Slides forQuantum Computing and Communications – An

Engineering Approach

Chapter 3Measurements

Sándor ImreFerenc Balázs

Motto

"There are two possible outcomes: If the resultconfirms the hypothesis, then you’ve made a

measurement. If the result is contrary to thehypothesis, then you’ve made a discovery.“

Enrico Fermi

General Measurements

"WHY must I treat the measuring device classically?? What willhappen to me if I don’t??"

Eugene Wigner

3rd Postulate using ket notations

• Measurement statistic

• Post measurement state

• Completeness relation

Projective Measurements(Neumann Measurements)

“Projective geometry has opened up for us with the greatestfacility new territories in our science, and has rightly beencalled the royal road to our particular field of knowledge.”

Felix Klein

Measurement operators and the 3rd Postulatein case of projective measurements

• Set of two orthogonal states

• To find we need to solve

• We are looking in the form of

• Similarly

Measurement operators and the 3rd Postulatein case of projective measurements

• Checking the Completeness relation

• Practical notation

• Conclusion

Measurement operators and the 3rd Postulatein case of projective measurements

• belong to a special set of operators calledprojectors

• Properties

Measurement operators and the 3rd Postulatein case of projective measurements

• 3rd Postulate with projectors

Measurement operators and the 3rd Postulatein case of projective measurements

• Direct construction approach

• Indirect construction approach

Measurement using the computationalbasis states

• Let us check what we have learned by means of asimple example

• Basis vectors and

• Measurement statistic

Measurement using the computationalbasis states

• Post measurements states

• Remark: Orthogonal states can always be distinguished viaconstructing appropriate measurement operators (projectors).This is another explanation why orthogonal (classical) states canbe copied as was stated in Section 2.7 because in possession ofthe exact information about such states we can build a quantumcircuit producing them.

Observable and projectivemeasurements

• Any observable can be represented by means of aHermitian operator whose eigenvalues refer to thepossible values of that observable

• Expected value of such an observable can becalculated in an easy way

Repeated projective measurement

• What happens when we repeat a projectivemeasurement on the same qreqister?

• Post measurement state after the first measurement

• Post measurement state after the secondmeasurement

CHSH inequality with entangledparticles

• We return to Bell inequalities and investigate CHSHinequality in the nano-world. We replace books withentangled pairs

• Alice and Bob check the following observables

CHSH inequality with entangledparticles

CHSH inequality with entangledparticles

• Finally we obtain which is obviouslygreater than the classical result 2.

• Experiments shore up the quantum model instead ofthe classical one!

• Hidden variables do not exist!

Positive Operator Valued Measurements

Good quotation is requested. If you have one please, send toimre@hit.bme.hu

Motivations to use POVM

• In certain cases either we are not interested in thepost measurement state or we are not able at all toget it (e.g. a photon hits the detector).

• Important properties of

POVM and the 3rd Postulate

• Measurement statistic

• Post measurement state

Unknown and/or indifferent!

• Completeness relation

Can non-orthogonal states bedistinguished?

• The answer is definitely NOT because

• Remark: No set of measurement operators exist which is able todistinguish non-orthogonal states unambiguously. This is anotherexplanation why non-orthogonal states can not be copied as wasstated earlier, because of lack of exact information about suchstates we can not build a quantum circuit to produce them.

POVM construction example

• Our two non-orthogonal states are:

• Based on the indirect construction approach we try toensure that

• To achieve this we need

POVM construction example

• To fulfil the completeness relation

• Completeness relation OK since

D2 must be positive semi-definite• is very promising, but unfortunately wrong!

How to apply POVM operators - 0 and1 has the same importance

• It requires

How to apply POVM operators -minimising uncertainty

• Goal: to minimise

• Assuming

How to apply POVM operators -minimising uncertainty

Copyright © 2005 John Wiley & Sons Ltd.

How to apply POVM operators - falsealarm vs. not happen alarm

• It is assumed that detecting 1 correctly is much moreimportant

Copyright © 2005 John Wiley & Sons Ltd.

Generalisation of POVM

Relations among the measurement types

• Projective measurement can be regarded as a specialPOVM.

• Clearly speaking POVM is a generalised measurementwithout the interest of post-measurement state +construction rules.

• Neumark’s extension: any generalised measurementcan be implemented by means of a projectivemeasurement + auxiliary qbits + unitary transform.

Relations among the measurement types

Copyright © 2005 John Wiley & Sons Ltd.

Quantum computing-based solution ofthe game with marbles

• We exploit entanglement and the orthogonalitybetween Bell states

"Victorious warriors win first and then go to war, while defeated warriors goto war first and then seek to win.” Sun Tzu

Quantum computing-based solution ofthe game with marbles

Copyright © 2005 John Wiley & Sons Ltd.