Quantum teleportation Masatsugu Sei Suzuki Physics...

1

Quantum teleportation Masatsugu Sei Suzuki

Physics Department, SUNY at Binghamton (Date: November 10, 2015)

Quantum teleportation is a process by which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly in principle) from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for superluminal transport or communication. And because it disrupts the quantum system at the sending location, it cannot be used to violate the no-cloning theorem by producing two copies of the system. Quantum teleportation is unrelated to the kind of teleportation commonly used in fiction, as it does not transport the system itself, does not function instantaneously, and does not concern rearranging particles to copy the form of an object. Thus, despite the provocative name, it is best thought of as a kind of communication, rather than a kind of transportation. The seminal paper first expounding the idea was published by C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters in 1993. Since then, quantum teleportation has been realized in various physical systems. Presently, the record distance for quantum teleportation is 143 km (89 mi) with photons, and 21m with material systems. On September 11th, 2013, the "Furusawa group at the University of Tokyo has succeeded in demonstrating complete quantum teleportation of photonic quantum bits by a hybrid technique for the first time worldwide." ((Optica vol.2, p.832)) H. Takesue et al. (October, 2015))

“Quantum teleportation is an essential quantum operation by which we can transfer an unknown quantum state to a remote location with the help of quantum entanglement and classical communication. Since the first experimental demonstrations using photonic qubits and continuous variables, the distance of photonic quantum teleportation over free-space channels has continued to increase and has reached >100 km. On the other hand, quantum teleportation over optical fiber has been challenging, mainly because the multifold photon detection that inevitably accompanies quantum teleportation experiments has been very inefficient due to the relatively low detection efficiencies of typical telecom-band single-photon detectors. Here, we report on quantum teleportation over optical fiber using four high-detection-efficiency superconducting nanowire single-photon detectors (SNSPDs). These SNSPDs make it possible to perform highly efficient multifold photon measurements, allowing us to confirm that the quantum states of input photons were successfully teleported over 100 km of fiber with an average fidelity of 83.7 ± 2.0%.”

2

Fig. 1. Experimental setup. (a) Setup for generating time-bin entangled photon pairs. ATT,

attenuator; EDFA, erbium-doped fiber amplifier; PPLN, periodically poled lithium niobate waveguide; SHG, second-harmonic generation; SPDC, spontaneous parametric downconversion. (b) Quantum teleportation setup. Yellow and gray solid lines indicate the optical fibers and electrical lines, respectively. SNSPD, superconducting nanowire single-photon detector; MZI, unbalanced Mach–Zehnder interferometer; DSF, dispersion-shifted fiber; TIA, time interval analyzer.

_____________________________________________________________________________

1. Q

Fig. S

pqu

[J. Audre2002).] p ________ We consi

The state

Before A

Quantum tel

chematic diarticle of anuantum state

etsch (Editorp.155 Fig.6.3

__________

ider the four

[2

1)(12

[2

1)(12

e of the parti

11 a

Alice makes a

leportation

iagram of qn entangled e from partic

r) Entangled 3

___________

r Bell-states

21zz

21zz

cle 1, the pa

1 b

a measureme

(I)

quantum telepair from

cle 1 can be

World (Wil

__________

given by

21zz

21zz

article Alice w

ent, the state

3

eportation. Wan EPR soutransferred (

ley-VCH Ve

___________

]

]

wishes to tel

e of the three

When Aliceurce (for Ei(teleported)

erlag GmbH

__________

leport, is sim

e particles is

e and Bob einstein-Podoto particle 3

& Co, KGa

___________

mply

s given by

each receiveolsky-Rosen).

aA, Weinheim

__________

e one ), the

m,

_____

4

)(2

1

)(2

1

][2

1)(

)(

321321

321321

323211

)(2311123

zzzzzzb

zzzzzza

zzzzzbza

zbza

where

][2

13232

)(23 zzzz .

Alice makes a special type of measurement called a Bell-state measurement. (a) The state of Bell basis:

0

1

1

0

2

10,1][

2

12121

)(12 zzzz

0

1

1

0

2

10,0][

2

12121

)(12 zzzz

where 0,10,1 mj , 0,00,0 mj .

(b) The additional basis (it is still called Bell basis)

1

0

0

1

2

1][

2

12121

)(12 zzzz

5

1

0

0

1

2

1][

2

12121

)(12 zzzz

Then we have

)(2

1

)(2

1

)(12

)(1221

)(12

)(1221

zz

zz

1,1)(2

1

1,1)(2

1

)(12

)(1221

)(12

)(1221

zz

zz

where 1,11,1 mj , 1,11,1 mj

Using the above Bell basis, the state 123

)(2

1

)(2

1

321321

321321123

zzzzzzb

zzzzzza

can be expressed as follows.

6

)(2

1)(

2

1

)(2

1))(

2

1

))(2

1)((

2

1

))(2

1)((

2

1

))(2

1)((

2

1

))(2

1)((

2

1

33

)(1233

)(12

33

)(1233

)(12

3

)(12

)(123

)(12

)(12

3

)(12

)(123

)(12

)(12

3

)(12

)(123

)(12

)(12

3

)(12

)(123

)(12

)(12123

zbzazbza

zbzazbza

zbzb

zaza

zbzb

zaza

If Alice's Bell-state measurement on particles 1 and 2 collapses the two particle state to the state

)(12 , for example,

))(33

)(12 zbza

then the particle 3, Bob's particle, is forced to be in the state

33333 ( zbzazbza

which is exactly the state, up to an overall phase, of the particle before the measurement. This is a dramatic illustration of the spooky action at a distance of entangled states that so troubled Einstein.

Fig. A

co

B

((Towns

Show

Bob can particle b ((Solutio

If Ali

state (12

the partic

which is We consi

Alice measur

(3

bza

onfused with

Bob instantly

end, Examp

w that if Alic

put his partiby 180° abou

on)) ice's Bell-sta

)2 , for exam

(3

)(12 za

cle 3, Bob's p

3 zb

exactly the s

ider the rota

res with the

)3zb , wh

h the classic

y get a copy s

ple 5.3))

ce's Bell-state

icle into the ut the y axis.

ate measurem

mple,

)33

zb

particle, is fo

33zaz

state, up to a

tion operato

e Bell’s sta

hich is the

cal informati

state 3 .

e measurem

state particle.

ment on part

orced to be i

an overall ph

r around the

7

ate )(12 .

same as

ion line. Aft

ent yield par

e 1 was in in

ticles 1 and 2

in the state

hase, of the p

e x axis.

After Bob

1 . The en

fter Alice me

rticles 1 and

nitially by ro

2 collapses th

particle befo

measures, w

ntangled lin

easures with

d 2 in the stat

otating the sp

the two parti

ore the measu

we get the

ne should no

h the Bell’s

te )(12 , th

pin state of th

cle state to t

urement.

state

ot be

state,

hen

his

the

8

2cos

2sin

2sin

2cos

ˆ2

sin12

cos)(ˆ

yx iR

and

01

10

2cos

2sin

2sin

2cos

)(ˆ

xR

Then we have

)(01

10)(ˆ

333 zbzab

a

a

bRx

______________________________________________________________________________ ((Example-1)) Rotation operator (I) The operator

)ˆexp()(ˆ xx Si

R

rotates spin states by an angle counterclockwise about the x axis. (a) Show that this rotation operator can be expressed in the form

2sinˆ2

2cos1)(ˆ xx S

iR

.

(b) Use matrix mechanics to determine which of the results )(12 and )(

12 of Alice's

Bell-state measurement yields a state for Bob's particle that is rotated into the state by the

rotation operator )(ˆ xR .

((Solution))

9

)([2

1][

2

1

][2

1])[

2

1

33

)(1233

)(12

33

)(1233

)(12123

zbzazbza

zbzazbza

with

][2

12121

)(12 zzzz

][2

12121

)(12 zzzz

______________________________________________________________________________ (a)

2sinˆ1

2cos)(ˆ xx iR

(b)

0

0ˆ

2sinˆ1

2cos)(ˆ

i

iiiR xxx

If Alice's Bell state measurement results in her tangling particles 1 and 2 in the state )(12

,

][33

)(12 zbza

then Bob's particle is in the state

33zazb .

Using similar procedures, we have the following results

33zbza for )(

12

33zbza for )(

12

b

Thus usin

R

for )(12

Fig. A

b

Note that

R

for )(12

________((Examp

3azb

3azb

ng matrix m

zbRz )[(ˆ3

, which is t

Alice measur

3azb

(3

bzai

t

zaRz )[(ˆ

, which is n

__________ple-2))

3z for

3z for

echanics, we

za ]33

the state

res with the

3z . Thro

)3zb , wh

zbz ]33

not the same

___________Rotation o

r )(12

r )(12

e have

i

i

0

0

up to an ov

e Bell’s sta

ough the

hich is the sa

i

i

0

0]

e as .

__________operator (II

10

b

ai

a

b

verall phase.

ate )(12 .

unitary op

ame as 1 .

a

ib

a

___________I)

After Bob

perator (ˆzR

a

b

__________

measures, w

)( , we

___________

we get the

get the

__________

state

state

_____

Deter

particle t

5.14, you

the state

((Solutio

R

or

R

Then we

R

for )(12

2. A

rmine which

that is rotate

u can simply

by a 18

on))

2

cos)(ˆ zR

iRz 0

)(ˆ

have

aRz )[(ˆ3

Approach fr

h of the resu

ed in the sta

y verify that

0° rotation a

2

sinˆ12

zi

i

0

b ]

33

om the redu

ults of Alic

ate by th

t the remain

about the z a

2

0

02

i

e

e

b

a

i

i

0

0

uced density

11

e's Bell-stat

he operator R

ning state of

axis.

2

0

i.

b

ai

a

y operator

te measurem

)(ˆ zR . If yo

f the Bob's p

ment yields

ou have wor

particle is in

a state for B

rked out Pro

ndeed rotated

Bob's

oblem

d into

12

We consider the pure particle state 123 which is related to the quantum teleportation. The

density operator for this pure state is given by

123123ˆ

where

)([2

1][

2

1

][2

1][

2

1

)(

33

)(1233

)(12

33

)(1233

)(12

)(2311123

zbzazbza

zbzazbza

zbza

and

]3,2,3,2,[2

1)(23 zzzz

Note that there are four Bell’s states for particles 1 and 2, which are defined by

0

1

1

0

2

1][

2

12121

)(12 zzzz

1

0

0

1

2

1][

2

12121

)(12 zzzz .

Note that

122 ba . (normalization).

The density operator can be obtained as

13

00000000

02/2/002/)(2/)(0

02/2/002/)(2/)(0

00000000

00000000

02/)(2/)(002/2/0

02/)(2/)(002/2/0

00000000

ˆ

22**

22**

**22

**22

bbbaba

bbbaba

ababaa

ababaa

Tracing out particle 1, the reduced density operators are obtained as

0000

0110

0110

0000

2

1

0000

00

00

0000

2

1

0000

00

00

0000

2

1

0000

00

00

0000

2

1ˆ

2222

2222

22

22

22

22

23

baba

baba

bb

bb

aa

aa

This reduced operator is the same as that of the density operator for the Bell’s state. Note that for the Bell's two-particle entangled state,

0

1

1

0

2

1]3;2;3;2;[

2

1)(23 zzzz

we have the density operator given by

14

0000

0110

0110

0000

2

1

0110

0

1

1

0

2

1

ˆ )(23

)(2323

.

Tracing over particle 2 furthermore, we have

10

01

2

1

10

00

2

1

00

01

2

1ˆ3

which is equivalent to a completely un-polarized state. So Bob (particle 3) has no information about the state of the particle Alice is attempting to teleport. On the other hand, if Bob waits until he receives the result of Alice’s Bell state measurement, Bob can then maneuver his particle into

the state that Alice’s particle was in initially.

((Mathematica))

15

3. Approach from the quantum qubits

Clear "Global` " ;

exp : exp . Complex re , im Complex re, im ;

11

2

011

0

; 21

2

0110

;

11

2

1001

;

21

2

1001

;

1ab

; 2a

b; 3

ba

; 4b

a;

12312

KroneckerProduct 1, 112


12



Simplify;

K1 Transpose 123 1 ;

K2 Transpose 123 . a a1, b b1 ;

Outer Times, K1, K2 1 FullSimplify;

MatrixForm

0 0 0 0 0 0 0 0

0 a a12

a a12

0 0 a b12

a b12

0

0 a a12

a a12

0 0 a b12

a b12

0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 a1 b2

a1 b2

0 0 b b12

b b12

0

0 a1 b2

a1 b2

0 0 b b12

b b12

0

0 0 0 0 0 0 0 0

16

Suppose Alice and Bob share a pair of qubits in the entangled state

BABAB 1100(2

100

Alice needs to communicate to Bob one qubit of information

10

Fig. Quantum teleportation scheme and corresponding circuit.

(P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information, Springer-Verlag, 2007). p.228.

The initial state of the system of three qubits is given by

17

]1100(11100(0[2

1

1100(2

1)10(

00)0(

3

BABABABA

BABA

B

The first two qubits are at the Alice’s location and the last bit is at the Bob’s location. Alice applies the CNOT transformation to her two qubits, with the control qubit being the quibit to be teleported to Bob.

)]1001(1)1100(0[2

1

]101011110000[2

1)1(3

BABABABA

BABABABA

where

AACNOTU 0000ˆ

AA

CNOTU 1010ˆ

AA

CNOTU 1101ˆ

AA

CNOTU 0111ˆ

She then applies the Hadamard transformation to the first qubit.

)10(2

10ˆ H , )10(

2

11ˆ H

Then we get

18

)01(11)10(10

)01(01)10(00[2

1

]110011100001[2

1

]111010101000[2

1

)]1001)(10()]1100)(10[([2

1

)]1001)(10()1100)(10([2

1)1(3

BBABBA

BBABBA

BABABABA

BABABABA

BABABABA

BABABABA

Finally, Alice measures the two qubits in her possession. The measurement outcome. For the

measurement of A00 Alice, the state of Bob's qubit is equivalent to the original state

BB 101

So Bob does not change, which is indicated by the identity operator I ,

BBI 10ˆ1

For the measurement of A01 Alice, the state of Bob's qubit is given by

BB 012

If Bob applies the X transformation to his qubit, the state becomes

12 01

10ˆ

X


BB 103

19

If Bob applies the Z transformation to his qubit, the state becomes

13 10

01ˆ

Z


)01(4BB

If Bob applies the XZ ˆˆ transformation to his qubit, the state becomes

14 01

10ˆˆ

XZ

Here note that we use the operators I , X , Z , and XZ ˆˆ , where

10

01I ,

01

10X ,

10

01Z ,

01

10

01

10

10

01ˆˆXZ .

_____________________________________________________________________________ REFERENCES John S. Townsend , A Modern Approach to Quantum Mechanics, second edition (University

Science Books, 2012). J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, second edition (Addison-Wesley,

New York, 2011). David H.McIntyre Quantum Mechanics A Paradigms Approach (Pearson Education, Inc.,

2012). S. Weinberg; Lectures on Quantum Mechanics (Cambridge University Press, 2013). J. Audretsch (Editor) Entangled World (Wiley-VCH Verlag GmbH & Co, KGaA, Weinheim,

2002). A.D. Aczel, Entanglement the great mystery in physics (Four Walls Eight Windows,

New York,2001). S.M. Barnett Quantum Information (Oxford University Press, 2009).

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G. Greenstein and A.G. Zajonc, The Quantum Challenge (Jones and Bartlett Publishers, Boston, 1997).

APPENDIX ((Mathematica)) Bell’s states

Clear"Global`";

exp_ :

exp . Complexre_, im_ Complexre, im;

1 10;

2 01;

B1

1

2KroneckerProduct1, 1

KroneckerProduct2, 2 MatrixForm

12

0012

B2

1



21

______________________________________________________________________________

012

12

0

B3

1



12

00

12

B4

1



012

12

0

Quantum teleportation Masatsugu Sei Suzuki Physics...

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