Classical capacity

In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel.

Background

Mixed states and quantum channels

A mixed quantum state is a unit trace,

positive operator known as a density operator, and is often denoted

by

ρ

{\displaystyle \rho }

,

σ

{\displaystyle \sigma }

,

ω

{\displaystyle \omega }

, etc. The simplest model for a quantum channel

is a classical-quantum channel

which sends the classical letter

x

{\displaystyle x}

at the transmitting end to a quantum state

ρ

x

{\displaystyle \rho _{x}}

at the receiving

end, with noise possibly introduced in between. The receiver's task is to perform a measurement to determine the

input of the sender. If the states

ρ

x

{\displaystyle \rho _{x}}

are perfectly

distinguishable from one another (i.e., if they have orthogonal supports such

that

Tr

⁡

ρ

x

ρ

x

′

=

0

{\displaystyle \operatorname {Tr} \rho _{x}\rho _{x^{\prime }}=0}

for

x

≠

x

′

{\displaystyle x\neq x^{\prime }}

) and the channel is noiseless, then perfect decoding is trivially possible. If the states

ρ

x

{\displaystyle \rho _{x}}

all

commute with each other then the channel is effectively classical.

The situation becomes nontrivial only when the states

ρ

x

{\displaystyle \rho _{x}}

have overlapping support and do not necessarily commute.

Quantum measurements

The most general way to describe a quantum measurement is with a

positive operator-valued measure, whose elements are typically denoted as

{

Λ

m

}

m

{\displaystyle \left\{\Lambda _{m}\right\}_{m}}

. These operators should satisfy

positivity and completeness in order to form a valid POVM:

Λ

m

≥

0

∀

m

{\displaystyle \Lambda _{m}\geq 0\ \ \ \ \forall m}

∑

m

Λ

m

=

I

.

{\displaystyle \sum _{m}\Lambda _{m}=I.}

The probabilistic interpretation of quantum mechanics states that if someone

measures a quantum state

ρ

{\displaystyle \rho }

using a measurement device corresponding to

the POVM

{

Λ

m

}

{\displaystyle \left\{\Lambda _{m}\right\}}

, then the probability

p

(

m

)

{\displaystyle p\left(m\right)}

for obtaining outcome

m

{\displaystyle m}

is equal to

p

(

m

)

=

Tr

⁡

Λ

m

ρ

,

{\displaystyle p(m)=\operatorname {Tr} \Lambda _{m}\rho ,}

and the post-measurement state is

ρ

m

′

=

1

p

(

m

)

Λ

m

1

/

2

ρ

Λ

m

1

/

2

,

{\displaystyle \rho _{m}^{\prime }={\frac {1}{p(m)}}\Lambda _{m}^{1/2}\rho \Lambda _{m}^{1/2},}

if the person measuring obtains outcome

m

{\displaystyle m}

.