Post on 06-Jan-2016
description
QUBIT VERSUS BIT
23 August 2004, Cambridge
Lev Vaidman
quant-ph/0406024
Zion MitraniAmir Kalev
Phys. Rev. Lett. 92, 217902 (2004),
QUBIT BIT
TO READ TO READ
QUBIT BIT
TO WRITE TO WRITE
NO!
N N N
N/2 NN
N/2
DENSE CODING
N NKNOWN QUBITS
Teleportation
UNKNOWN QUBIT
2 BITS
Teleportation
UNKNOWN QUBIT
2 BITS
2
We cannot store and retrieve more than one bit in a qubit HOLEVO
What can we do with a qubit that we cannot do with a bit?
We cannot store and retrieve more than one bit in a qubit HOLEVO
What can we do with a qubit that we cannot do with a bit?
Tasks with 2 possible outcomes
1
32
4i
eveni
N
i
1
... N
oddi
N
i
1
?or
5
We know integeri
N
i
1
We cannot store and retrieve more than one bit in a qubit HOLEVO
What can we do with a qubit that we cannot do with a bit?
Tasks with 2 possible outcomes
1
32
4i
eveni
N
i
1
... N
oddi
N
i
1
?or
5
We know integeri
N
i
1
We cannot store and retrieve more than one bit in a qubit HOLEVO
What can we do with a qubit that we cannot do with a bit?
Tasks with 2 possible outcomes
1
32
4i
eveni
N
i
1
... N
oddi
N
i
1
?or
5
We know integeri
N
i
1
Measurement of the parity of the integral of a classical field Galvao and Hardy,Phys. Rev. Lett. 90, 087902 (2003)
1
32
4i ... N
5
evenN
i i
N
i i 11
11
1 2
B
A
N
i i dxx)(1
1
N
i iinteger
ii
Measurement of the integral of a classical field
integerdxxIB
A )(
A
B
Measurement of the integral of a classical field
integerdxxIB
A )(
Binary representation of I
mod2)mod2()mod2(... 2mod2
2mod22
2
II IIII
)mod2(2mod2mod2 2mod22 IIII
2
3
dd 1
22 dd
223 dd
2mod2
2I
2
2
2
mod23
I
.
. .
.
.
. …1 0 1
A
B
2
3
dd 1
22 dd
223 dd
2mod2
2I
2
2
2
mod23
I
.
. .
.
.
.
0 0 0 0 0 … 1 0 1
n
12 n
dnd
2
1-n
2
mod2In
What can we do with bits passing one at a time?
A
B
( )xB
AdxxI )(
What can we do with bits passing one at a time?
A
B
or( )x
We can “write” a real number in a bit as the probability of its flip
dxxdp )(]flip[
B
AdxxI )(
dxxedxxdp )()(1]flipno[
Idxxdxx eeedpp )()(]flipno[
]flipno[ln1 pI 11 I
NeI
Uncertainty in measurement with bits
11 I
NeI
MI Optimization for M
2.1 N
MI 3.1
B
Adxx)(
The number of bits for finding MI is 2~ M
Uncertainty in measurement with bits
11 I
NeI
MI Optimization for M
2.1 N
MI 3.1
B
Adxx)(
The number of bits for finding MI is 2~ M
The number of qubits for finding MI is M2log~
Quantum method yields precise result for integer I if MN 2log
Measurement of the integral of a classical field
integernondxxIB
A )(
Measurement of the integral of a classical field
integernondxxIB
A )(
dd A
B
I N
1
N qubits
N
II
Measurement of the integral of a classical field
integernondxxIB
A )(
dd A
B
N
1
N entangled qubits
N
II
Peres and Scudo PRL 86 4160 (2001)
But the digital method works much better!
mdxxIB
A)(
ddnn 12
2
~2cos)
~( nn
np
2
~
21
.
. .
.
.
.
A
B
Inn 12
n~
2
~
2
~
32
21
N
1nn1-n2mI
~
~~
N
n
IInIIp
12
)~
(2cos)|~(
N
M
2
10
Quantum uncertainty
N
M
2
10~
Classical uncertainty N
M~
)mmIII ~(~
Information about I in N qubits is in N ,..., 21
Can we use a single particle in a superposition of N different states instead?
2
~
21
.
. .
.
.
.
Inn 12
n~
2
~
2
~
32
21
.
.
.
Information about I in N qubits is in N ,..., 21
Can we use a single particle in a superposition of N different states instead?
2
~
21
.
. .
.
.
.
Inn 12
n~
2
~
2
~
32
21
.
.
.
No. Hilbert space is too small : states2N states2ofinstead N
Measurement of the integral of a classical field with a single particle in a superposition of states
NK 2
)(x
)(int xK
kH
Measurement of the integral of a classical field with a single particle in a superposition of states
NK 2
)(x
)(int xK
kH
Measurement of the integral of a classical field with a single particle in a superposition of states
NK 2
mI ~~ Measurement yields
The probability of the error )( Ip III~
N qubits Single particle NK 2
Binary representation of k
k
N qubits Single particle
1101001
for0
for22
j
j
j K
I
K
kI
K
IN
j
jN
j j
22211
N
jN
1in )(
2
1
N
jj
j
K
kIi
Nkje
N
1
2
1
2
fin ),(2
1
K
kI2
Binary representation of k
k
N qubits Single particle
1101001
for0
for22
j
j
j K
I
K
kI
K
IN
j
jN
j j
22211
N
jN
1in )(
2
1
N
jj
j
K
kIi
Nkje
N
1
2
1
2
fin ),(2
1
K
kI2
Interaction
N qubits Single particle NK 2 states
Measurement of the integral of a classical field with N bits running together
A
Bdxxdp )(]count[
countsNI
N
MI
N
II
2
10
counts
Quantum methods
How to read a string of length out of strings
using a single particle?
NK 2 1N
How to read a string of length out of strings
using a single particle?
NK 2 1N
10
How to read a string of length out of strings
using a single particle?
NK 2 1N
10
Bits instead N2logWe need at least
What else can we do with the quantum phase ?