outcome
-so if there is an eaves dropper 25% chance the qubit state changes

5) Check the remaining qubits and if about 25% are different to what they should be there is likely an eavesdropper

6) if there are discrepancies but are smaller than 25% then it is possible that they are just due to channel noise

Question 8

What is the equation to find proabbaility, p, that number of qubits, n, were not tampered with at a certainty, c,

Answer 8

( 1 - ( p / 4 ) ) ^ n <= 1 - C

Question 9

What is the information reconciliation part of the BB84 protocol

Answer 9

-Some error may be due to error correction
- to solve this. Alice and Bob partition their block sizes to size k
-announce parity of each block. If they agree, do nothing
-if parity difference, use binary search to locate the error and fix it.
–this means announcing each half of the block, and the parity of the halves of that block etc.
-everytime a block’s parity is announced, Alice and Bob must drop the last bit of the block to stop information leakage

Question 10

How is block size k chosen in information reconciliation

Answer 10

chosen such that the probability, p, of blocks having a different parity is 1/3

Question 11

what are disadvantages to using parity to check for errors

Answer 11

if a block has two bit flips 1-> 0 and 0->1 then the parity doesn’t change despite and error

Question 12

name methods you need to remember for BB84 to improve security and account for errors

Answer 12

• Information reconciliation (detail)
• Privacy amplification (just know it exists)