In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurement described by PVMs (called projective measurements).
In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system.
POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory.[1] They are extensively used in the field of quantum information.
Let
l{H}
(X,M)
M
X
F
M
l{H}
\psi\inl{H}
E\mapsto\langleF(E)\psi\mid\psi\rangle,
M
F(X)=\operatorname{I}l{H
\{Fi\}
l{H}
n | |
\sum | |
i=1 |
Fi=\operatorname{I}.
A POVM differs from a projection-valued measure in that, for projection-valued measures, the values of
F
In quantum mechanics, the key property of a POVM is that it determines a probability measure on the outcome space, so that
\langleF(E)\psi\mid\psi\rangle
E
|\psi\rangle
Fi
i
\rho
Prob(i)=\operatorname{tr}(\rhoFi)
where
\operatorname{tr}
|\psi\rangle
Prob(i)=\operatorname{tr}(|\psi\rangle\langle\psi|Fi)=\langle\psi|Fi|\psi\rangle
\{\Pii\}
N | |
\sum | |
i=1 |
\Pii=\operatorname{I}, \Pii\Pij=\deltai\Pii.
The probability formulas for a PVM are the same as for the POVM. An important difference is that the elements of a POVM are not necessarily orthogonal. As a consequence, the number of elements
n
N
See main article: Naimark's dilation theorem.
Note: An alternate spelling of this is "Neumark's Theorem"
Naimark's dilation theorem[4] shows how POVMs can be obtained from PVMs acting on a larger space. This result is of critical importance in quantum mechanics, as it gives a way to physically realize POVM measurements.[5]
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if
\{Fi\}
n | |
i=1 |
l{H}A
dA
\{\Pii\}
n | |
i=1 |
l{H}A'
dA'
V:l{H}A\tol{H}A'
i
Fi=V\dagger\PiiV.
For the particular case of a rank-1 POVM, i.e., when
Fi=|fi\rangle\langlefi|
|fi\rangle
V=
n | |
\sum | |
i=1 |
|i\rangleA'\langlefi|A
\Pii=|i\rangle\langlei|A'
dA'=n
In the general case, the isometry and PVM can be constructed by defining[6] [7]
l{H}A'=l{H}A ⊗ l{H}B
\Pii=\operatorname{I}A ⊗ |i\rangle\langlei|B
V=
n | |
\sum | |
i=1 |
\sqrt{Fi}A ⊗ {|i\rangle}B.
dA'=ndA
In either case, the probability of obtaining outcome
i
Prob(i)=\operatorname{tr}\left(V\rhoAV\dagger\Pii\right)=\operatorname{tr}\left(\rhoAV\dagger\PiiV\right)=\operatorname{tr}(\rhoAFi)
This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry
V
U
U
V|i\rangleA=U|i\rangleA'
i
dA
The recipe for realizing the POVM described by
\{Fi\}
n | |
i=1 |
\rho
l{H}A'
U
\{\Pii\}
n | |
i=1 |
The post-measurement state is not determined by the POVM itself, but rather by the PVM that physically realizes it. Since there are infinitely many different PVMs that realize the same POVM, the operators
\{Fi\}
n | |
i=1 |
W
Mi=W\sqrt{Fi}
\dagger | |
M | |
i |
Mi=Fi
VW=
n | |
\sum | |
i=1 |
{Mi}A ⊗ {|i\rangle}B
|\psi\rangleA
UW
UW(|\psi\rangleA|0\rangleB)=
n | |
\sum | |
i=1 |
Mi|\psi\rangleA|i\rangleB,
|\psi\rangleA
|\psi'\rangleA=
| ||||||||||||
|
i0
\rhoA
\rho'A=
{M | |
i0 |
\rho
\dagger | |
M | |
i0 |
\over{\rm
tr}(M | |
i0 |
\rho
\dagger)} | |
M | |
i0 |
W
\dagger | |
M | |
i |
Mi=Fi
Mi
Another difference from the projective measurements is that a POVM measurement is in general not repeatable. If on the first measurement result
i0
i1
Prob(i1|i0)=
{\operatorname{tr}(M | |
i1 |
M | |
i0 |
\rho
\dagger | |
M | |
i0 |
\dagger) | |
M | |
i1 |
\over{\rm
tr}(M | |
i0 |
\rho
\dagger)} | |
M | |
i0 |
M | |
i0 |
M | |
i1 |
Suppose you have a quantum system with a 2-dimensional Hilbert space that you know is in either the state
|\psi\rangle
|\varphi\rangle
|\psi\rangle
|\varphi\rangle
\{|\psi\rangle\langle\psi|,|\varphi\rangle\langle\varphi|\}
|\psi\rangle
|\varphi\rangle
The task of unambiguous quantum state discrimination (UQSD) is the next best thing: to never make a mistake about whether the state is
|\psi\rangle
|\varphi\rangle
\{|\psi\rangle\langle\psi|,|\psi\perp\rangle\langle\psi\perp|\}
|\psi\perp\rangle
|\psi\rangle
|\psi\perp\rangle\langle\psi\perp|
|\varphi\rangle
|\psi\rangle\langle\psi|
\{|\varphi\rangle\langle\varphi|,|\varphi\perp\rangle\langle\varphi\perp|\}
|\varphi\perp\rangle
|\varphi\rangle
This is unsatisfactory, though, as you can't detect both
|\psi\rangle
|\varphi\rangle
F\psi=
1 | |
1+|\lang\varphi|\psi\rang| |
|\varphi\perp\rangle\langle\varphi\perp|
F\varphi=
1 | |
1+|\lang\varphi|\psi\rang| |
|\psi\perp\rangle\langle\psi\perp|
F?=\operatorname{I}-F\psi-F\varphi=
2|\lang\varphi|\psi\rang| | |
1+|\lang\varphi|\psi\rang| |
|\gamma\rangle\langle\gamma|,
|\gamma\rangle=
1{\sqrt{2(1+|\lang\varphi|\psi\rang|)}}(|\psi\rangle+e | |
i\arg(\lang\varphi|\psi\rang) |
|\varphi\rangle).
Note that
\operatorname{tr}(|\varphi\rangle\langle\varphi|F\psi)=\operatorname{tr}(|\psi\rangle\langle\psi|F\varphi)=0
\psi
|\psi\rangle
\varphi
|\varphi\rangle
The probability of having a conclusive outcome is given by
1-|\lang\varphi|\psi\rang|,
|\psi\rangle
|\varphi\rangle
Since the POVMs are rank-1, we can use the simple case of the construction above to obtain a projective measurement that physically realises this POVM. Labelling the three possible states of the enlarged Hilbert space as
|resultψ\rangle
|resultφ\rangle
|result?\rangle
UUQSD
|\psi\rangle
UUQSD|\psi\rangle=\sqrt{1-|\lang\varphi|\psi\rang|}|resultψ\rangle+\sqrt{|\lang\varphi|\psi\rang|}|result?\rangle,
|\varphi\rangle
UUQSD|\varphi\rangle=\sqrt{1-|\lang\varphi|\psi\rang|}|resultφ\rangle+e-i\arg(\lang\varphi|\psi\rang)\sqrt{|\lang\varphi|\psi\rang|}|result?\rangle.
This POVM has been used to experimentally distinguish non-orthogonal polarisation states of a photon. The realisation of the POVM with a projective measurement was slightly different from the one described here.[13] [14]