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
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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.
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