Concept in quantum information theory
In
quantum information theory and
quantum optics , the Schrödinger–HJW theorem is a result about the realization of a
mixed state of a
quantum system as an
ensemble of
pure quantum states and the relation between the corresponding purifications of the
density operators . The theorem is named after physicists and mathematicians
Erwin Schrödinger ,
[1]
Lane P. Hughston ,
Richard Jozsa and
William Wootters .
[2] The result was also found independently (albeit partially) by
Nicolas Gisin ,
[3] and by Nicolas Hadjisavvas building upon work by
Ed Jaynes ,
[4]
[5] while a significant part of it was likewise independently discovered by
N. David Mermin .
[6] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem ,
[7] the HJW theorem , and the purification theorem .
Purification of a mixed quantum state
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}}
.
[8]
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 }
". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state.
[9] Because all of them admit a decomposition in the form given above, given any pair of purifications
|
Ψ
⟩
,
|
Ψ
′
⟩
∈
H
S
⊗
H
A
{\displaystyle |\Psi \rangle ,|\Psi '\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}}
, there is always some unitary operation
U
:
H
A
→
H
A
{\displaystyle U:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A}}
such that
|
Ψ
′
⟩
=
(
I
⊗
U
)
|
Ψ
⟩
.
{\displaystyle |\Psi '\rangle =(I\otimes U)|\Psi \rangle .}
Theorem
Consider a mixed quantum state
ρ
{\displaystyle \rho }
with two different realizations as ensemble of pure states as
ρ
=
∑
i
p
i
|
ϕ
i
⟩
⟨
ϕ
i
|
{\textstyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|}
and
ρ
=
∑
j
q
j
|
φ
j
⟩
⟨
φ
j
|
{\textstyle \rho =\sum _{j}q_{j}|\varphi _{j}\rangle \langle \varphi _{j}|}
. Here both
|
ϕ
i
⟩
{\displaystyle |\phi _{i}\rangle }
and
|
φ
j
⟩
{\displaystyle |\varphi _{j}\rangle }
are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state
ρ
{\displaystyle \rho }
reading as follows:
Purification 1:
|
Ψ
S
A
1
⟩
=
∑
i
p
i
|
ϕ
i
⟩
⊗
|
a
i
⟩
{\displaystyle |\Psi _{SA}^{1}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle }
;
Purification 2:
|
Ψ
S
A
2
⟩
=
∑
j
q
j
|
φ
j
⟩
⊗
|
b
j
⟩
{\displaystyle |\Psi _{SA}^{2}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes |b_{j}\rangle }
.
The sets
{
|
a
i
⟩
}
{\displaystyle \{|a_{i}\rangle \}}
and
{
|
b
j
⟩
}
{\displaystyle \{|b_{j}\rangle \}}
are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix
U
A
{\displaystyle U_{A}}
such that
|
Ψ
S
A
1
⟩
=
(
I
⊗
U
A
)
|
Ψ
S
A
2
⟩
{\displaystyle |\Psi _{SA}^{1}\rangle =(I\otimes U_{A})|\Psi _{SA}^{2}\rangle }
.
[10] Therefore,
|
Ψ
S
A
1
⟩
=
∑
j
q
j
|
φ
j
⟩
⊗
U
A
|
b
j
⟩
{\textstyle |\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes U_{A}|b_{j}\rangle }
, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.
References
^ Schrödinger, Erwin (1936). "Probability relations between separated systems".
Proceedings of the Cambridge Philosophical Society . 32 (3): 446–452.
Bibcode :
1936PCPS...32..446S .
doi :
10.1017/S0305004100019137 .
^ Hughston, Lane P.; Jozsa, Richard; Wootters, William K. (November 1993). "A complete classification of quantum ensembles having a given density matrix".
Physics Letters A . 183 (1): 14–18.
Bibcode :
1993PhLA..183...14H .
doi :
10.1016/0375-9601(93)90880-9 .
ISSN
0375-9601 .
^ Gisin, N. (1989). “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62, 363–371.
^ Hadjisavvas, Nicolas (1981). "Properties of mixtures on non-orthogonal states".
Letters in Mathematical Physics . 5 (4): 327–332.
Bibcode :
1981LMaPh...5..327H .
doi :
10.1007/BF00401481 .
^ Jaynes, E. T. (1957). "Information theory and statistical mechanics. II".
Physical Review . 108 (2): 171–190.
Bibcode :
1957PhRv..108..171J .
doi :
10.1103/PhysRev.108.171 .
^ Fuchs, Christopher A. (2011). Coming of Age with Quantum Information: Notes on a Paulian Idea . Cambridge:
Cambridge University Press .
ISBN
978-0-521-19926-1 .
OCLC
535491156 .
^
Mermin, N. David (1999). "What Do These Correlations Know about Reality? Nonlocality and the Absurd".
Foundations of Physics . 29 (4): 571–587.
arXiv :
quant-ph/9807055 .
Bibcode :
1998quant.ph..7055M .
doi :
10.1023/A:1018864225930 .
^ Nielsen, Michael A.; Chuang, Isaac L.,
"The Schmidt decomposition and purifications" , Quantum Computation and Quantum Information , Cambridge: Cambridge University Press, pp. 110–111 .
^ Watrous, John (2018).
The Theory of Quantum Information . Cambridge: Cambridge University Press.
doi :
10.1017/9781316848142 .
ISBN
978-1-107-18056-7 .
^ Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem".
Foundations of Physics Letters . 19 (1): 95–102.
arXiv :
quant-ph/0305068 .
Bibcode :
2006FoPhL..19...95K .
doi :
10.1007/s10702-006-1852-1 .
ISSN
0894-9875 .