POVM

Summary

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

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Let   denote a Hilbert space and   a measurable space with   a Borel σ-algebra on  . A POVM is a function   defined on   whose values are positive bounded self-adjoint operators on   such that for every  

 

is a non-negative countably additive measure on the σ-algebra   and   is the identity operator.[2]

In quantum mechanics, the key property of a POVM is that it determines a probability measure on the outcome space, so that   can be interpreted as the probability of the event   when measuring a quantum state  .

In the simplest case, in which   is a finite set,   is the power set of   and   is finite-dimensional, a POVM is equivalently a set of positive semi-definite Hermitian matrices   that sum to the identity matrix,[3]: 90 

 

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

In the discrete case, the POVM element   is associated with the measurement outcome  , such that the probability of obtaining it when making a quantum measurement on the quantum state   is given by

 ,

where   is the trace operator. When the quantum state being measured is a pure state   this formula reduces to

 .

The discrete case of a POVM generalizes the simplest case of a PVM, which is a set of orthogonal projectors   that sum to the identity matrix:

 

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

Naimark's dilation theorem

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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]: 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   is a POVM acting on a Hilbert space   of dimension  , then there exists a PVM   acting on a Hilbert space   of dimension   and an isometry   such that for all  ,

 

For the particular case of a rank-1 POVM, i.e., when   for some (unnormalized) vectors  , this isometry can be constructed as[5]: 285 

 

and the PVM is given simply by  . Note that here  .

In the general case, the isometry and PVM can be constructed by defining[6][7]  ,  , and

 

Note that here  , so this is a more wasteful construction.

In either case, the probability of obtaining outcome   with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM:

 

This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry   into a unitary  , that is, finding   such that

 

for   from 1 to  . This can always be done.

The recipe for realizing the POVM described by   on a quantum state   is then to embed the quantum state in the Hilbert space  , evolve it with the unitary  , and make the projective measurement described by the PVM  .

Post-measurement state

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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   alone do not determine what the post-measurement state will be. To see that, note that for any unitary   the operators

 

will also have the property that  , so that using the isometry

 

in the second construction above will also implement the same POVM. In the case where the state being measured is in a pure state  , the resulting unitary   takes it together with the ancilla to state

 

and the projective measurement on the ancilla will collapse   to the state[3]: 84 

 

on obtaining result  . When the state being measured is described by a density matrix  , the corresponding post-measurement state is given by

 .

We see therefore that the post-measurement state depends explicitly on the unitary  . Note that while   is always Hermitian, generally,   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   was obtained, the probability of obtaining a different result   on a second measurement is

 ,

which can be nonzero if   and   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

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Bloch sphere representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states   and  . 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   or the state  , and you want to determine which one it is. If   and   are orthogonal, this task is easy: the set   will form a PVM, and a projective measurement in this basis will determine the state with certainty. If, however,   and   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.[3]: 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   or  , at the cost of sometimes having an inconclusive result. It is possible to do this with projective measurements.[8] For example, if you measure the PVM  , where   is the quantum state orthogonal to  , and obtain result  , then you know with certainty that the state was  . If the result was  , then it is inconclusive. The analogous reasoning holds for the PVM  , where   is the state orthogonal to  .

This is unsatisfactory, though, as you can't detect both   and   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 [8][9]

 
 
 

where

 

Note that  , so when outcome   is obtained we are certain that the quantum state is  , and when outcome   is obtained we are certain that the quantum state is  .

The probability of having a conclusive outcome is given by

 

when the quantum system is in state   or   with the same probability. This result is known as the Ivanović-Dieks-Peres limit, named after the authors who pioneered UQSD research.[10][11][12]

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  ,  , and  , we see that the resulting unitary   takes the state   to

 

and similarly it takes the state   to

 

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.[13][14]

See also

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References

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  1. ^ Peres, Asher; Terno, Daniel R. (2004). "Quantum information and relativity theory". Reviews of Modern Physics. 76 (1): 93–123. arXiv:quant-ph/0212023. Bibcode:2004RvMP...76...93P. doi:10.1103/RevModPhys.76.93. S2CID 7481797.
  2. ^ Davies, Edward Brian (1976). Quantum Theory of Open Systems. London: Acad. Press. p. 35. ISBN 978-0-12-206150-9.
  3. ^ a b c M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, (2000)
  4. ^ I. M. Gelfand and M. A. Neumark, On the embedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213.
  5. ^ a b A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993.
  6. ^ J. Preskill, Lecture Notes for Physics: Quantum Information and Computation, Chapter 3, http://theory.caltech.edu/~preskill/ph229/index.html
  7. ^ J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 2.3, https://cs.uwaterloo.ca/~watrous/TQI/
  8. ^ a b J.A. Bergou; U. Herzog; M. Hillery (2004). "Discrimination of Quantum States". In M. Paris; J. Řeháček (eds.). Quantum State Estimation. Springer. pp. 417–465. doi:10.1007/978-3-540-44481-7_11. ISBN 978-3-540-44481-7.
  9. ^ Chefles, Anthony (2000). "Quantum state discrimination". Contemporary Physics. 41 (6). Informa UK Limited: 401–424. arXiv:quant-ph/0010114v1. Bibcode:2000ConPh..41..401C. doi:10.1080/00107510010002599. ISSN 0010-7514. S2CID 119340381.
  10. ^ Ivanovic, I.D. (1987). "How to differentiate between non-orthogonal states". Physics Letters A. 123 (6). Elsevier BV: 257–259. Bibcode:1987PhLA..123..257I. doi:10.1016/0375-9601(87)90222-2. ISSN 0375-9601.
  11. ^ Dieks, D. (1988). "Overlap and distinguishability of quantum states". Physics Letters A. 126 (5–6). Elsevier BV: 303–306. Bibcode:1988PhLA..126..303D. doi:10.1016/0375-9601(88)90840-7. ISSN 0375-9601.
  12. ^ Peres, Asher (1988). "How to differentiate between non-orthogonal states". Physics Letters A. 128 (1–2). Elsevier BV: 19. Bibcode:1988PhLA..128...19P. doi:10.1016/0375-9601(88)91034-1. ISSN 0375-9601.
  13. ^ B. Huttner; A. Muller; J. D. Gautier; H. Zbinden; N. Gisin (1996). "Unambiguous quantum measurement of nonorthogonal states". Physical Review A. 54 (5). APS: 3783–3789. Bibcode:1996PhRvA..54.3783H. doi:10.1103/PhysRevA.54.3783. PMID 9913923.
  14. ^ R. B. M. Clarke; A. Chefles; S. M. Barnett; E. Riis (2001). "Experimental demonstration of optimal unambiguous state discrimination". Physical Review A. 63 (4). APS: 040305(R). arXiv:quant-ph/0007063. Bibcode:2001PhRvA..63d0305C. doi:10.1103/PhysRevA.63.040305. S2CID 39481893.
  • POVMs
    • K. Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983).
    • A.S. Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland Publ. Cy., Amsterdam (1982).
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  • Interactive demonstration about quantum state discrimination