Ismétlés. Mérések – átjáró a kvantum és a klasszikus világ között

Ismétlés

Mérések – átjáró a kvantum és a klasszikus világ között

Motto

"There are two possible outcomes: If the result confirms the hypothesis, then you’ve made a measurement. If the result is

contrary to the hypothesis, then you’ve made a discovery.“

Enrico Fermi

General Measurements"WHY must I treat the measuring device

classically?? What will happen 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 greatest facility new territories in our science,

and has rightly been called the royal road to our particular field of knowledge.”

Felix Klein

Measurement operators and the 3rd Postulate in 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 Postulate in case of projective measurements

• Checking the Completeness relation

• Practical notation

• Conclusion

Measurement operators and the 3rd Postulate in case of projective measurements

• belong to a special set of operators called projectors

• Properties

Measurement operators and the 3rd Postulate in case of projective measurements

• 3rd Postulate with projectors

Measurement operators and the 3rd Postulate in case of projective measurements

• Direct construction approach

• Indirect construction approach

Measurement using the computational basis states

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

• Basis vectors and

• Measurement statistic

Measurement using the computational basis states

• Post measurements states

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

Observable and projective measurements

• Any observable can be represented by means of a Hermitian operator whose eigenvalues refer to the possible values of that observable

• Expected value of such an observable can be calculated in an easy way

Repeated projective measurement

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

• Post measurement state after the first measurement

• Post measurement state after the second measurement

Schrödinger’s cat

"When I hear about Schrödinger’s cat, I reach for my gun."Stephen Hawking

Source: http://www.cbs.dtu.dk/staff/dave/roanoke/schrodcat01.gif

EPR paradox

• Einstein: “spooky action at a distance“• Eintein: “God does not play dice with the universe.”

• Bhor’s answer: "Quit telling God what to do!"• Born’s answer: "If God has made the world a perfect

mechanism, He has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success".

The EPR paradox and the Bell inequality

"’Obvious’ is the most dangerous word in mathematics."Eric Temple Bell

• EPR paradox: Entanglement seems to contradict to limited speed of information transfer (light).

• Einstein: Therefore quantum mechanics provides an incomplete description of the Nature. Hidden variables!!!!

• Bell inequality: gives unambiguously different results in case hidden variables exist or not.

• Problem: hard to test it in practice.• Solution: Clauser-Horne-Schimony-Holt (CHSH) inequality

CHSH inequality if hidden variables exist 1

• The game: Alice and Bob investigate book titles whether certain letters are contained or not: a,b,c,d

• ‘winning’ function

Causally disconnected

rooms with identical book shelves

Alice Bob

a or b c or d

CHSH inequality if hidden variables exist 2

CHSH inequality with entangled particles

• We return to Bell inequalities and investigate CHSH inequality in the nano-world. We replace books with entangled pairs

• Alice and Bob check the following observables

CHSH inequality with entangled particles

CHSH inequality if hidden variables do not exist

CHSH inequality with entangled particles

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

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

• Consequence: HIDDEN VARIABLES DO NOT EXIST!• What may be wrong with our local realism picture?

– Locality – Realism– Logic

Positive Operator Valued Measurements

Motivations to use POVM

• In certain cases either we are not interested in the post measurement state or we are not able at all to get 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 be distinguished?

• The answer is definitely NOT because

• Remark: No set of measurement operators exist which is able to distinguish non-orthogonal states unambiguously. This is another explanation why non-orthogonal states can not be copied as was stated earlier, because of lack of exact information about such states 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 to ensure 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 and 1 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 - false alarm vs. not happen alarm

• It is assumed that detecting 1 correctly is much more important

Copyright © 2005 John Wiley & Sons Ltd.

Generalisation of POVM

Relations among the measurement types

• Projective measurement can be regarded as a special POVM.• Clearly speaking POVM is a generalised measurement

without the interest of post-measurement state + construction rules.

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

Relations among the measurement types

Copyright © 2005 John Wiley & Sons Ltd.

Quantum computing-based solution of the game with marbles

• We exploit entanglement and the orthogonality between Bell states

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

Quantum computing-based solution of the game with marbles

Copyright © 2005 John Wiley & Sons Ltd.