11th FIS Discussion Session:
QUANTUM INFORMATION
Andrei Khrennikov & Jonathan D.H. Smith
1. QUANTUM INFORMATION ---MYTHS AND REALITY QUANTUM INFORMATION
Andrei Khrennikov & Jonathan D.H. Smith
(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.arxiv.org/abs/quant-ph/0302065 , 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 Bells 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 (see also Khrennikov A.Yu., Information dynamics in cognitive, psychological, social, and anomalous phenomena. Kluwer, Dordreht, 2004) which contains unconventional interpretation of violations of Bells 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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