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.

Definition

In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite Hermitian matrices ${\displaystyle \{F_{i}\}}$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$ that sum to the identity matrix,^[2]: 90

${\displaystyle \sum _{i=1}^{n}F_{i}=\operatorname {I} .}$

In quantum mechanics, the POVM element ${\displaystyle F_{i}}$ is associated with the measurement outcome ${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state ${\displaystyle \rho }$ is given by

${\displaystyle {\text{Prob}}(i)=\operatorname {tr} (\rho F_{i})}$,

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

${\displaystyle {\text{Prob}}(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle }$.

The simplest case of a POVM generalises the simplest case of a PVM, which is a set of orthogonal projectors ${\displaystyle \{\Pi _{i}\}}$ that sum to the identity matrix:

${\displaystyle \sum _{i=1}^{N}\Pi _{i}=\operatorname {I} ,\quad \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}.}$

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

In general, POVMs can also be defined in situations where the number of elements and the dimension of the Hilbert space is not finite:

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

${\displaystyle E\mapsto \langle F(E)\psi \mid \psi \rangle \ ({\text{for every }}E\in M)}$

is a non-negative countably additive measure on the σ-algebra ${\displaystyle M}$.

Its key property is that it determines a probability measure on the outcome space, so that ${\displaystyle \langle F(E)\psi \mid \psi \rangle }$ can be interpreted as the probability (density) of outcome ${\displaystyle E}$ when making a measurement on the quantum state ${\displaystyle |\psi \rangle }$.

This definition should be contrasted with that of the projection-valued measure, which is similar, except that for projection-valued measures, the values of ${\displaystyle F}$ are required to be projection operators.

Naimark's dilation theorem

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

Naimark's dilation theorem^[3] 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.^[4]: 285

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 ${\displaystyle \{F_{i}\}_{i=1}^{n}}$ is a POVM acting on a Hilbert space ${\displaystyle {\mathcal {H}}_{A}}$ of dimension ${\displaystyle d_{A}}$, then there exists a PVM ${\displaystyle \{\Pi _{i}\}_{i=1}^{n}}$ acting on a Hilbert space ${\displaystyle {\mathcal {H}}_{A'}}$ of dimension ${\displaystyle d_{A'}}$ and an isometry ${\displaystyle V:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A'}}$ such that for all ${\displaystyle i}$,

${\displaystyle F_{i}=V^{\dagger }\Pi _{i}V.}$

One way to construct such a PVM and isometry^[5][6] is to let ${\displaystyle {\mathcal {H}}_{A'}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}$, ${\displaystyle \Pi _{i}=\operatorname {I} _{A}\otimes |i\rangle \langle i|_{B}}$, and

${\displaystyle V=\sum _{i=1}^{n}{\sqrt {F_{i}}}_{A}\otimes {|i\rangle }_{B}.}$

The probability of obtaining outcome ${\displaystyle i}$ with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM:

${\displaystyle {\begin{aligned}{\text{Prob}}(i)&=\operatorname {tr} \left(V\rho _{A}V^{\dagger }\Pi _{i}\right)\\&=\operatorname {tr} \left(V\rho _{A}V^{\dagger }\left[\operatorname {I} _{A}\otimes |i\rangle \langle i|_{B}\right]\right)\\&=\operatorname {tr} \left(\rho _{A}\left(\sum _{j=1}^{n}{\sqrt {F_{j}}}_{A}^{\dagger }\otimes {\langle j|}_{B}\right)\operatorname {I} _{A}\otimes |i\rangle \langle i|_{B}\left(\sum _{k=1}^{n}{\sqrt {F_{k}}}_{A}\otimes {|k\rangle }_{B}\right)\right)\\&=\operatorname {tr} \left(\rho _{A}{\sqrt {F_{i}}}_{A}\operatorname {I} _{A}{\sqrt {F_{i}}}_{A}\right)\\&=\operatorname {tr} (\rho _{A}F_{i})\end{aligned}}}$

This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry ${\displaystyle V}$ into a unitary ${\displaystyle U}$, that is, finding ${\displaystyle U}$ such that

${\displaystyle V=U(\operatorname {I} _{A}\otimes |0\rangle _{B}).}$

This can always be done. The recipe for realizing the POVM measurement described by ${\displaystyle \{F_{i}\}_{i=1}^{n}}$ on a quantum state ${\displaystyle \rho }$ is then to prepare an ancilla in the state ${\displaystyle |0\rangle _{B}}$, evolve it together with ${\displaystyle \rho }$ through the unitary ${\displaystyle U}$, and make the projective measurement on the ancilla described by the PVM ${\displaystyle \{|i\rangle \langle i|_{B}\}_{i=1}^{n}}$.

Note that in this construction the dimension of the larger Hilbert space ${\displaystyle {\mathcal {H}}_{A'}}$ is given by ${\displaystyle d_{A'}=nd_{A}}$. This is not the minimum possible, as a more complicated construction gives ${\displaystyle d_{A'}=n}$ for rank-1 POVMs.^[4]: 285

An example: unambiguous quantum state discrimination

Bloch sphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states ${\displaystyle |\psi \rangle =|0\rangle }$ and ${\displaystyle |\varphi \rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}}$. Note that on the Bloch sphere orthogonal states are antiparallel.

Suppose you have a quantum system with a 2-dimensional Hilbert space that you know is in either the state ${\displaystyle |\psi \rangle }$ or the state ${\displaystyle |\varphi \rangle }$, and you want to determine which one it is. If ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\varphi \rangle }$ are orthogonal, this task is easy: the set ${\displaystyle \{|\psi \rangle \langle \psi |,|\varphi \rangle \langle \varphi |\}}$ will form a PVM, and a projective measurement in this basis will determine the state with certainty. If, however, ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\varphi \rangle }$ 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.^[2]: 87 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 ${\displaystyle |\psi \rangle }$ or ${\displaystyle |\varphi \rangle }$, at the cost of sometimes having an inconclusive result. It is possible to do this with projective measurements.^[7] For example, if you measure the PVM ${\displaystyle \{|\psi \rangle \langle \psi |,|\psi ^{\perp }\rangle \langle \psi ^{\perp }|\}}$, where ${\displaystyle |\psi ^{\perp }\rangle }$ is the quantum state orthogonal to ${\displaystyle |\psi \rangle }$, and obtain result ${\displaystyle |\psi ^{\perp }\rangle \langle \psi ^{\perp }|}$, then you know with certainty that the state was ${\displaystyle |\varphi \rangle }$. If the result was ${\displaystyle |\psi \rangle \langle \psi |}$, then it is inconclusive. The analogous reasoning holds for the PVM ${\displaystyle \{|\varphi \rangle \langle \varphi |,|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|\}}$, where ${\displaystyle |\varphi ^{\perp }\rangle }$ is the state orthogonal to ${\displaystyle |\varphi \rangle }$.

This is unsatisfactory, though, as you can't detect both ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\varphi \rangle }$ 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 ^[7][8]

${\displaystyle F_{\psi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|}$

${\displaystyle F_{\varphi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\psi ^{\perp }\rangle \langle \psi ^{\perp }|}$

${\displaystyle F_{?}=\operatorname {I} -F_{\psi }-F_{\varphi }.}$

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

The probability of having a conclusive outcome is given by

${\displaystyle 1-|\langle \varphi |\psi \rangle |,}$

when the quantum system is in state ${\displaystyle |\psi \rangle }$ or ${\displaystyle |\varphi \rangle }$ with the same probability. This result is known as the Ivanovic-Dieks-Peres limit, named after the authors who pioneered UQSD research.^[9][10][11]

Using the above construction we can obtain a projective measurement that physically realises this POVM. The square roots of the POVM elements are given by

${\displaystyle {\sqrt {F_{\psi }}}={\frac {1}{\sqrt {1+|\langle \varphi |\psi \rangle |}}}|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|}$

${\displaystyle {\sqrt {F_{\varphi }}}={\frac {1}{\sqrt {1+|\langle \varphi |\psi \rangle |}}}|\psi ^{\perp }\rangle \langle \psi ^{\perp }|}$

${\displaystyle {\sqrt {F_{?}}}={\sqrt {\frac {2|\langle \varphi |\psi \rangle |}{1+|\langle \varphi |\psi \rangle |}}}|\gamma \rangle \langle \gamma |,}$

where

${\displaystyle |\gamma \rangle ={\frac {1}{\sqrt {2(1+|\langle \varphi |\psi \rangle |)}}}(|\psi \rangle +e^{i\arg(\langle \varphi |\psi \rangle )}|\varphi \rangle ).}$

Labelling the three possible states of the ancilla as ${\displaystyle |{\text{result$ ?}}\rangle } , ${\displaystyle |{\text{result ψ}}\rangle }$, ${\displaystyle |{\text{result φ}}\rangle }$, and initializing it on the state ${\displaystyle |{\text{result$ ?}}\rangle } , we see that the resulting unitary ${\displaystyle U_{\text{UQSD}}}$ takes the state ${\displaystyle |\psi \rangle }$ together with the ancilla to

${\displaystyle U_{\text{UQSD}}(|\psi \rangle |{\text{result$ ?}}\rangle )={\sqrt {1-|\langle \varphi |\psi \rangle |}}|\varphi ^{\perp }\rangle |{\text{result ψ}}\rangle +{\sqrt {|\langle \varphi |\psi \rangle |}}|\gamma \rangle |{\text{result ?}}\rangle ,}

and similarly it takes the state ${\displaystyle |\varphi \rangle }$ together with the ancilla to

${\displaystyle U_{\text{UQSD}}(|\varphi \rangle |{\text{result$ ?}}\rangle )={\sqrt {1-|\langle \varphi |\psi \rangle |}}|\psi ^{\perp }\rangle |{\text{result φ}}\rangle +e^{-i\arg(\langle \varphi |\psi \rangle )}{\sqrt {|\langle \varphi |\psi \rangle |}}|\gamma \rangle |{\text{result ?}}\rangle .}

A measurement on the ancilla 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, using the path degree of freedom as an ancilla. The realisation of the POVM with a projective measurement was slightly different from the one described here.^[12][13]

See also

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

References

• POVMs
  • K. Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983).
  • E.B.Davies, Quantum Theory of Open Systems, Academic Press (1976).
  • A.S. Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland Publ. Cy., Amsterdam (1982).

External links

• Interactive demonstration about quantum state discrimination