QUBIT VERSUS BIT

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?



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




1

32

4i

eveni

N

i

1

... N

oddi

N

i

1

?or

5

We know integeri

N

i

1




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


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


integernondxxIB

A )(


integernondxxIB

A )(

dd A

B

I N

1

N qubits

N

II


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


NK 2

)(x

)(int xK

kH


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



NK 2 1N

10



NK 2 1N

10

Bits instead N2logWe need at least

What else can we do with the quantum phase ?

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QUBIT VERSUS BIT