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greatly disliked what he called the “spooky action at a distance”
16
effect needed
to explain these surprising quantum spin correlations.
Because entanglement is entirely nonclassical, it may not be surprising that
a quantum computer acting on entangled states can lead to results beyond the
power of a classical computer. We can easily see how such entangled states can
arise in quantum computation. Consider the action of a quantum CNOT gate
on a two-qubit state (see box on Quantum Entanglement). When the two-qubit
states are just simple products of single-particle 1 and 0 states, we obtain the
exact analog of the classical result. But if one of the qubits is in a superposi-
tion state of 1 and 0, acting on this state with a quantum CNOT gate yields an
entangled two-qubit state just like the example of the pion decaying into an
electron and positron. It is this nonclassical feature of quantum mechanics that
gives quantum computers their extraordinary properties.
Quantum entanglement
For a pion decaying at rest to a positron-electron pair (e
+
e
-
), the positron and the electron move away
from each other in opposite directions as shown in Fig. 15.16 (a). Since the pion has zero spin, the net spin
of the positron-electron pair must also be zero because of conservation of angular momentum. However,
the spin state of either the positron or the electron is not definitely known and the spin state of the pair is
said to be entangled. The entangled wave function for the pair is shown in
Fig. 15.16
(b). If we measure the
positron spin to be
↑
e
then we know immediately that the spin of the electron must be
↓
e
and vice versa.
Since the positron and the electron are moving apart, this quantum correlation of spin measurements can
be over long distances.
Quantum computation often involves entangled states. These can
arise from the action of a quantum CNOT gate on a two qubit state. The
equations a, b, and c below show the action of a CNOT gate on three dif-
ferent two qubit states. With qubit input 1 on the upper control line of
the gate, a qubit input 0 to the bottom line is flipped to a 1 as shown in
(a). With qubit input 0 to the upper control line, a qubit input 0 to the
bottom line is left unchanged as in (b). However, if the qubitinput on
the control line is a superposition (1 + 0) the action of the CNOT gate on
a 0 qubit input to the bottom line produces the entangled state shown
in (c).
π
0
e
+
e
-
Fig. 15.16. Addition - Pion decay:
π
ο
→
+−
ee
a) Pion decaying at rest to a positron-
electron pair (e
+
e
-
).
(
)
Ψ
ee
↑↓ −↓ ↑
For cases (a) and (b) the action is straightforward and each particle
is in a definite spin state before and after the CNOT gate. Acting with a
CNOT gate on a two qubit superposition input state on the upper control
line produces an entangled state in which neither particle is in a definite
spin state as shown in (c).
+−
+
−
+
−
e
e
e
e
b) Entangled spin state of the positron-
electron pair resulting from the pion
decay.
1
1
1
2
(b) O
1
O
2
CNO
(a) 1
1
O
2
CNO
O
1
O
2
(c) (1
1
+ O
1
) O
2
CNO
(1
1
1
2
+ O
1
O
2
)
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