量子通信 期中考试

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Tele 9757 Midterm 2013 Do NOT separate the grid from the rest of the paper This entire paper must be handed in at end of exam 1. Which of the following is not a postulate of quantum mechanics :

a. All composite systems can be reduced to separable states *** b. Physical symmetries act on Hilbert spaces via unitary transformation c. Observables are represented by self-adjoint operators acting on states d. The Hilbert space of composite systems is derived via the tensor product e. The expected value of an operator A is given by

sAs

acos2. A quantum state is described . What statement is always true if . 2bsina. a1,b1/b. a1/c.

2 2,b1/2

a21,banything

d. aanything,banything e. bb1 ***

*cos3. The polarization of a photon can be described. What is true about the angle

sina. That  is always 90 or 0

o

o

b. c. d.

 is always 45o

 can never be measured ***  can always be measured

e. none of above

4. If the states

m1andm2represent two measurement outcomes for a photon, what statement

m1and m2

represents the closest match to the conditions that must be imposed upon

a. Their inner product must always equal ½ b. They only need to be normalized

c. Their inner product must be unconstrained d. They must form an orthonormal basis *** e. They only need to be orthogonal

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s15. Lets, which of the following conditions on s1 and s2 represents right-handed circular

s2polarization?

a. b. c. d.

s2/s1 must be real s1/s21 s10 s2/s1i ***

e. None of the above

6. In teleporting particle C from Alice (where Particle A is) to Bob (where particle B is), what occurs?

a. Particle entanglement initially set between A and C is destroyed and B and A become

entangled

b. Particle entanglement initially set between A and B is destroyed and C and A become

disentangled

c. Particle entanglement initially set between A and B is destroyed and C and A become

entangled ***

d. Particle entanglement initially set between A and B and C is destroyed and C and A

become entangled e. None of above

7. In teleporting particle C from Alice (where Particle A is) to Bob (where particle B is), the

probability of measuring a particular Bell state of particles A and C is?

a. 0% b. 25% *** c. 50% d. 75%

e. It is unknown

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8. In teleporting particle C from Alice (where Particle A is) to Bob (where particle B is), the final step

is

a. Bob applies one of four unitary transformations to particle B, where the transformation

depends on quantum information sent to Bob from Alice

b. Alice applies one of four unitary transformations to particle A, where the transformation

depends on classical information sent to Alice from Bob

c. Bob applies one of four unitary transformations to particle B, where the transformation

depends on classical information sent to Bob from Alice

***

d. Bob applies one of two unitary transformations to particle B, where the transformation

depends on classical information sent to Bob from Alice e. None of the above

9. In teleporting particle C from Alice (where Particle A is) to Bob (where particle B is), after Alice

has made her measurement the state of her two particles will be

a. In one of the Bell states of the composite system A and B

b. In a linear superposition of all the Bell states of composite system A and C c. In one the Bell states of the composite system C and B

d. In a linear superposition of all Bell the states of the composite system A and B e. None of the above ***

10. Consider the final step of teleportation. If the state that was teleported is initially given by

a01, and after classical communication from Alice Bob knows his photon is in the

state

b01, what must he do to complete teleportation?

11b 10a. Bob must invoke the transformation b. Bob must invoke the transformation 01b ***

10c. Bob must invoke the transformation 11a 1011d. Bob must invoke the transformation b

11e. Bob must do nothing

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11. One of the statements below if true would violate the ‘No cloning’ theorem – which one?

a. b. c. d. e.

ss sss ss ss

sss

***

12. Which statement most closely states the most general form of the ‘No Cloning’ theorem

a. No quantum state of the form cos45can be copied

sin451b. No quantum state of the form can be copied

0c. No photon quantum state can be copied d. No arbitrary quantum state can be copied*** e. No Bell state can be copied

13. A quantum transformation is represented by a matrix U. What conditions are imposed on the

matrix U?

a. Column vectors must have inner product=1, and normalized b. Column vectors must have inner product=1 and matrix trace=1 c. Column vectors must be orthogonal and matrix trace=1

d. Column vectors must be normalized and all elements of matrix must be real e. None of above***

14. In Quantum Key Distribution (QKD) which statement is false?

a. We use quantum communication to develop a one time pad or a one time ‘key’ that can

be used for encrypting messages b. We do not apply quantum communications to the message itself – just to the generation

of the key c. An important component of QKD is the ‘No cloning’ Theorem d. Entanglement can be used to deliver QKD

e. QKD depends on the all channels being a quantum channel ***

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15. In the BB84 protocol, bit errors detected by Alice and Bob indicate that for sure

a. The key has been fullyintercepted by an adversary b. The key has been partiallyintercepted by an adversary c. The key has been fully hidden from an adversary

d. The XoR'ed key has been fully hidden from an adversary e. None of the above ***

16. In the BB84 protocol, what is Eve's probability of forwarding a photon to Bob without him detecting

that it has been measured by her

a. 0 b. 0.25*** c. 0.5 d. 0.75 e. 1

17. The Renyi entropy

a. Quantifies the information not known about a key b. Quantifies the capacity of a quantum channel c. Quantifies the capacity of a classical channel d. Quantifies the information not known about a bit *** e. None of above

18. If p0 = probability of a 0 and p1 = probability of a 1, the Renyi Entropy is at its maximum when

a. p0=1/2, p1=1 b. p0=0, p1=1/2 c. p0=1/2, p1=1/2 *** d. p0=1/2, p1=1/4 e. none of above

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19. Which of the following is not an allowed quantum state?

a.

s11 2i11 2ib.

sc.

11s ***

2i1d. s

0e.

s1i 2i20. Which of the following is an allowed transformation on a state

10a. 

01/2b. 10

02c.

111 21011 1011i*** i12d.

e.

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21. If a photon is in state soutcome of

130ei1 what is the probability of finding a measurement 221 ifmeasurement matrix is given by M0,1?

a.

3 2b. 2/3 c. 1/4 d.

2

e. none of above***

22. If a photon is in the state s1301 and a possible measurement outcome is223101, what is the probability of finding that measurement outcome? 22a. 1 b. 1/2 c. 1/4 d. 1/8 e. 0***

23. Which unitary transform maps the 1111 state to the state s? 212ia.

11U

1i10U ***

0ib.

11c. U

1id.

10U

0i10U

118

e.

24. A transformation H performs the following transformation

110122 .

11101220Which of the following represents the unitary operator,UH, that implements H ?

a. UH1012010121

b.

UHUHUH1012010120 0

c.

1011101221012d.

01011 2e. None of the above***

25. Which of the following states represents a product state?

a.

sss10110 210011 210011 2b.

c.

d.

ss10010 *** 210110 2e.

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26. Which of the following states represents an entangled state?

1a. si02

100b. s112

011c.

s1020 *** 11d. s102

10e. None of the above

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27. Let

100,1and consider the mapping

01110122 .

11101220Which of the following matrices implements the above mapping?

a.

111 *** 211111 210101 102111 102b.

c.

d.

e.

111 1i2eicossin28. Let 1,2i. The tensor product of these two states is

ecossineicossina. 2sinb. c. d.

ei2cos2 2coscoscos22eicossinsinsin22eicossin***

TTeicossineicossine2icossin

e2icossin

Teicos2sin2e. None of above

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eicossin24

29. Given the basis 1,2i in C, a basis in C is

ecossina. b. c. d.

11,12,21,221,11,21,211,12,11,12 ***  

T11eicossineicos2sin2e2icossin

e. None of above

011030. Consider the two Pauli matrices x,z,xz what?

1001a. b. c. d.

xx zz zx

zI2

e. None of above ***

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