Download Projective Measurements

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
Alice
Bob
Causally
disconnected
rooms with identical
book shelves
a or b
• ‘winning’ function
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
"Victorious warriors win first and then go to war, while defeated warriors go
to war first and then seek to win.” Sun Tzu
• We exploit entanglement and the orthogonality between Bell
states
Quantum computing-based solution of
the game with marbles
Copyright © 2005 John Wiley & Sons Ltd.