possible to construct higher-dimensional states by taking direct products
of lower-dimensional states. However not all higher-dimensional states can be
constructed this way. There will always exist states that cannot be expressed as
a direct product. Such states are called entangled states. This nomenclature
is due to Erwin Schr¨odinger who first discovered the implication of such states
in 1935 [61].
For example, consider two generic qubits
|ψ
1
i = α
1
|0i + β
1
|1i, |ψ
2
i = α
2
|0i + β
2
|1i. (4.6)
If you form the direct product, you get
|ψ
1
i ⊗ |ψ
2
i =
“
α
1
β
1
#
⊗
“
α
2
β
2
#
=
α
1
α
2
α
1
β
2
β
1
α
2
β
1
β
2
. (4.7)
This is called a product state. Now the most general 2-qubit state is a
superposition of the form
|φi
2
= c
0
|0i + c
1
|1i + c
2
|2i + c
3
|3i. (4.8)
Equation 4.7 is of a special form:
c
0
c
3
= c
1
c
2
. (4.9)
Not all states satisfy this property. Those states which do NOT are called
entangled states. Equation 4.9 is the criterion for a 2-qubit state to be a
product state.
For example, the state
1
√
2
(|00i + |11i) is entangled while
1
√
2
(|00i + |01i)
is not. A state like |00i + |10i + |11i is partially entangled.
Box 4.1: Bell States
The classic examples of entangled states are the Bell states, so named
in honor of John Bell [5] whose famous arguments resolved the Einstein–
Podolsky–Rosen paradox [31] involving entangled states. They are also re-
ferred to as EPR states for this reason. These states exhibit maximum corre-
lation or anticorrelation between their components:
|β
00
i =
1
√
2
(|00i + |11i) ; (4.10a)
|β
01
i =
1
√
2
(|00i − |11i) ; (4.10b)
|β
10
i =
1
√
2
(|01i + |10i) ; (4.10c)
Properties of Qubits 69
|β
11
i =
1
√
2
(|01i − |10i) (4.10d)
In the states |β
00
i and |β
01
i, the spin values of each component are always
the same (correlated), while they are always opposite (anticorrelated) for the
other two states.
Verify that these states are mutually orthogonal. They can thus be used
as a basis for the 2-qubit Hilbert space H
2
.
When you have more than two qubits, you can have entanglement between
all or some of the component qubits. In a 3-qubit system, for example, you
could have entanglement between all three:
|ψ
3
i =
1
√
3
(|010i + |101i) , (4.11)
which is one of the so-called GHZ states (after Greenberger, Horne and
Zeilinger [39]). Note for this particular state that each of the component qubits
are anticorrelated, with the first and third having the opposite anticorrelation
as the second.
You could have entanglement between two qubits alone, for example:
|ψ
12
i =
1
√
3
(|000i + |110i) (4.12)
One can imagine more possible combinations of partial entanglement. Thus
for larger dimensional systems, entanglement becomes more complicated.
Entangled states are just some among the possible states of higher di-
mensional quantum systems. Why do we single them out for a special name
and status? What does it mean for a state to be entangled? We have already
pointed out that entangled states have properties that make them correlated
to each other. When two (or more) systems are in an entangled state, each
component system does not have a definite state. This is what it means to
say that the superposition cannot be written as a product of states of the
component systems.
Let us examine the meaning of correlations in the context of a two-qubit
system in the entangled spin state
|ψi = |β
00
i =
1
√
2
(|00i + |11i.
Assume we have a beam of atom pairs in this state, and that we separate
each pair carefully without changing the state and send one atom each to
Alice and Bob, who proceed to measure the S
z
value on their atom. Each
has equal probability of having a value ±1/2. Suppose Alice measures a value
+1/2 on her atom. This means its state has collapsed to |0i. But this is
70 Introduction to Quantum Physics and Information Processing
possible only if the combined state collapses to |00i, so that Bob’s atom also
collapses to |0i. This happens even without Bob making a measurement on
his atom. If Bob now measures S
z
, he will get a value +1/2. Similarly, had
Alice obtained −1/2, Bob would also measure the same value. There is perfect
correlation between the spins of the two particles. Alice and Bob can verify
this by making measurements on a large number of qubit pairs in the same
state and comparing the values. As another example, if the state were the
so-called singlet state
|β
11
i =
1
√
2
(|01i − |10i,
and the same experiment is performed, then there is perfect anticorrelation
between the spins of the two qubits.
In contrast, suppose that the spins were in the state
|ψi
2
=
1
√
2
(|00i + |10i).
It’s easy to see that this state can be expressed as
|ψi
2
=
1
√
2
(|0i + |1i)|0i,
decomposed into a product of states of each spin. In this un-entangled state,
each spin does possess a definite state. The superposition in the state of the
first spin is merely a basis state in another basis: the S
x
basis. Here there is
no correlation between spin measurements made by Alice and those obtained
by Bob
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