From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 13

As a final remark, which will be useful in the following, we notice that, using
the coordinate representation:

a=

√^1

2

(ˆx+ipˆ),

a†=

1

√

2

(ˆx−ipˆ). (1.36)

As we will see in the next subsection, these operators are necessary to describe
the quantum 1D harmonic oscillator.

Example 1.2.5. Composite systems.
For completeness, we very briefly review what happens when the system we con-
sider is composed ofNindependent degrees of freedom/particles^9 .Inthiscase,
the Hilbert space of the total system is given by the tensor product of the Hilbert
space of each particle:H=⊗Nj=1Hj.systemsStatesinHof the form:

|ψ〉=|ψ〉 1 ⊗|ψ 2 〉⊗···⊗|ψN〉 with|ψj〉∈Hj (1.37)

are said to beseparable. A state that cannot be written as so, is calledentangled.
Entanglement is a truly quantum property: it encodes the possibility of knowing
some properties of one subsystem by measuring observables on the other part.
Consider for example two spin-1/2 particles,AandB, in the singlet state (we
omit the symbol of tensor product):

|ψ〉=

|+〉A|−〉B√+|−〉A|+〉B

2

. (1.38)

In this state, the spins of neitherAnorBare defined, but are opposite. When an
experimentalist, say Alice, performs a measurement on the systemAof the spin
along the third component, she finds +/2or−/2 with 50% probability but Bob,
without performing any measurement on its qubitB, can infer that the latter has
the opposite value of the spin.
This phenomenon, that Einstein himself defined as “that spooky action at a
distance”, is at the origin of much work and interesting discussions in the history
of development of QM (the EPR paradox, the theory of hidden variables and Bell’s
inequality, for example) and it is now at the heart of recent applications in the
field of quantum information and quantum computation, such as teleportation.
The interested reader can look at[ 32 ].
As a final remark, let us observe that different states might encode a differ-
ent level of entanglement, the singlet state given above being an example of a

(^9) We are assuming that the particles are distinguishable.