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# POVM

In functional analysis and quantum measurement theory, a POVM (Positive Operator Valued Measure) is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics. The need for the POVM formalism arises from the fact that projective measurements on a larger system will act on a subsystem in ways that cannot be described by projective measurement on the subsystem alone. They are used in the field of quantum information.

In rough analogy, a POVM is to a projective measurement what a density matrix is to a pure state. Density matrices can describe part of a larger system that is in a pure state (see purification of quantum state); analogously, POVMs on a physical system can describe the effect of a projective measurement performed on a larger system.

## Definition

In the simplest case, a POVM is a set of Hermitian positive semidefinite operators $\left\{F_i\right\}$ on a Hilbert space H that sum to unity,

$sum_\left\{i=1\right\}^n F_i = operatorname\left\{I\right\}_H.$

This formula is similar to the decomposition of a Hilbert space into a set of orthogonal projectors,

$sum_\left\{i=1\right\}^N E_i = operatorname\left\{I\right\}_H,$

and if ij,

$E_i E_j = 0. quad$

An important difference is that the elements of a POVM are not necessarily orthogonal, with the consequence that the number of elements in the POVM, n, can be larger than the dimension, N, of the Hilbert space they act in.

In general, POVMs can be defined in situations where outcomes can occur in a non-discrete space. The relevant fact is that measurements determine probability measures on the outcome space:

Definition. Let (X, M) be measurable space; that is M is a σ-algebra of subsets of X. A POVM is a function F defined on M whose values are bounded non-negative self-adjoint operators on a Hilbert space H such that F(X) = IH and for every ξ $in$ H,

$E mapsto langle F\left(E\right) xi mid xi rangle$

is a non-negative countably additive measure on the σ-algebra M.

This definition should be contrasted with that for the projection-valued measure, which is very similar, except that, in the projection-valued measure, the F are required to be projection operators.

## POVMs and measurement

As in the theory of projective measurement, the probability the outcome associated with measurement of operator $F_i$ occurs is,

$P\left(i\right)=Tr\left(F_irho\right),$
where $rho$ is the density matrix describing the state of the measured system.

An element of a POVM can always be written as,

$F_i = M^dagger_i M_i,$
for some operator $M_i$, known as a Kraus Operator. The state of the system after the measurement $rho\text{'}$ is transformed according to,
$rho\text{'} = \left\{M_i rho M_i^dagger over \left\{rm tr\right\}\left(M_i rho M_i^dagger\right)\right\}.$

## Neumark's dilation theorem

An alternate spelling of this is Naimark's Theorem

Neumark's dilation theorem is the classification result for POVM's. It states that a POVM can be "lifted" by an operator map of the form V*(·)V to a projection-valued measure. In the physical context, this means that measuring a POVM consisting of a set of n > N operators acting on a N-dimensional Hilbert space can always be achieved by performing a projective measurement on a Hilbert space of dimension n then consider the reduced state.

In practice, however, obtaining a suitable projection-valued measure from a given POVM is usually done by coupling to the original system an ancilla. Consider a Hilbert space $H_A$ that is extended by $H_B$. The state of total system is $rho_\left\{AB\right\}$ and $rho_A=Tr_B\left(rho_\left\{AB\right\}\right)$. The probability the projective measurement $hat\left\{pi\right\}_i$ succeeds is,

$P\left(i\right)=Tr_A\left(Tr_B\left(hat\left\{pi\right\}_irho_\left\{AB\right\}\right)\right).$
An implication of Neumark's theorem is that the associated POVM in subspace A, $F_i$, must have the same probability of success.
$P\left(i\right)=Tr_A\left(F_irho_A\right)\right).$

## An example: Unambiguous quantum state discrimination

The task of unambiguous quantum state discrimination (UQSD) is to discern conclusively which state, of given set of pure states, a quantum system (which we call the input) is in. The impossibility of perfectly discriminating between a set of non-orthogonal states is the basis for quantum information protocols such as quantum cryptography, quantum coin-flipping, and quantum money. This example will show that a POVM has a higher success probability for performing UQSD than any possible projective measurement.

First let us consider a trivial case. Take a set that consists of two orthogonal states $|psirang$ and $|psi^Trang$. A projective measurement of the form,

$hat\left\{A\right\}= a|psi^Tranglangpsi^T| + b|psiranglangpsi|,$
will result in eigenvalue a only when the system is in $|psi^Trang$ and eigenvalue b only when the system is in $|psirang$. In addition, the measurement always discriminates between the two states (i.e. with 100% probability). This latter ability is unnecessary for UQSD and, in fact, is impossible for anything but orthogonal states. Now consider a set that consists of two states $|psirang$ and $|phirang$ in two-dimensional Hilbert space that are not orthogonal. i.e.,
$|langphi|psirang| = cos\left(theta\right),$
for $theta > 0$. These states could a system, such as the spin of spin-1/2 particle (e.g. an electron), or the polarization of a photon. Assuming that the system has an equal likelihood of being in each of these two states, the best strategy for UQSD using only projective measurement is to perform each of the following measurements,
$hat\left\{pi\right\}_\left\{psi^T\right\}= |psi^Tranglangpsi^T|,$
$hat\left\{pi\right\}_\left\{phi^T\right\}= |phi^Tranglangphi^T|,$
50% of the time. If $hat\left\{pi\right\}_\left\{phi^T\right\}$ is measured and results in an eigenvalue of 1, than it is certain that the state must have been in $|psirang$. However, an eigenvalue of zero is now an inconclusive result since this can come about from the system could being in either of the two states in the set. Similarly, a result of 1 for $hat\left\{pi\right\}_\left\{psi^T\right\}$ indicates conclusively that the system is in $|phirang$ and 0 is inconclusive. The probability that this strategy returns a conclusive result is,
$P_\left\{proj\right\}=frac\left\{1-|langphi|psirang|^2\right\}\left\{2\right\}.$
In contrast, a strategy based on POVMs has a greater probability of success given by,
$P_\left\{POVM\right\}=1-|langphi|psirang|.$
This is the minimum allowed by the rules of quantum indeterminacy and the uncertainty principle. This strategy is based on a POVM consisting of,
$hat\left\{F\right\}_\left\{psi\right\}=frac\left\{1-|phiranglangphi|\right\}\left\{1+|langphi|psirang|\right\}$
$hat\left\{F\right\}_\left\{phi\right\}=frac\left\{1-|psiranglangpsi|\right\}\left\{1+|langphi|psirang|\right\}$
$hat\left\{F\right\}_\left\{inconcl.\right\}= 1-hat\left\{F\right\}_\left\{psi\right\}-hat\left\{F\right\}_\left\{phi\right\},$
where the result associated with $hat\left\{F\right\}_\left\{i\right\}$ indicates the system is in state i with certainty.

These POVMs can be created by extending the two-dimensional Hilbert space. This can be visualized as follows: The two states fall in the x-y plane with an angle of θ between them and the space is extended in the z-direction. (The total space is the direct sum of spaces defined by the z-direction and the x-y plane.) The measurement first unitarily rotates the states towards the z-axis so that $|psirang$ has no component along the y-direction and $|phirang$ has no component along the x-direction. At this point, the three elements of the POVM correspond to projective measurements along x-direction, y-direction and z-direction, respectively.

For a specific example, take a stream of photons, each of which are polarized along either along the horizontal direction or at 45 degrees. On average there are equal numbers of horizontal and 45 degree photons. The projective strategy corresponds to passing the photons through a polarizer in either the vertical direction or -45 degree direction. If the photon passes through the vertical polarizer it must have been at 45 degrees and vice versa. The success probability is $\left(1-1/2\right)/2=25%$. The POVM strategy for this example is more complicated and requires another optical mode (known as an ancilla). It has a success probability of $1-1/sqrt\left\{2\right\}=29.3%$.

## References

• POVMs
• J.Preskill, Lecture Note for Physics: Quantum Information and Computation, http://theory.caltech.edu/people/preskill
• K.Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983)
• E.B.Davies, Quantum Theory of Open Systems, Academic Press (1976).
• Neumark's theorem
• A. Peres. Neumark’s theorem and quantum inseparability. Foundations of Physics, 12:1441–1453, 1990.
• A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993.
• I. M. Gelfand and M. A. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213.
• Unambiguous quantum state-discrimination
• I. D. Ivanovic, Phys. Lett. A 123 257 (1987).
• D. Dieks, Phys. Lett. A 126 303 (1988).
• A. Peres, Phys. Lett. A 128 19 (1988).

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