Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
come, applying some appropriate quantum gates to iment with two people, Alice and Bob. In the ex-
his qubit Bob will recover the state ψ . periment Alice observes X or X while Bob meas-
1 2
| i
Afew importantremarks aboutquantum telepor- uresX orX .ConsiderthequantityX X +X X +
3 4 1 3 2 3
tation are in the line. First, quantum teleportation X X X X . It is equal to
2 4 1 4
−
does not involve any transfer of matter or energy.
(X +X )X +(X X )X = 2 2.
Alice’s particle has not been physically moved to 1 2 3 2 − 1 4 ± ≤
Bob; only its state has been transferred. Second, af- Regardless of the distributions of X , taking expec-
i
ter the teleportation Bob’s qubitwill beon the tele- tationonbothsidesoftheaboveinequalitywearrive
ported state, while Alice’s qubit will become some at the famous Bell inequality,
undefinedpartofanentangledstate.Inotherwords,
E(X X )+E(X X )+E(X X )
what the teleportation does is that a qubit was de- 1 3 2 3 2 4
(9)
stroyed inoneplace butinstantaneously resurrected E(X X ) 2.
1 4
in another. Teleportation does not copy any qubits, − ≤
TheviolationofBell’sinequalitydemonstratesen-
and hence is consistent with the no-cloning theo-
tanglement effect in quantum mechanics. In fact,
rem(whichforbidsthecreation ofidenticalcopies of
quantum experiments yield a quantum version of
an arbitrary unknown quantum state; see Wootters
the inequality. Consider that a quantum system of
and Zurek (1982)). Third, in order to teleportate
two qubits is prepared in a Bell state
a qubit, Alice has to inform Bob of her measure-
ment by sending him two classical bits of informa- 01 10
ψ = | i−| i.
tion. These two classical bits do not carry complete
| i √2
information about the qubit being teleported. If the
Alice takes the first qubit of ψ while Bob gets its
two bits are intercepted by an eavesdropper, he or
| i
second qubit. Define four observables with eigenval-
she may know exactly what Bob needs to do in or-
ues 1,
der to recover the desired state. However, this infor-
±
mation isuseless iftheeavesdroppercannotinteract X =σ , X =σ ,
1 z 2 x
withtheentangledparticleinBob’spossession.Also
on the first qubit and
the requirement of sending two bits of information
viaclassicalchannelpreventsquantumteleportation σ +σ σ σ
X = z x , X = z − x ,
from transmittinginformation faster than thespeed 3 − √2 4 √2
of light.
onthesecond qubit,whereσ and σ arePauli ma-
x z
3.3.3 Bell’s inequality The Bell test experiments trices given by (7). Again Alice performs measure-
are designed to investigate the validity of the en- ments on X or X while Bob measures X or X .
1 2 3 4
tanglement effect in quantum mechanics through The quantum expectations of X X , X X , X X ,
1 3 2 3 2 4
Bell’s inequality. Over the past four decades many X X in the state ψ are calculated below:
1 4
| i
physicalexperimentsonquantumsystemswerecon-
1 1
ducted to check the validity of Bell’s inequality and E (X X )= , E (X X )= ,
ψ 1 3 ψ 2 3
√2 √2
resulted in some violation of the inequality. For ex-
ample, Aspect, Grangier and Roger (1981, 1982a, E (X X )= 1 , E (X X )= 1 .
ψ 2 4 ψ 1 4
1982b) provided overwhelming support to the vi- √2 −√2
olation of Bell’s inequality. The experimental re-
Here theobservableproductis in the senseof tensor
sultsareoften invoked astheproofofquantumnon-
product. Thus we obtain a value in the quantum
locality andlack of realism that noparticle has defi-
framework for the analog quantity on the left-hand
niteformuntilitismeasuredandmeasuringaquan-
side of the Bell’s inequality (9)
tumentity caninstantaneouslyinfluenceanotherfar
away. SeeAspect,Grangier andRoger (1981,1982a, E ψ(X 1X 3)+E ψ(X 2X 3)+E ψ(X 2X 4)
1982b), Bohm (1951), Bell (1964), Clauser et al.
E (X X )=2√2,
(1969) and Einstein, Podolsky and Rosen (1935). − ψ 1 4
Below we describe the CHSH version of the Bell’s which exceeds 2 and hence violates the Bell’s in-
inequality (Clauser et al. (1969)). equality. In fact, the quantum version of the Bell’s
11
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
inequality is the Tsirelson’s inequality (Tsirelson, ponents, each of which may be thought of as a sin-
1980) which shows that in any quantum state ρ, gle argument to function f(x). Because of quantum
E (X X )+E (X X )+E (X X ) nature, a single circuit U f applied once to the su-
ρ 1 3 ρ 2 3 ρ 2 4
(10) perposition state is actually performed on each of
E (X X ) 2√2. the components of the superposition, and the whole
ρ 1 4
− ≤ range of the values of function f(x) is stored in the
3.4 Quantum Parallelism