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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%.”
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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
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The state
Before A
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chematic diarticle of anuantum state
etsch (Editorp.155 Fig.6.3
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ider the four
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1)(12
[2
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e of the parti
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Alice makes a
leportation
iagram of qn entangled e from partic
r) Entangled 3
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r Bell-states
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cle 1, the pa
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a measureme
(I)
quantum telepair from
cle 1 can be
World (Wil
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given by
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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
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(3
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cle 3, Bob's p
3 zb
exactly the s
ider the rota
res with the
)3zb , wh
h the classic
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ple 5.3))
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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
KroneckerProduct 2, 2
12
KroneckerProduct 1, 312
KroneckerProduct 2, 4
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
For the measurement of A10 Alice, the state of Bob's qubit is given by
BB 103
19
If Bob applies the Z transformation to his qubit, the state becomes
13 10
01ˆ
Z
For the measurement of A11 Alice, the state of Bob's qubit is given by
)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).
20
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
2KroneckerProduct1, 2
KroneckerProduct2, 1 MatrixForm
21
______________________________________________________________________________
012
12
0
B3
1
2KroneckerProduct1, 1
KroneckerProduct2, 2 MatrixForm
12
00
12
B4
1
2KroneckerProduct1, 2
KroneckerProduct2, 1 MatrixForm
012
12
0