The mysterious world of
quantum computing
____________________________________________
Rajendra K. Bera, PhD
Honorary Professor, International Institute of Information Technology, Bangalore
Chief Mentor, Acadinnet Education Services India Pvt. Ltd., Bangalore
IEEE Workshop on Modern Computing Trends
Basaveshwar Engineering College, Bagalkot
October 16, 2011
Classical mechanics
_________________________________________
A perplexing aspect of quantum mechanics is that it defies an intuitive
understanding. It is so different from classical physics as built by Newton,
Maxwell, and Einstein. Laws of classical physics are deterministic in the
sense that given, say, Newton’s laws of motion, and initial conditions
(position and momentum) at some instant t = 0 for a system and a time
history of the force(s) acting on the system, we can, in principle, accurately
predict the state of that system at any time in the past or the future. In
principle, at least, we can measure the state of the system (position and
momentum) without disturbing it.
2
Quantum mechanics
_________________________________________
In quantum mechanics, the situation is completely different. The counterpart of Newton’s laws of motion for a quantum system is the Schrödinger’s
equation, and the state of the system is described by something called the
“wave function”,, which no one understands intuitively. It is so abstract
that we understand it only in a mathematical sense.
It has not been possible for physicists (or anyone else for that matter) to
understand the wave function in any other way. If we try to measure the
state of a quantum system, hell breaks loose; we have no way of deterministically predicting what the result of a measurement will be! And , even
in principle, there is no way we can measure a quantum system without
disturbing it. That is why physics is divided into two parts: classical
physics, and quantum physics.
3
Quantum measurement is a mystery
_________________________________________
No one knows what transitory changes a quantum system undergoes when
it is measured. We do know, however, that while we cannot make a
deterministic prediction of the result of a measurement, we can make an
amazingly accurate probabilistic prediction of it. I and a former student of
mine, Vikram Menon, have come up with a hypothesis to explain this very
unusual aspect of quantum systems. You can look up our paper at arXiv:
Bera, R.K., and Menon, V., A new interpretation of superposition, entanglement, and measurement in
quantum mechanics, arXiv:0908.0957v1 [quant-ph], 07 August 2009, at http://arxiv.org/abs/0908.0957.
The probabilistic aspect of quantum mechanics is intriguing because
Schrödinger’s equation has no built in probabilities; indeed it produces only
deterministic results! So where did the probabilities come in?
4
Measurement is probabilistic
_________________________________________
The probabilities came in because a bunch of physicists, sometime in the 1920s,
said so! (This became known as the Copenhagen interpretation of quantum
mechanics.) They looked at available experimental data, and they found that the
results of measurements carried out on quantum systems follow an unusual
probabilistic pattern.
Just as Isaac Newton observed that material things are gravitationally attracted to
each other (but only “God” knows why) and stated it as a fundamental law of
nature, so did Max Born* state this probabilistic aspect of quantum systems as a law
of quantum mechanics.
It is extremely important to note that laws of nature are like the man-made axioms
in mathematics. We do not know (and can never know) why the laws are as they are.
Only “God” can enlighten you. We can only marvel at the intellectual genius of
those physicists who are able to read the mind of “God”.
shared the Nobel Prize in Physics, 1954 (with Walther Bothe) “for his fundamental research in
quantum mechanics, especially for his statistical interpretation of the wavefunction”.
*Born
5
Axioms of quantum mechanics
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Here are the laws (or postulates or axioms) of quantum mechanics stated informally.
 Quantum mechanics describes a physical system through a mathematical
object called the state vector (or the wave-function) |. | is complex (i.e., it
has real and imaginary parts) and a vector of unit length.
 | evolves in a deterministic manner according to the linear Schrödinger
equation:

2 2

   V   i
2m
t
where    is the wave function, is thereduced Planck's constant, 2 is the
Laplacianoperator hat
t describes how the wave functionchangesfromone place
to another,V describes theforcesactingon theparticle,m is themass of theparticle
being described, and  t describes how the wave functionchangesits shape with time.
| remains a unit vector during its evolution, only its orientation changes.
6
Axioms of quantum mechanics (contd)
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 Any measurement made on a quantum system leads to the (non-unitary)
irreversible collapse of its wave function to a new state described by a
probability rule.
 The state space of a composite quantum system is the tensor product of the
state spaces of the component quantum systems.
It is only when measurements are made that indetermination and probabilities come
into quantum theory. Otherwise, things are very deterministic. The collapse of the
wave function involves no forces of any kind but it does involve loss of information.
Warning: Unless you understand linear algebra and complex variable theory, you will not understand
quantum mechanics.
7
Axioms of quantum mechanics (contd)
_________________________________________
The axioms mean, e.g., that the wave function of a system, say, with two possible
states |F and |G can be described by the linear combination
| = a|F + b|G
where a and b are complex constants. After the ‘measurement’, either | = |F or |
= |G and these alternatives occur with certain probabilities as noted below:
probability |F : probability |G = |a|2 : |b|2,
|a|2 + |b|2 = 1.
The collapse of the wave function seems to happen instantly unlike the ordinary time
evolution of quantum states (according to the Schrödinger’s equation). We still do
not understand the physical mechanism, which causes the collapse.
Complex linear superposition of states and collapse of wave functions are unusual
features of quantum mechanics.
8
Axioms of quantum mechanics (contd)
_________________________________________
If | has n possible states |1, |2, , |n, it can be described at any instant in
time by some unique linear combination as
| = a1 |1 + a2 |2 +  + an |n
where a1, a2, , an, are complex constants, which may change with time. After a
‘measurement’, | will collapse to | = |i with the index i occurring with certain
probabilities as noted below:
prob |1 : prob |2 :  : prob |n = |a1|2 : | |a2|2 :  : |an|2,
|a1|2 + |a2|2 +  + |an|2 = 1.
When | changes, either due to Schrödinger evolution or measurement, only the ais
change such that the sum of the |ai|2 remains 1. | therefore remains a unit vector,
and only its orientation changes.
9
Weirdness of quantum mechanics
_________________________________________
Here is an imperfect example of how weird
things can be.
What did you see?
Question: Just before you realized what the
picture represents, was your brain in a state
of superposition? Were you in two minds?
Hopefully, you now have some understanding
of what one means by superposition and wavefunction collapse in quantum mechanics.
My wife and my mother-in-law.
(W.E. Hill (1915))
http://www.at-ristol.org.uk/Optical/Wife_main.htm
10
Unitary transformation in QM
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The evolution of the wave function, alternatively, can be described in matrix form
due to Heisenberg, which is the form used in quantum computing. In this form, the
evolution of a closed quantum system is described by a unitary transformation. That
is, the state |(t1) of the system at time t1 is related to the state |(t2) of the system
at time t2 by a unitary operator U which depends only on the times t1 and t2,
|(t2) = U |(t1).
A linear operator U whose inverse is its adjoint (conjugate transpose, U†) is called
unitary, that is, U†U = UU† = I, where U†  (U*)T  (UT)*. By definition, unitary
operators are invertible. Also, by definition, a unitary operator does not change the
length of the state vector it acts upon; it only changes that vector’s orientation. This
means that if
|(t1) = a1|a + b1|b with |a1|2 + |b1|2 = 1 then
|(t2) = a2|a + b2|b with |a2|2 + |b2|2 = 1.
11
Quantum computing
_________________________________________
Quantum computing is about computing with quantum systems using the rules
of quantum mechanics rather than the rules of classical mechanics. The
important quantum mechanical phenomena that come into play in the building
of a quantum computer are:
 Superposition
 Entanglement
 Decoherence
On a quantum computer, programs are executed by unitary evolution of an
input that is given by the state of the system. Since all unitary operators are
invertible, we can always reverse or ‘uncompute’ a computation on a quantum
computer. (We can do this using classical physics also. See: Bennett, C. H.,
The Thermodynamics of Computation – a Review, International Journal of
Theoretical Physics, Vol. 21, No. 12, 1982, pp. 905-940.)
12
Quantum measurement
_________________________________________
In Slide 10, you got a glimpse of the weirdness of QM. I shall now elaborate.
Quantum mechanics differs from classical mechanics in the way we interpret the
mathematical results. This interpretation is intimately tied with the law related to the
measurement of quantum systems. That law is probabilistic in nature. It says that when
a quantum system is measured, the wave function |, in general, collapses to a new
state according to a probabilistic rule. That is, if
| = a|F + b|G
then after the ‘measurement’, either | = |F or | = |G and these alternatives occur
with certain probabilities.
probability |F : probability |G = |a|2 : |b|2.
( |a|2 + |b|2 = 1 ).
A quantum measurement never produces | = a|F + b|G; a classical measurement
does, given an appropriate measuring system.
13
Quantum superposition
_________________________________________
Superposition: Let us look at the picture “My wife, and my mother-in-law”. You cannot really
define a linear combination (superposition) of
| = a|F + b|G = a|my wife + b|my mother-in-law,
in a physical sense, yet you know that in some “complex” sense the two people are superposed
in the picture (i.e., they exist simultaneously at the same place and time). You recognize the
superposition at an intellectual level, but not at the measurement level (vision); you see the
picture “collapsing” to one or the other person. In quantum mechanics, the measurement
operator is like a prism—it splits the wavefunction into its component parts (akin to white light
being split into its rainbow components by a prism). In our example, the “prism” would split the
picture into |my wife and |my mother-in-law and the probability with which we will see one or
the other is given by
Prob. of seeing |my wife : prob. of seeing |my mother-in-law = |a|2 : |b|2.
( |a|2 + |b|2 = 1 ).
You will not see some weird hybrid form of “my wife” and “my mother-in-law”.
14
Quantum entanglement
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Entanglement: This is a strange state of being in which two particles are so deeply connected
that they share the same existence, even when light years apart. Indeed, distance has no meaning
for entangled particles. If the state of one is changed, the state of the other is instantly adjusted
to be consistent with quantum mechanical rules. If a measurement is made on one, the other will
automatically collapse. Entanglement is a form of quantum superposition. There is no easy
explanation of entangled correlations. There is no counterpart of entanglement in classical
mechanics.
Entanglement is a joint characteristic of two or more quantum particles.
Einstein called such action-at-a-distance ‘spooky’.
“I cannot seriously believe in [the quantum theory] because it cannot be reconciled
with the idea that physics should represent a reality in time and space, free from
spooky actions at a distance.” So wrote Einstein to Max Born in March 1947.
[Quote taken from http://www.science-frontiers.com/sf114/sf114p12.htm, William R. Corliss, Science Frontiers #114, Nov-Dec
1997.]
15
Quantum decoherence
_________________________________________
Decoherence: It is the spontaneous interaction between a quantum system and its environment.
This interaction destroys quantum superposition. The reason why quantum computers still have
a long way to go beyond laboratory experimentation is that superposition and entanglement are
extremely fragile states. Any interaction with the environment and the particles decohere.
Preventing decoherence from taking hold before a calculation is completed remains the biggest
challenge in building quantum computers.
16
Physical laws are mathematical
_________________________________________
When we say F = ma expresses Newton’s second law of motion, what we
mean is that if you interpret F as representing a force (a vector with 3-scalar
components), m representing the mass of a material body, and a representing
the acceleration of that material body, then we can very accurately compute
the motion of that material body. F = ma means nothing until we give it an
interpretation. Surprisingly, all the important laws of physics can be precisely
stated in mathematical form. This fact led the 1963 Nobel Laureate in
physics, Eugene Paul Wigner, to comment in wonder, The Unreasonable
Effectiveness of Mathematics in the Natural Sciences.*
It turns out that without knowing mathematics, you cannot develop a deep
understanding of physics. This is particularly true for quantum mechanics.
*Eugene
P. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications
in Pure and Applied Mathematics, Vol. 13, No. 1 (February 1960), available at http://www.physik.uniwuerzburg.de/fileadmin/tp3/QM/wigner.pdf.
17
Interpretation of Schrödinger’s equation
_________________________________________
Recall Schrödinger’s equation
2 2


   V   i
.
2m
t
In quantum mechanics we really do not know what the wave function  means. Yet,
in Schrödinger’s equation, when we say, that is, make the following interpretation:
time:
position:
momentum:
energy:
t  t,
r  r,
p  i,
E  i( / t ),
we make a connection with the physical world. Note
that we are interpreting two mathematical operators as
physical variables: the operator i is interpreted to
represent the physical variable momentum p and the
operator i(/t) as the physical variable energy E! Now
you can understand why quantum mechanics is so
difficult to understand. It requires a lot of imagination
to connect the real world with mathematics.
18
Why quantum systems are quantized
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We now have an interesting situation. Let
2 2
ˆ
ˆ is theHamiltonian operator.
H
  V , where H
2m
Then Schrödinger’s equation becomes, in new symbols:
ˆ   E .
H
Since the Hamiltonian operator can, alternatively, be written as a square matrix, we
have an equation of the form we know as an eigenvalue problem in linear algebra:
Ax = x.
This equation has the trivial solution, x = 0, but it also has non-trivial solutions for
certain discrete values of  (which we call the eigenvalues of A). So now you see why,
for a finite quantum system, energy is quantized. The E values are the eigenvalues of
the Hamiltonian operator! A quantum system is quantized because the Schrödinger’s
equation describes an eigenvalue problem.
19
Heisenberg’s uncertainty principle
_________________________________________
We now find another intriguing aspect of quantum mechanics. This comes from the
fact that operators, do not always commute. Thus, pr  rp, because
 ir  r(i).
This means that the position r and momentum p that you measure of a particle will
depend on the sequence in which you measure them. This is the reason why you
cannot measure both position and momentum of a quantum particle with absolute
accuracy. And this fact is responsible for the famous Heisenberg’s uncertainty
principle in quantum mechanics, which says:
p q  ħ/2,
where q is the error in the measurement of any coordinate and p is the error in its
canonically conjugate momentum. In quantum mechanics, position and momentum of
a particle are complementary variables.
20
Heisenberg’s uncertainty principle (contd)
_________________________________________
The deep significance of the uncertainty principle is that we cannot observe a
quantum system without changing it. The independent observer, watching from the
sidelines without influencing the observed phenomenon, simply does not exist. The
uncertainty principle essentially says that even in principle it is not possible to know
enough about the present to make a complete prediction about the future. This is not
so in Newtonian mechanics.
The measurement limits imposed by the uncertainty principle cannot be overcome by
refining measurement technology; it is a limit imposed by Nature as we penetrate into
the subatomic world. Classical and quantum particles are entirely different entities.
The principle sets limits on precision technologies, e.g., metrology and lithography.
However, 
21
Heisenberg’s uncertainty principle (contd)
_________________________________________
 if the particle is prepared entangled with a quantum memory (such as an optical
delay line) and the observer has access to the particle stored in the quantum memory,
it is possible to predict the outcomes for both measurement choices precisely.1 This is
a more general uncertainty relation, formulated in terms of entropies. This new
relation has recently been verified experimentally.2,3
1M.
Berta, et al, The uncertainty principle in the presence of quantum memory, Nature Physics, Vol. 6, Issue 9, September 2010,
pp. 659-662.
2Shuan-Feng Li, et al, Experimental investigation of the entanglement-assisted entropic uncertainty principle, Nature Physics,
Vol. 7, Issue 10, October 2011, pp. 752-756.
3Robert Prevedel, et al, Experimental investigation of the uncertainty principle in the presence of quantum memory and its
application to witnessing entanglement, Nature Physics, Vol. 7, Issue 10, October 2011, pp. 757-761.
22
No cloning, no deletion
_________________________________________
That quantum operators are unitary, presents another unusual consequence. One is
that if you do not know the state of a quantum system (even if it is a single particle)
then you cannot make an exact copy of it. This is known as the no-cloning theorem1.
(You can, of course, prepare a particle in any desired state and make as many copies
of it as you like.)
The other is that unless a quantum system collapses, you cannot delete information in
a quantum system. This is known as the no-deletion theorem2.
These results are connected with the quantum phenomenon called entanglement.
Quantum algorithms make very clever use of quantum superposition and quantum
entanglement.
______________________
1
Wootters, W. K., and Zurek, W. H., A single quantum cannot be cloned, Nature, Vol. 299, 28 October 1982, pp. 802-3,
http://pm1.bu.edu/~tt/qcl/pdf/wootterw198276687665.pdf.
2Pati,
A. K., and Braunstein, S. L., Impossibility of deleting an unknown quantum state, Nature, Vol. 404, 9 March 2000, pp.
164-5, http://www-users.cs.york.ac.uk/~schmuel/papers/pb00.pdf.
23
Ingredients of quantum algorithms
_________________________________________
Let me now tell you something about quantum algorithms. All quantum algorithms
make clever use of choosing and sequencing unitary operators (i.e., moving the
system according to Schrödinger’s equation), making measurements (i.e., collapsing
the system), and for non-trivial quantum algorithms, making very imaginative use of
superposition and entanglement.
A single qubit (the quantum analogue of the classical bit) is the simplest quantum
system we can think of. Mathematically, a qubit is described as a unit vector | =
a|0 + b|1, parameterized by two complex numbers a and b, satisfying |a|2 + |b|2 = 1.
While the qubit can be in either state |0 or state |1 (analogous to the 0-1 states of a
classical bit), it can also be in a superposed state of a|0 + b|1 (which a classical bit
can never be in).
24
Manipulating single qubits
_________________________________________
Any unitary operator M used to manipulate the state of a qubit can be represented
by a linear combination of 4 unitary operators (I, X, Y, Z)  (0, 1, 2, 3), i.e.,
M =  I +  X +  Y +  Z,
where , , ,  are complex constants.
Operation
Operator Labels
Post operation qubit state
Identity
0, I
|0  |0; |1  |1
Negation
1, X
|0  |1; |1  |0
ZX
2, Y
|0  |1; |1  |0
Phase shift
3, Z
|0  |0; |1  |1
Note the very important Hadamard gate below.
Hadamard
When a measurement on a qubit is made,
the state of that qubit will collapse, in a
probabilistic manner, to either | = |0 or
| = |1 according to the rule:
probability |0 : probability |1 = |a|2 : |b|2.
( |a|2 + |b|2 = 1 ).
H = (X + Z)/2
|0  (|0 + |1)/2  |+
|1  (|0  |1)/2  |
The operators (I, X, Y, Z) are called Pauli
matrices, and in their alternative symbolic
form (0, 1, 2, 3) they are called sigma
matrices. The operators are 22 matrices.
The Hadamard gate is a very important and
often used gate.
25
The amazing H-gate
_________________________________________
The action of H gate on a qubit is such that
and
|0  (|0 + |1)/2
(|0 + |1)/2  |0
|1  (|0  |1)/2
(|0  |1)/2  |1
_________________________________________________
After a qubit in state |0 or |1 has been acted upon by a H gate, the state of
the qubit is an equal superposition of |0 and |1. Thus the qubit goes from a
deterministic state to a truly random state, i.e., if the qubit is now measured,
we will measure |0 or |1 with equal probability.
We see that H is its own inverse, that is, H1 = H or H2 = I. Therefore, by
applying H twice to a qubit we change nothing. This is amazing.
By applying a randomizing operation to a random state produces a
deterministic outcome!
26
The Cnot-gate
_________________________________________
The Cnot gate acts on a qubit-pair such that
|00  |00, |01  |01, |10  |11, |11  |10
Note that the state of the first qubit, called the control qubit, does not change while the state of
the second qubit, called the target qubit, changes only if the control qubit is in state |1.
This signifies the exclusive-or (XOR) operation (that is, the output is “true”
if and only if exactly one of the operands has a value of “true”).
In quantum computing we rely on the following facts:
(1) All quantum gates are reversible.
(2) Any unitary operation on n qubits can be implemented exactly by
stringing together operations composed of 1-qubit Pauli operators and 2qubit controlled-NOT gates.
27
Entangling qubits
_________________________________________
An application of the H-gate on the first qubit followed by the Cnot-gate to a 2-qubit
system (with the first qubit as the control qubit) gives the following results when the
system’s initial state is |00, |01, |10, and |11, respectively:
Cnot (H I ) (|00) = Cnot (|00 + |10)/√2 = (|00
Cnot (H I ) (|01) = Cnot (|01  |11)/√2 = (|01
Cnot (H I ) (|10) = Cnot (|00 + |10)/√2 = (|00
Cnot (H I ) (|11) = Cnot (|01  |11)/√2 = (|01
+ |11)/√2
 |10)/√2
+ |11)/√2
 |10)/√2
The resulting states (in blue) are called Bell states (after John Bell).
They are interesting because they are all entangled states, that is, they cannot be
attained by manipulating each qubit using the 1-qubit Pauli operators alone. If the
states of the entangled particles are used to encode bits, then the entangled joint state
represents what is called an ebit. Its state is always distributed between two qubits.
The states of these qubits are correlated, but undetermined until measured.
28
3-qubit Toffoli gate
_________________________________________
The T gate (named after Tommaso Toffoli who invented it) acts on a qubit-triplet
such that
|000  |000
|001  |001
|010  |010
|011  |011
|100  |100
|101  |101
|110  |111
|111  |110
It can be viewed as a controlled-controlled-NOT gate, which negates the last of three
bits, if and only if the first two are 1. The Toffoli gate is its own inverse.
Toffoli gates can be constructed using six Cnot gates and several 1-qubit gates.*
*See,
e.g., Vivek V. Shinde and Igor L. Markov, On the CNOT-cost of TOFFOLI gates, 15 March 2008, arXiv, available at
http://arxiv.org/abs/0803.2316; also as Quant. Inf. Comp. 9(5-6):461-486 (2009).
29
Swapping algorithm
_________________________________________
The state of two qubits can be swapped by applying the Cnot gate thrice.
Examples
|01  |01
 |11
 |10
with the first qubit as control
with the second qubit as control
with the first qubit as control
|10  |11
 |01
 |01
with the first qubit as control
with the second qubit as control
with the first qubit as control
|11  |10
 |10
 |11
with the first qubit as control
with the second qubit as control
with the first qubit as control
The Cnot gate has no effect on |00.
30
Computing xy (x AND y)
_________________________________________
Take 3 qubits, each prepared in state |0, i.e., | = |000. The first qubit is the
placeholder for x, the second for y, and the third for the result of x  y. To create the
4 possible inputs of x and y, apply the Hadamard gate to the first two qubits
|1 = (H H I) |000 = (1/2) (|0 + |1) (1/2) (|0 + |1) |0
= (1/2) (|000 + |010 + |100 + |110).
An application of the Toffoli gate, T, now produces
|2 = T|1 = (1/2) (|000 + |010 + |100 + |111).
Notice that the 3-qubit system is put into equal superposition of the four possible
results of x  y. The result of ANDing the first two qubits appears in the third qubit.
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Computing x+y
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Take 3 qubits, each prepared in state |0. Compute x  y. So we have
|2 = (1/2) (|000 + |010 + |100 + |111).
Now apply the Cnot gate to the first two qubits, to get
Cnot I |2 = (1/2) (|000 + |010 + |110 + |101)
where the second (blue) qubit is the sum and the third (red) qubit is the carry bit.
Note that the carry bit in the adder is the result of an AND operation. The carry and
AND are really the same thing. The sum bit comes from an XOR gate (that is, the
Cnot operation).
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Interpretations of quantum mechanics
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Interpretations of quantum mechanics
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There are several interpretations of quantum mechanics because the wave
function is an abstract mathematical object. Neither its origin nor its
underlying structure has been disclosed in the laws of quantum mechanics.
In particular, the mechanisms for superposition, entanglement, and measurement have not been elucidated. Hence, they too are open to interpretation.
We shall briefly describe four interpretations—(1) the Copenhagen interpretation, (2) Bohm’s interpretation, (3) Everett’s many world interpretation,
and (4) Bera- Menon interpretation.
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Copenhagen interpretation
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In the Copenhagen interpretation (~ 1927), one cannot describe a quantum system
independently of a measuring apparatus. Indeed, it is meaningless to ask about the
state of the system in the absence of a classical measuring system. The role of the
observer is central since it is the observer who decides what he wants to measure.
In this interpretation, a particle’s position is essentially meaningless; measurement
causes a collapse of the wave function and the collapsed state is randomly picked to
be one of the many possibilities allowed for by the system’s wave function; the
fundamental objects handled by the equations of quantum mechanics are not actual
particles that have an extrinsic reality but “probability waves” that merely have the
capability of becoming “real” when an observer makes a measurement. Entanglement is treated as a mysterious phenomenon.
The Copenhagen interpretation is also known as the ‘shut up and calculate’ interpretation (the phrase is due to David Mermin)*.
*See
Mermin, N. D., Could Feynman have said this? Physics Today, May 2004, pp. 10-11,
http://www.physicstoday.org/resource/1/phtoad/v57/i5/p10_s1.
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Bohm’s interpretation
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In Bohm’s interpretation (1952), the whole universe is entangled; its parts cannot be
separated. Entanglement is not a mystery; it is mediated by a very special unknown
anti-relativistic quantum information field (pilot wave, derivable from Schrödinger’s
equation) that does not diminish with distance and that binds the whole universe
together. It is an all pervasive field that is instantaneous; it is not physically
measurable but manifests itself in terms of non-local (unmediated, instantaneous, and
unaffected by the nature of the intervening medium) correlations.
In this interpretation, an electron, e.g., has a well-defined position and momentum at
any instant. However, the path an electron follows is guided by the interaction of its
own pilot wave with the pilot waves of other entities in the universe.
In fact, Bohm treats measurement as an objective process in which the measuring
apparatus and what is observed interact in a well-defined way. At the conclusion of
this interaction, the quantum system enters into one of a set of ‘channels’, each of
which corresponds to the possible results of the measurement while the other
channels become inoperative. In particular, there is no ‘collapse’ of the wave
function, yet the wave function behaves as if it had collapsed to one of the channels.
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Everett’s many world interpretation
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Everett’s interpretation (1956) is perhaps the most bizarre and yet perhaps the
simplest (it is free of the measurement problem because Everett omits the measurement postulate) and, instead, requires us to believe that we inhabit one of an infinite
number of parallel worlds!
He assumes that when a quantum system in a given world is faced with a choice, i.e.,
choosing between alternatives as in a measurement, the system splits into a number
of systems (worlds) equal to the number of options available. Thus, the world of a
qubit in state (|0 + |1)/2 will split into two worlds if the qubit is measured. The
two worlds will be identical to each other except for the different option chosen by
the qubit—in one it will be in state |0 and in the other it will be in state |1. Each
world will also carry its own copy of the observer(s), and each observer copy will
see the specific outcome that appears in his respective world. Of course, the worlds
can overlap and interact in the overlapping regions.
Decoherence, that is, (spontaneous) interactions between a quantum system and its
environment will cause the worlds to separate into non-interacting worlds.
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Non-uniqueness of interpretations
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What we find in the various interpretations is that while the formalism of quantum
mechanics is widely accepted, there is no single interpretation of it that is agreeable
to everyone. The disagreements essentially stem from the incompatibility that exists
between two evolutionary paths a quantum system follows—the Schrödinger’s
equation, and the “collapse” mode of measurement.
Indeed, without the measurement postulate telling us what we can observe, the
equations of quantum mechanics would be just pure mathematics that would have no
physical meaning at all. Note also that any interpretation can come only after an
investigation of the logical structure of the postulates of quantum mechanics is made.
Let me explain what we mean by an interpretation in the context of quantum
mechanics.
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Form and meaning are separate
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For example, Newtonian mechanics does not define the structure of matter. How we
interpret or model the structure of matter is largely an issue separate from Newtonian
mechanics. However, any model of the structure of matter we propose is expected to
be such that it is compatible with Newton’s laws of motion in the realm where
Newtonian mechanics rules. If it is not, then Newtonian mechanics as we know it
would have to be abandoned or modified or the model of the structure of matter
would have to be abandoned or modified. One may also have a partial interpretation
and leave the rest in abeyance till further insight strikes us and leads us to a complete
or a new interpretation.
A question such as whether a particular result deduced from Newton’s laws of
motion is deducible from a given model of material structure is therefore not
relevant.
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Form and meaning are separate (contd)
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Likewise, as long as an interpretation (or model) of superposition, entanglement, and
measurement does not require the axioms of quantum mechanics to be altered, none
of the predictions made by quantum mechanics would be incompatible with that
interpretation. This assertion is important because in our (Bera-Menon) interpretation
we make no comments on the Hamiltonian (in the Schrödinger’s equation), which
captures the detailed dynamics of a quantum system. Quantum mechanics does not
tell us how to construct the Hamiltonian. In fact, real life problems seeking solutions
in quantum mechanics need to be addressed in detail by physical theories built within
the framework of quantum mechanics. The postulates of quantum mechanics provide
only the scaffolding around which detailed physical theories are to be built.
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A new interpretation (Bera-Menon, 2009)
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In our interpretation, we provide a sub-Planck-scale view of the wave function,
superposition, entanglement, and measurement without affecting the postulates of
quantum mechanics. The sub-Planck scale is chosen to provide us with the freedom
to construct mechanisms for our interpretation that are not necessarily bound by the
laws of quantum mechanics (just as atomic structure is not bound by Newtonian
mechanics). In particular our interpretation does not have to satisfy the Schrödinger
wave equation because quantum mechanics is not expected to rule in the sub-Planck
scale. The high point of our interpretation is that it is able to explain the measurement postulate as the inability of a classical measuring device to measure at a
precisely predefined time.
Ref. Bera, R.K., and Menon, V., A new interpretation of superposition, entanglement, and measurement in
quantum mechanics, arXiv:0908.0957v1 [quant-ph], 07 August 2009, at http://arxiv.org/abs/0908.0957.
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A new interpretation (Superposition)
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In our interpretation we assume that the sub-Planck scale structure of the wave function
is such that the wave function is in only one state at any instant but oscillates between its
various “superposed” component states (eigenstates). (There is no expenditure of energy
in maintaining the oscillations.)
State |1
State |0
T0
T1
Tc
That is, the superposed states appear as time-sliced in a cyclic manner such that the time
spent by an eigenstate in a cycle is related to the complex amplitudes (a, b) appearing in
the qubit’s wave function, | = a |0 + b |1. The cycle time Tc of the qubit’s oscillation
between states |0 and |1 is much smaller than the Planck time (<< 10-43 sec). It is not
necessary for us to know the value of Tc. We only assert that it is a universal constant.
Within a cycle, the time spent by the particle in state |0 is T0 = |a|2 Tc and in state |1 is T1
= |b|2 Tc so that Tc = T0 + T1.
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A new interpretation (Measurement)
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Our hypothesized measurement mechanism acts instantaneously (through
entanglement) but the instant of actual measurement occurs randomly over a small
but finite interval tm, which is much greater than Planck time (otherwise its actual
value is immaterial), from the time the measurement apparatus is activated. In
particular, we regard measurement as the joint product of the quantum system and
the macroscopic classical measuring apparatus. To avoid bias, we assume that the
device can choose any instant in the interval tm with equal probability.
Thus the source of indeterminism built into quantum mechanics is interpreted here
as occurring due to the classical measuring device’s inability to measure at a
precisely predefined time. We do not explain how the collapse of the wave function
occurs when a measurement is made, only why the measurement outcome is probabilistic. Once a measurement is made, the wave function assumes the collapsed
state.
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A new interpretation (Entanglement)
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Entangled states binding two or more qubits appear in our interpretation as the synchronization of the sub-Planck level oscillation of the participating qubits, as shown below for
the two-qubit system Bell states,
1: |01
1: |00
Particle 1
Particle 1
Particle 2
Particle 2
2: |10
2: |11
Tc
Tc
(|00 ± |11)/√2
(|01 ± |10)/√2
A measurement on one of the entangled qubits will collapse both simultaneously to the
respective state they are in at the instant of measurement (such as 1 or 2 in the Figure).
We do not know how Nature might accomplish the required synchronization.
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A new interpretation (Entanglement) (contd)
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It is, of course, clear that our interpretation cannot violate the uncertainty principle
since the latest measurement on a system collapses the system according to the
measurement postulate. Thus, there can be no direct correlation between any earlier
results of measurement on the system, and the succeeding measurement.
Unlike the Copenhagen interpretation, in our interpretation it is not meaningless to
ask about the state of the system in the absence of a measuring system.
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There is no quantum world. There is only an abstract physical
description. It is wrong to think that the task of physics is to find out
how nature is. Physics concerns what we can say about nature.
— Niels Bohr
This view is very different from that of Einstein’s who believed that the
job of physical theories is to ‘approximate as closely as possible to
the truth of physical reality.’
Thank you!
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