POVM

 ( Quantum Information Theory ) 

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. They are extensively used in the field of quantum information.

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Definition Let $\backslash mathcal\{H\}$ denote a Hilbert space and $(X,\; M)$ a measurable space with $M$ a Borel σ-algebra on $X$. A POVM is a function $F$ defined on $M$ whose values are positive bounded self-adjoint operators on $\backslash mathcal\{H\}$ such that for every $\backslash psi\; \backslash in\; \backslash mathcal\{H\}$

$E\; \backslash mapsto\; \backslash langle\; F(E)\; \backslash psi\; \backslash mid\; \backslash psi\; \backslash rangle,$

is a non-negative countably additive measure on the σ-algebra $M$ and $F(X)\; =\; \backslash operatorname\{I\}\_\{\backslash mathcal\{H\}\}$ is the identity operator.

Edward Brian Davies (1976). 9780122061509, Acad. Press. ISBN 9780122061509

In quantum mechanics, the key property of a POVM is that it determines a probability measure on the outcome space, so that $\backslash langle\; F(E)\; \backslash psi\; \backslash mid\; \backslash psi\; \backslash rangle$ can be interpreted as the probability of the event $E$ when measuring a quantum state $|\; \backslash psi\; \backslash rangle$.

In the simplest case, in which $X$ is a finite set, $M$ is the power set of $X$ and $\backslash mathcal\{H\}$ is finite-dimensional, a POVM is equivalently a set of positive semi-definite Hermitian matrix $\backslash \{F\_i\backslash \}$ that sum to the identity matrix,M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, (2000)

$\backslash sum\_\{i=1\}^n\; F\_i\; =\; \backslash operatorname\{I\}.$

A POVM differs from a projection-valued measure in that, for projection-valued measures, the values of $F$ are required to be orthogonal projections.

In the discrete case, the POVM element $F\_i$ is associated with the measurement outcome $i$, such that the probability of obtaining it when making a quantum measurement on the quantum state $\backslash rho$ is given by

$\backslash text\{Prob\}(i)\; =\; \backslash operatorname\{tr\}(\backslash rho\; F\_i)$,

where $\backslash operatorname\{tr\}$ is the trace operator. When the quantum state being measured is a pure state $|\backslash psi\backslash rangle$ this formula reduces to

$\backslash text\{Prob\}(i)\; =\; \backslash operatorname\{tr\}(|\backslash psi\backslash rangle\backslash langle\backslash psi|\; F\_i)\; =\; \backslash langle\backslash psi|F\_i|\backslash psi\backslash rangle$.

The discrete case of a POVM generalizes the simplest case of a PVM, which is a set of orthogonal projectors $\backslash \{\backslash Pi\_i\backslash \}$ that sum to the identity matrix:

$\backslash sum\_\{i=1\}^N\; \backslash Pi\_i\; =\; \backslash operatorname\{I\},\; \backslash quad\; \backslash Pi\_i\; \backslash Pi\_j\; =\; \backslash delta\_\{i\; j\}\; \backslash Pi\_i.$

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 $n$ of the POVM can be larger than the dimension of the Hilbert space they act in. On the other hand, the number of elements $N$ of the PVM is at most the dimension of the Hilbert space.

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Naimark's dilation theorem

Note: An alternate spelling of this is "Neumark's Theorem"

Naimark's dilation theoremI. M. Gelfand and M. A. Neumark, On the embedding of normed rings into the ring of operators in Hilbert space, Rec. Math. Mat. N.S. 12(54) (1943), 197–213. 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.A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993.

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 $\backslash \{F\_i\backslash \}\_\{i=1\}^n$ is a POVM acting on a Hilbert space $\backslash mathcal\{H\}\_A$ of dimension $d\_A$, then there exists a PVM $\backslash \{\backslash Pi\_i\backslash \}\_\{i=1\}^n$ acting on a Hilbert space $\backslash mathcal\{H\}\_\{A\text{'}\}$ of dimension $d\_\{A\text{'}\}$ and an isometry $V\; :\; \backslash mathcal\{H\}\_A\; \backslash to\; \backslash mathcal\{H\}\_\{A\text{'}\}$ such that for all $i$,

$F\_i\; =\; V^\backslash dagger\; \backslash Pi\_i\; V.$

For the particular case of a rank-1 POVM, i.e., when $F\_i\; =\; |f\_i\backslash rangle\backslash langle\; f\_i|$ for some (unnormalized) vectors $|f\_i\backslash rangle$, this isometry can be constructed as

$V\; =\; \backslash sum\_\{i=1\}^n\; |i\backslash rangle\_\{A\text{'}\}\backslash langle\; f\_i|\_\{A\}$

and the PVM is given simply by $\backslash Pi\_i\; =\; |i\backslash rangle\backslash langle\; i|\_\{A\text{'}\}$. Note that here $d\_\{A\text{'}\}\; =\; n$.

In the general case, the isometry and PVM can be constructed by defining /ref> $\backslash mathcal\{H\}\_\{A\text{'}\}\; =\; \backslash mathcal\{H\}\_\{A\}\backslash otimes\; \backslash mathcal\{H\}\_\{B\}$, $\backslash Pi\_i\; =\; \backslash operatorname\{I\}\_A\; \backslash otimes\; |i\backslash rangle\backslash langle\; i|\_B$, and

$V\; =\; \backslash sum\_\{i=1\}^n\; \backslash sqrt\{F\_i\}\_A\; \backslash otimes$>

Note that here $d\_\{A\text{'}\}\; =\; nd\_A$, so this is a more wasteful construction. In either case, the probability of obtaining outcome $i$ with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM: $\backslash text\{Prob\}(i)\; =\; \backslash operatorname\{tr\}\backslash left(\; V\; \backslash rho\_A\; V^\backslash dagger\; \backslash Pi\_i\; \backslash right)$ = \operatorname{tr}\left( \rho_A V^\dagger \Pi_i V \right) = \operatorname{tr}(\rho_A F_i) This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry $V$ into a unitary $U$, that is, finding $U$ such that $V|i\backslash rangle\_A\; =\; Ui\backslash rangle\_\{A\text{'}\}$ for $i$ from 1 to $d\_A$. This can always be done. The recipe for realizing the POVM described by $\backslash \{F\_i\backslash \}\_\{i=1\}^n$ on a quantum state $\backslash rho$ is then to embed the quantum state in the Hilbert space $\backslash mathcal\{H\}\_\{A\text{'}\}$, evolve it with the unitary $U$, and make the projective measurement described by the PVM $\backslash \{\backslash Pi\_i\backslash \}\_\{i=1\}^n$. Post-measurement state 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 $\backslash \{F\_i\backslash \}\_\{i=1\}^n$ alone do not determine what the post-measurement state will be. To see that, note that for any unitary $W$ the operators $M\_i\; =\; W\backslash sqrt\{F\_i\}$ will also have the property that $M\_i^\backslash dagger\; M\_i\; =\; F\_i$, so that using the isometry $V\_W\; =\; \backslash sum\_\{i=1\}^n\; \{M\_i\}\_A\; \backslash otimes\; \{$

i\rangle}_B in the second construction above will also implement the same POVM. In the case where the state being measured is in a pure state $$

\psi\rangle_A, the resulting unitary $U\_W$ takes it together with the ancilla to state $U\_W($

i\rangle_B, and the projective measurement on the ancilla will collapse $$

\psi\rangle_A to the state $$

\psi \rangle}} on obtaining result $i\_0$. When the state being measured is described by a density matrix $\backslash rho\_A$, the corresponding post-measurement state is given by $\backslash rho\text{'}\_A\; =\; \{M\_\{i\_0\}\; \backslash rho\; M\_\{i\_0\}^\backslash dagger\; \backslash over\; \{\backslash rm\; tr\}(M\_\{i\_0\}\; \backslash rho\; M\_\{i\_0\}^\backslash dagger)\}$. We see therefore that the post-measurement state depends explicitly on the unitary $W$. Note that while $M\_i^\backslash dagger\; M\_i\; =\; F\_i$ is always Hermitian, generally, $M\_i$ does not have to be Hermitian. Another difference from the projective measurements is that a POVM measurement is in general not repeatable. If on the first measurement result $i\_0$ was obtained, the probability of obtaining a different result $i\_1$ on a second measurement is $\backslash text\{Prob\}(i\_1$

i_0) = {\operatorname{tr}(M_{i_1}M_{i_0} \rho M_{i_0}^\dagger M_{i_1}^\dagger) \over {\rm tr}(M_{i_0} \rho M_{i_0}^\dagger)}, which can be nonzero if $M\_\{i\_0\}$ and $M\_\{i\_1\}$ are not orthogonal. In a projective measurement these operators are always orthogonal and therefore the measurement is always repeatable. An example: unambiguous quantum state discrimination Suppose you have a quantum system with a 2-dimensional Hilbert space that you know is in either the state $$

and $>\backslash varphi\backslash rangle$ are not orthogonal, this task is impossible, in the sense that there is no measurement, either PVM or POVM, that will distinguish them with certainty. The impossibility of perfectly discriminating between non-orthogonal states is the basis for quantum information protocols such as quantum cryptography, quantum coin flipping, and quantum money. The task of unambiguous quantum state discrimination (UQSD) is the next best thing: to never make a mistake about whether the state is $$

is the state orthogonal to $>\backslash varphi\backslash rangle$. This is unsatisfactory, though, as you can't detect both $$

and $>\backslash varphi\backslash rangle$ with a single measurement, and the probability of getting a conclusive result is smaller than with POVMs. The POVM that gives the highest probability of a conclusive outcome in this task is given by (2026). 9783540444817, Springer. . ISBN 9783540444817 $F\_\{\backslash psi\}=\backslash frac\{1\}\{1+$

|\varphi^\perp\rangle\langle\varphi^\perp|

$F\_\{\backslash varphi\}=\backslash frac\{1\}\{1+|\backslash lang\backslash varphi|\backslash psi\backslash rang|\}|\backslash psi^\backslash perp\backslash rangle\backslash langle\backslash psi^\backslash perp|$

$F\_?=\; \backslash operatorname\{I\}-F\_\{\backslash psi\}-F\_\{\backslash varphi\}=\; \backslash frac\{2|\backslash lang\backslash varphi|\backslash psi\backslash rang|\}\{1+|\backslash lang\backslash varphi|\backslash psi\backslash rang|\}\; |\backslash gamma\backslash rangle\backslash langle\backslash gamma|,$

where

$|\backslash gamma\backslash rangle\; =\; \backslash frac1\{\backslash sqrt\{2(1+|\backslash lang\backslash varphi|\backslash psi\backslash rang|)\}\}(|\backslash psi\backslash rangle+e^\{i\backslash arg(\backslash lang\backslash varphi|\backslash psi\backslash rang)\}|\backslash varphi\backslash rangle).$

Note that $\backslash operatorname\{tr\}(|\backslash varphi\backslash rangle\backslash langle\backslash varphi|F\_\{\backslash psi\})\; =\; \backslash operatorname\{tr\}(|\backslash psi\backslash rangle\backslash langle\backslash psi|F\_\{\backslash varphi\})\; =\; 0$, so when outcome $\backslash psi$ is obtained we are certain that the quantum state is $|\backslash psi\backslash rangle$, and when outcome $\backslash varphi$ is obtained we are certain that the quantum state is $|\backslash varphi\backslash rangle$.

The probability of having a conclusive outcome is given by

$1-|\backslash lang\backslash varphi|\backslash psi\backslash rang|,$

when the quantum system is in state $|\backslash psi\backslash rangle$ or $|\backslash varphi\backslash rangle$ with the same probability. This result is known as the Ivanović-Dieks-Peres limit, named after the authors who pioneered UQSD research.

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 $|\backslash text\{result\; \psi \}\backslash rangle$, $|\backslash text\{result\; \phi \}\backslash rangle$, and $|\backslash text\{result\; ?\}\backslash rangle$, we see that the resulting unitary $U\_\backslash text\{UQSD\}$ takes the state $|\backslash psi\backslash rangle$ to

$U\_\backslash text\{UQSD\}|\backslash psi\backslash rangle\; =\; \backslash sqrt\{1-|\backslash lang\backslash varphi|\backslash psi\backslash rang|\}|\backslash text\{result\; \psi \}\backslash rangle\; +\; \backslash sqrt$ |\text{result ?}\rangle,

and similarly it takes the state $|\backslash varphi\backslash rangle$ to

$U\_\backslash text\{UQSD\}|\backslash varphi\backslash rangle\; =\; \backslash sqrt\{1-|\backslash lang\backslash varphi|\backslash psi\backslash rang|\}|\backslash text\{result\; \phi \}\backslash rangle\; +\; e^\{-i\backslash arg(\backslash lang\backslash varphi|\backslash psi\backslash rang)\}\backslash sqrt$|\text{result ?}\rangle.

A projective measurement then gives the desired results with the same probabilities as the POVM.

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.

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See also

• SIC-POVM
• Quantum measurement
• Mathematical formulation of quantum mechanics
• Density matrix
• Quantum operation
• Projection-valued measure
• Vector measure

• POVMs
• K. Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983).
• Alexander Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland Publ. Cy., Amsterdam (1982).

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External links

• Interactive demonstration about quantum state discrimination

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