11th FIS Discussion Session:
QUANTUM INFORMATION
Andrei Khrennikov & Jonathan D.H. Smith
1. QUANTUM INFORMATION ---MYTHS AND REALITY
(Andrei Khrennikov)
Quantum information is science about processing of information by exploring
some distinguishing features of quantum systems, e.g., electrons, photons,
ions. Last years gurus of quantum information promised a lot, may be even
too much. In quantum computing it was promised that NP-problems would be
solved in polynomial time. In quantum cryptography there were claims that
protocols would have 100% security. This gave the possibility to sell
quantum mechanics the second time � in new century and with new sauce. Huge
grants were distributed in USA, EU and Japan. At the moment it is too early
to say anything definite about the final output of this great project.
In quantum computing there were created a few quantum algorithms and
developed devices, �quantum pre-computers�, with a few quantum registers.
However, difficulties could not be more ignored. By some reason it was
impossible to create numerous quantum algorithms which could be applied to
various problems. Up to now the whole project is based on 2-3 algorithms
and among them the only one, namely, the algorithms for finding prime
factors, can be interesting for real applications. There is a general
tendency to consider this situation with quantum algorithms as an
occasional difficulty. But as years pass, one might start to think that
there is something fundamentally wrong. The same feelings are induced by
development of quantum hardware. It seems that the complexity of the
problem of creation of a device with a large number N of quantum registers
increases extremely nonlinearly with increasing N.
In quantum cryptography the situation is in some sense opposite to quantum
computing. There were tremendous successes in development of technologies
for production and transmission of quantum information, especially pairs of
entangled photons. We emphasize that at the moment photons give the most
real basis of quantum cryptography. It is doubtful that there would be
created systems for quantum cryptography which would be based on e.g.
electrons. It is not easy to imagine refrigerators with electrons which are
used for transportation of quantum information. On the other hand, the
claim on 100% security of quantum protocols is far from to be totally
justified.
Any careful analysis of this situation implies immediately that the whole
project �Quantum Information� should be based on more solid foundations.
We recall that quantum mechanics by itself is a huge building having the
sand-fundament �the orthodox Copenhagen interpretation. On one hand, there
was created the advanced mathematical formalism (calculus of probabilities
in the complex Hilbert space) giving predictions which are supported by all
existing experimental data. On the other hand, it is still unclear why this
formalism works so well and moreover it is not clear what it really
predicts, because by the orthodox Copenhagen interpretation (which is the
conventional interpretation) quantum mechanics is not about physical
reality by itself, but about just our observations (of what?). All unsolved
problems of quantum foundations are essentially amplified in the quantum
information project. Problems which were of a purely philosophic interest
during one hundred years became technological and business problems.
Therefore �Quantum Information� gives a new great chance for
reconsideration of quantum foundations, see, e.g., electronic proceedings
of conferences at:
<http://www.vxu.se/msi/forskn/publications.html>http://www.vxu.se/msi/forskn/publications.html
, http://www.arxiv.org/abs/quant-ph/0302065
,
<http://www.arxiv.org/abs/quant-ph/0101085>http://www.arxiv.org/abs/quant-ph/0101085
. Whether such a chance will be used depends on many scientific,
psychological and market factors. Unfortunately, at the time being there is
the tendency to ignore fundamental difficulties and reduce everything to
technological problems. Of course, development of quantum technologies, in
particular manipulation with individual quantum systems, is the extremely
interesting project. But I hope that it could be done essentially more if
quantum computing and cryptography would be also considered as new tools
for testing the foundations of quantum mechanics.
First of all we should come back to the greatest debate of 20th century,
namely debate between Einstein and Bohr on completeness of quantum
mechanics. It is commonly accepted that quantum mechanics is complete: the
psi-function provides the most complete description of the state of quantum
system. It is impossible to find a more detailed description of quantum
reality � to find a model with hidden variables. This is the basis of the
orthodox Copenhagen interpretation and nowadays this is the basis of
quantum cryptography. If one were able to find a model with hidden
variables which reproduce quantum statistics, then the total security of
quantum protocols would be questioned! In probabilistic terms this is the
problem of so called irreducible quantum randomness. In the opposition to
classical randomness, it is claimed (since von Neumann) that quantum
randomness could not be reduced to the conventional ensemble randomness.
Thus I would like to propose to discuss �Quantum Foundations in Light of
Quantum Information� or �Quantum Information in Light of Quantum
Foundations.� One of the possibilities is to start with Bell�s inequality,
since its violations play the fundamental role in foundations of quantum
information. One of the possible starting points might be
<http://www.arxiv.org/abs/quant-ph/0006016>http://www.arxiv.org/abs/quant-ph/0006016
(see also Khrennikov A.Yu., Information dynamics in cognitive,
psychological, social, and anomalous phenomena. Kluwer, Dordreht, 2004)
which contains unconventional interpretation of violations of Bell�s
inequality.
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2. QUANTUM INFORMATION: BASICS AND OPEN QUESTIONS
(Jonathan D H Smith)
To approach quantum information theory, it is best to contrast it with
classical information theory. Laying aside philosophical connotations that
often cause confusion, it will help to focus on the basic formulation for
some elementary and quite concrete examples. These examples may serve as
'reality checks' during the course of our discussion. In mathematical
terms, classical information theory just works with sets, or probability
distributions on those sets, while quantum information theory works with
linear algebra or matrices over the complex numbers.
+ CLASSICAL BITS
A classical bit is a set with two elements A and B, say 0 and 1 as binary
digits. (The term 'bit' is a contraction of 'binary digit,' although also
having the layman's sense of 'a small piece,' since one may build larger
stores of information by using lots of bits.) Concretely, A and B may
represent: a coin showing heads or tails, a switch being open or closed, a
magnetization of north or south, a spin oriented up or down, an abacus
pebble at the top or bottom of the frame, and so on.
Instead of a set with two elements, we may consider a probability
distribution on that set. This just means two non-negative numbers, PA
attached to the element A, and PB attached to the element B, such that PA
and PB add up to unity or 1. For example, if the coin is being flipped onto
a table, it may have probability PA of landing heads, and PB of landing
tails. (PA and PB add up to unity, since we assume the coin lands flat on
the table, and doesn't roll off under the refrigerator where we can't see it.)
To get back from probability distributions to elements, we may consider the
element A as the probability distribution with PA = 1 and PB = 0 , while
B corresponds to PA = 0 and PB = 1 . So the classical bit has become the
set of all the probability distributions (PA,PB), identifying the element A
with the distribution (1,0), and the element B with the distribution (0,1).
We may call (1,0) the 'pure state A', and (0,1) the 'pure state B.' While
the (fair) coin is flipping through the air, it is in the 'mixed state'
(1/2,1/2). When it lands on the table, the force of the impact bumps it
from the mixed state into one of the pure states.
One may also consider the pairs (PA,PB) as Cartesian coordinates of points
in the plane. Geometrically, the classical bit is then the straight line
segment connecting the two pure states (1,0) and (0,1).
+ QUANTUM BITS
The quantum information theory analog of the classical bit is the qbit
(pronounced like the biblical 'cubit'), short for 'quantum bit' (or maybe
'quantum binary digit'). A qbit is a 2x2 matrix QSTATE or
[ QAA QAB ]
[ QBA QBB ]
in which the entries Qxy are complex numbers, QAA and QBB are
non-negative real numbers adding up to 1, and the complex conjugate of QAB
is QBA.
Just as a classical bit is implemented physically by a flipping coin, a
qbit is implemented physically by a spinning electron (stationary at a
known location).
Explicitly, consider the Pauli matrices
[ 0 1 ]
[ 1 0 ]
or SX,
[ 0 -i ]
[ i 0 ]
or SY, and
[ 1 0 ]
[ 0 -1 ]
or SZ.
Then for Planck's constant HBAR which is about 10 to the -35 th power
joule-seconds, the spin of the electron in the direction of the X-axis is
HBAR/2 times the trace of the product of SX with QSTATE. (The trace of a
square matrix is the sum of its diagonal elements.) The spin in the
Y-direction is obtained similarly using SY, and the spin in the Z-direction
is obtained similarly using SZ. Just as the flipping coin can be observed
(heads or tails) once it has landed on the table, you get to measure the
spin of the electron in just one direction.
A qbit certainly contains at least a classical bit of information. If
QSTATE has QAA = 1 and all other entries zero, then you get a positive spin
in the Z-direction. If QSTATE has QBB = 1 and all other entries zero, then
you get a negative spin in the Z-direction.
In experiments on quantum computation, engineers can apparently now prepare
several electrons with prescribed spins. But a qbit is much more than a
classical bit. While you only have one degree of freedom moving back and
forth along the line segment (PA,PB) of the classical bit, there are three
degrees of freedom in the qbit QSTATE. The pair (QAA,QBB) is like the
classical bit, but QAB can be anywhere in the complex plane.
+ IS QUANTUM (INFORMATION) THEORY REALLY NECESSARY?
This question is an obvious topic for discussion in our forum. If you think
the answer is negative,
that maybe everything can be done using clever tricks with classical
probabilities, then a test case would be to describe the spinning electron.
Is your description as elegant and satisfactory
as the quantum description?
+ ENTANGLEMENT
To handle more information, you can build a store with several classical
bits. The (pure) state set for this store is the Cartesian or direct
product of many two-element sets, one for each individual classical bit.
If you have several qbits QSTATE1, ... , QSTATEn, then the state of the
complete store is given
as the tensor or Kronecker product of these matrices. For example, the
Kronecker product of
QSTATE1 with QSTATE2 is
[ QAA1*QSTATE2 QAB1*QSTATE2 ]
[ QBA1*QSTATE2 QBB1*QSTATE2 ]
as a matrix of four 2x2 blocks. If you measure the first qbit here, your
choices for the measurement of the second or any further qbits are limited.
These later qbits have become 'entangled' with the first.
By contrast, knowing the first classical bit in your classical store puts
no limitations on the possibilities for the subsequent classical bits.
However, knowledge of the first marginal in a classical multivariate
probability distribution may limit the possibilities for the subsequent
marginals.
Is this classical effect enough to account for quantum entanglement?
+ QUANTUM COMMUNICATION
Take a two-part quantum system, and separate its parts, giving one to Alice
in Algeria
and the other to Bob in Botswana. Under certain circumstances, entanglement
may imply that a measurement made by Bob immediately tells him what Alice
is experiencing.
Can this be a communication protocol?
If so, what are its features and limitations?
+ QUANTUM COMPUTATION
Since a qbit can potentially store more than a single classical bit, people
nowadays view processing qbits as a highly parallel version of digital
processing. Before digital computers, there were analog computers. (For
example, integrating tables.) How does quantum computation compare with
analog computation? Is it just a special case (using spinning electrons
instead of friction wheels and slides)?
Turing machines provide a good formal model for classical digital
computers. Is there an equally good formal model for quantum computation?
Can the output from a physically feasible quantum computer ever be more
than a single classical bit --a single yes or no answer?
More generally: What exactly is computable with quantum computation? (For
comparison, Church's Thesis says that Turing machines compute recursive
functions.)
Sometimes it is claimed that the human brain displays certain aspects of
quantum computation. Is this analogy helpful? How far does it go?
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Received on Mon May 15 17:14:16 2006
