Chapter 2
A Little Bit of Quantum Mechanics
In this chapter we present a brief review of quantum mechanics, necessary for the
development of this book. We suggest the textbook [121] for more details. We start
by the four postulates of quantum mechanics.
2.1 Postulates of Quantum Mechanics
Postulate 2.1.1 There exists a complex vector space with inner product which is
complete in the metric defined by the inner product (that is, a Hilbert space) associated with any isolated physical system. This Hilbert space is known as the state
space of the system. The system is completely described by its state vector, which is
a unit vector in system’s state space.
The second postulate of quantum mechanics tells us when the system evolves
over time.
Postulate 2.1.2 The evolution of a closed quantum system is described by a unitary
transformation. More precisely, the state |v of the system at time t 1 is related to the
state |w of the system at time t 2 by a unitary operator U which depends only on t 1
and t 2 : |w = U |v.
Postulate 2.1.3 Quantum measurements are described by a collection {M n } of measurement operators. These are operators acting on the state space of the system being
measured. The index n refers to the measurement outcomes that may occur in the
experiment. If the state of the quantum system is |v immediately before the measurement then the probability that result n occurs is given by p(n) = =v|M
†
n M n |v, and
the state of the system after measuring is M n |v/
√
p(n). The measurement operators satisfy
n
M
†
n M n = I (called the completeness equation), where I is the identity
operator.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_2
17
A Little Bit of Quantum Mechanics
In this chapter we present a brief review of quantum mechanics, necessary for the
development of this book. We suggest the textbook [121] for more details. We start
by the four postulates of quantum mechanics.
2.1 Postulates of Quantum Mechanics
Postulate 2.1.1 There exists a complex vector space with inner product which is
complete in the metric defined by the inner product (that is, a Hilbert space) associated with any isolated physical system. This Hilbert space is known as the state
space of the system. The system is completely described by its state vector, which is
a unit vector in system’s state space.
The second postulate of quantum mechanics tells us when the system evolves
over time.
Postulate 2.1.2 The evolution of a closed quantum system is described by a unitary
transformation. More precisely, the state |v of the system at time t 1 is related to the
state |w of the system at time t 2 by a unitary operator U which depends only on t 1
and t 2 : |w = U |v.
Postulate 2.1.3 Quantum measurements are described by a collection {M n } of measurement operators. These are operators acting on the state space of the system being
measured. The index n refers to the measurement outcomes that may occur in the
experiment. If the state of the quantum system is |v immediately before the measurement then the probability that result n occurs is given by p(n) = =v|M
†
n M n |v, and
the state of the system after measuring is M n |v/
√
p(n). The measurement operators satisfy
n
M
†
n M n = I (called the completeness equation), where I is the identity
operator.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_2
17