2.5 Quantum Operations
21
2.5.1 Depolarizing Channel
An important example of quantum operation is the depolarizing channel:
E(ρ) = (1 − p)ρ +
p
3
3
i=1
σ i ρσ
†
i ,
(2.43)
with Kraus operators K 0 =
√
1 − p 1 and K i =
√
p/3 σ i . Recall from Sect. 2.3
that every qubit error can be reproduced by applying a Pauli operator σ i . Thus, the
resulting state in (2.43) is unchanged with probability 1 − p or is affected by one
of the qubit errors with probability p/3 each. By applying the depolarizing channel
on the qubits used in a quantum information protocol, one can test the protocol’s
robustness against noise.
Note that, under the substitution p = 3q/4, the map in (2.43) can be recast as:
E(ρ) = (1 − q)ρ + q
1
2
,
(2.44)
i.e. it depolarizes a qubit with probability q by replacing it with the completely mixed
state 1/2.
2.6 Shannon and von Neumann Entropy
The uncertainty that an observer has about a physical system, i.e. the amount of
randomness characterizing the system from her perspective, is quantified by a certain
entropy measure.
Definition 2.6 (Shannon entropy) Let X be a random variable whose outcomes follow the probability distribution { p x }. The Shannon entropy of X (or of the distribution
{ p x }) is given by:
H (X ) = H ({ p x }) = −
x
p x log p x .
(2.45)
In this book the logarithm symbol “log” is always intended in base 2 and by
convention it holds: 0 log 0 = 0.
The Shannon entropy quantifies the uncertainty about X before we learn its value.
Equivalently, H (X ) are the bits of information gained after reading the outcome of
X . If X has only two possible outcomes, the Shannon entropy is often called binary
entropy and reads:
21
2.5.1 Depolarizing Channel
An important example of quantum operation is the depolarizing channel:
E(ρ) = (1 − p)ρ +
p
3
3
i=1
σ i ρσ
†
i ,
(2.43)
with Kraus operators K 0 =
√
1 − p 1 and K i =
√
p/3 σ i . Recall from Sect. 2.3
that every qubit error can be reproduced by applying a Pauli operator σ i . Thus, the
resulting state in (2.43) is unchanged with probability 1 − p or is affected by one
of the qubit errors with probability p/3 each. By applying the depolarizing channel
on the qubits used in a quantum information protocol, one can test the protocol’s
robustness against noise.
Note that, under the substitution p = 3q/4, the map in (2.43) can be recast as:
E(ρ) = (1 − q)ρ + q
1
2
,
(2.44)
i.e. it depolarizes a qubit with probability q by replacing it with the completely mixed
state 1/2.
2.6 Shannon and von Neumann Entropy
The uncertainty that an observer has about a physical system, i.e. the amount of
randomness characterizing the system from her perspective, is quantified by a certain
entropy measure.
Definition 2.6 (Shannon entropy) Let X be a random variable whose outcomes follow the probability distribution { p x }. The Shannon entropy of X (or of the distribution
{ p x }) is given by:
H (X ) = H ({ p x }) = −
x
p x log p x .
(2.45)
In this book the logarithm symbol “log” is always intended in base 2 and by
convention it holds: 0 log 0 = 0.
The Shannon entropy quantifies the uncertainty about X before we learn its value.
Equivalently, H (X ) are the bits of information gained after reading the outcome of
X . If X has only two possible outcomes, the Shannon entropy is often called binary
entropy and reads: