Quantum state purification

In quantum information theory, quantum state purification refers to the process of representing a mixed state as a pure quantum state of higher-dimensional Hilbert space. The purification allows the original mixed state to be recovered by taking the partial trace over the additional degrees of freedom. The purification is not unique, the different purifications that can lead to the same mixed states are limited by the Schrödinger–HJW theorem.

Purification is used in algorithms such as entanglement distillation, magic state distillation and algorithmic cooling.

Description

Let

H

S

{\displaystyle {\mathcal {H}}_{S}}

be a finite-dimensional complex Hilbert space, and consider a generic (possibly mixed) quantum state

ρ

{\displaystyle \rho }

defined on

H

S

{\displaystyle {\mathcal {H}}_{S}}

and admitting a decomposition of the form

ρ

=

∑

i

p

i

|

ϕ

i

⟩

⟨

ϕ

i

|

{\displaystyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|}

for a collection of (not necessarily mutually orthogonal) states

|

ϕ

i

⟩

∈

H

S

{\displaystyle |\phi _{i}\rangle \in {\mathcal {H}}_{S}}

and coefficients

p

i

≥

0

{\displaystyle p_{i}\geq 0}

such that

∑

i

p

i

=

1

{\textstyle \sum _{i}p_{i}=1}

. Note that any quantum state can be written in such a way for some

{

|

ϕ

i

⟩

}

i

{\displaystyle \{|\phi _{i}\rangle \}_{i}}

and

{

p

i

}

i

{\displaystyle \{p_{i}\}_{i}}

.

Any such

ρ

{\displaystyle \rho }

can be purified, that is, represented as the partial trace of a pure state defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space

H

A

{\displaystyle {\mathcal {H}}_{A}}

and a pure state

|

Ψ

S

A

⟩

∈

H

S

⊗

H

A

{\displaystyle |\Psi _{SA}\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}}

such that

ρ

=

Tr

A

⁡

(

|

Ψ

S

A

⟩

⟨

Ψ

S

A

|

)

{\displaystyle \rho =\operatorname {Tr} _{A}{\big (}|\Psi _{SA}\rangle \langle \Psi _{SA}|{\big )}}

. Furthermore, the states

|

Ψ

S

A

⟩

{\displaystyle |\Psi _{SA}\rangle }

satisfying this are all and only those of the form

|

Ψ

S

A

⟩

=

∑

i

p

i

|

ϕ

i

⟩

⊗

|

a

i

⟩

{\displaystyle |\Psi _{SA}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle }

for some orthonormal basis

{

|

a

i

⟩

}

i

⊂

H

A

{\displaystyle \{|a_{i}\rangle \}_{i}\subset {\mathcal {H}}_{A}}

. The state

|

Ψ

S

A

⟩

{\displaystyle |\Psi _{SA}\rangle }

is then referred to as the "purification of

ρ

{\displaystyle \rho }

".