T1: 3. Entanglement and Applications

Question 1

Define Bell states

Answer 1

The set of four maximally entangled two-qubit states which form an orthonormal basis.

Question 2

Give the general form of a Bell state

Answer 2

B_xy = 1/√2 (|0 y⟩ + (-1)^x|1 y ̅ ⟩)

Question 3

Why can bipartite systems not create Bell states from separable states?

Answer 3

The required local operation would only transform a separable state to a separable state. Similarly, a measurement by Alice and/or Bob would also result
in a separable state

Question 4

Describe super dense coding between A&B

Answer 4

If A&B share a Bell state and A acts an LO onit to become B_xy this does not affects B’s state. A sends the state and B projects on to a sum of |B_xy⟩⟨B_xy|. This returns the number (xy).

Question 5

In SD coding, what happens if A’s qubit is intercepted?

Answer 5

If E intercepts A’s qubit then E only has a density matrix independent of xy. E also has no access to B’s part of the system.

Question 6

State the no-cloning theorem

Answer 6

It is impossible to create a copy of an unknown state.

Question 7

Describe encryption for transmitting some number n.

Answer 7

Send F(n) with the receiver knowing the inverse function. F^-1 should be sufficiently difficult to find that by the time it is done, the information is redundant.

Question 8

How does RSA create a secure key?

Answer 8

It multiplies two high-digit primes: m=p*q. It is easy to compute m, but very difficult to determine p and q given m.

Question 9

Describe how a OTP encodes and decodes an n-bit number.

Answer 9

Choose a number k, which is the same length as n. Add the two numbers modulo 2 and send this message. Decrypt by adding the key modulo 2 to this message.

Question 10

Briefly describe the process of QKD

Answer 10

Alice chooses a basis from X or Z knowing and assigns a number to each basis state. She sends her qubit to Bob who randomly measures S_x or S_z. Alice and Bob shares their choices (X,Z basis or S_x,S_z measurement). If they agree they add the number to their key.

Question 11

Describe how Eve can intercept QKD

Answer 11

She can choose either S_z or S_x to always measure. If she matches with Alice, she will not affect the state and will know the outcome is part of the key.

If she does not match Alice she will change the state Bob receives. If Bob and Alice agree, Bob will correctly measure 50% of the time.

Question 12

How much of the correct QKD can Eve determine

Answer 12

Alice = Eve: 50%. This does not change the state so 100% of kept numbers, Eve will know.

Alice != Eve:50% Eve changes the state. Of kept numbers 50% will be correct. 50%*50%=25%

Total: 75% code

Question 13

How can Alice and Bob find out if Eve has intercepted their QKD?

Answer 13

Share a portion of the key and compare where theirs differ. This is where changes have been made to the state.

Question 14

What does ‘local’ mean in local realism?

Answer 14

Nothing can travel faster than the speed of light.

Question 15

What does ‘realism’ mean in local realism?

Answer 15

Measurements are deterministic; they properly describe the state of the system.

Question 16

What did Einstein believe local realism implied about Quantum Mechanics?

Answer 16

That it is not a complete theory: there must be some hidden variables which make it deterministic.

Question 17

Briefly describe the EPR experiment

Answer 17

Charlie stands perfectly between Alice and Bob and sends them the first and second qubit in a Bell state respectively. Alice randomly measures Q or R and Bob S or T. After a large number of results, the come together and calc expectation value. Compare to deterministic case.

Question 18

Why must Charlie be directly Between Alice and Bob in EPQ

Answer 18

To rule out any measurement affecting the other and having the information travel at speed v < c.

Question 19

What are the physical Q,R,S,T that Alice and Bob measure?

Answer 19

Q = σ_1⊗ I
R = σ_3⊗ I
S = I ⊗ -1/sqrt(2)(σ_1+ σ_3)
T = I ⊗ -1/sqrt(2)(σ_1- σ_3)

Question 20

What is the local realism response to violation of bell inequalities?

Answer 20

That QM is super-deterministic: the randomness in each state is precisely determined.

Question 21

Why can Bob and Alice simultaneously measure their observables?

Answer 21

Because they commute.

Question 22

Why does entanglement preserve speed of light communication.

Answer 22

Even if Alice makes a measurement which affects Bob’s state, Bob will not know; his reduced density matrix describes the same state unless info is CC to him.

Question 23

Define superdense coding

Answer 23

Using one qubit to transmit two bits of information

Question 24

Briefly give the procedure for teleportation

Answer 24

Alice has an unknown state and the first qubit in a Bell pair w/Bob. Alice entangles here qubits w/CNOT and then acts with a Hadamard on the first.

Alice has |x⟩⌦|y⟩ and Bob has Uxy |phi⟩

If Alice measures xy and CC to Bob, bob can act Udag on his state to recover |phi⟩