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Marek Czachor

Publications and source records attributed to Marek Czachor.

At least 55 records · Page 3Linked to original sources
arXiv2008

Teleportation seen from space-time

Formal similarity between Minkowski tetrads and Bell bases allows to think of metric tensors in terms of quantum teleportation protocols. The role of null tetrads for quantum information processing is different. They define qubits resistant to a special kind of noise that occurs if coding and decoding of quantum information is performed in different reference frames. These examples show that mutual links between quantum information and the 2-spinor calculus may be nontrivial and worthy of further studies.

quant-ph↗
arXiv2007

Geometric Analogue of Holographic Reduced Representation

Holographic reduced representations (HRR) are based on superpositions of convolution-bound $n$-tuples, but the $n$-tuples cannot be regarded as vectors since the formalism is basis dependent. This is why HRR cannot be associated with geometric structures. Replacing convolutions by geometric products one arrives at reduced representations analogous to HRR but interpretable in terms of geometry. Variable bindings occurring in both HRR and its geometric analogue mathematically correspond to two different representations of $Z_2\times...\times Z_2$ (the additive group of binary $n$-tuples with addition modulo 2). As opposed to standard HRR, variable binding performed by means of geometric product allows for computing exact inverses of all nonzero vectors, a procedure even simpler than approximate inverses employed in HRR. The formal structure of the new reduced representation is analogous to cartoon computation, a geometric analogue of quantum computation.

cs.AI↗
arXiv2007

Tensor-product vs. geometric-product coding

Quantum computation is based on tensor products and entangled states. We discuss an alternative to the quantum framework where tensor products are replaced by geometric products and entangled states by multivectors. The resulting theory is analogous to quantum computation but does not involve quantum mechanics. We discuss in detail similarities and differences between the two approaches and illustrate the formulas by explicit geometric objects where multivector versions of the Bell-basis, GHZ, and Hadamard states are visualized by means of colored oriented polylines.

quant-ph↗
arXiv2007

Elementary gates for cartoon computation

The basic one-bit gates (X, Y, Z, Hadamard, phase, pi/8) as well as the controlled cnot and Toffoli gates are reformulated in the language of geometric-algebra quantum-like computation. Thus, all the quantum algorithms can be reformulated in purely geometric terms without any need of tensor products.

quant-ph↗
arXiv2007

Cartoon Computation: Quantum-like computing without quantum mechanics

We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.

quant-ph↗
arXiv2006

Notes on Geometric-Algebra Quantum-Like Algorithms

In these notes we present preliminary results on quantum-like algorithms where tensor product is replaced by geometric product. Such algorithms possess the essential properties typical of quantum computation (entanglement, parallelism) but employ additional algebraic structures typical of geometric algebra -- structures absent in standard quantum computation. As a test we reformulate in Geometric Algebra terms the Deutsch-Jozsa problem.

quant-ph↗
arXiv2006

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

We analyze an example of a photon in superposition of different modes, and ask what is the degree of their entanglement with vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\infty$-representation and show in two ways that in two-mode states the modes are maximally entangled with vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with vacuum is present even for single-mode states. Still, the degree of entanglement is here difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

quant-ph↗
arXiv2006

Regularization as quantization in reducible representations of CCR

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincaré covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to Rényi's $α$. It is shown that the ``Shannon limit" $α\to 1$ plays here a role of correspondence principle with the standard regularized formalism.

hep-th↗
arXiv2005

Two-state dynamics for replicating two-strand systems

We propose a formalism for describing two-strand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions.

q-bio.PE↗
arXiv2005

Abstract DNA-type systems

An abstract DNA-type system is defined by a set of nonlinear kinetic equations with polynomial nonlinearities that admit soliton solutions associated with helical geometry. The set of equations allows for two different Lax representations: A von Neumann form and a Darboux-covariant Lax pair. We explain why non-Kolmogorovian probability models occurring in soliton kinetics are naturally associated with chemical reactions. The most general known characterization of soliton kinetic equations is given and a class of explicit soliton solutions is discussed. Switching between open and closed states is a generic behaviour of the helices. The effect does not crucially depend on the order of nonlinearity (i.e. types of reactions), a fact that may explain why simplified models possess properties occuring in realistic systems. We explain also why fluctuations based on Darboux transformations will not destroy the dynamics but only switch between a finite number of helical structures.

q-bio.PE↗
arXiv2005

Quantum optics in different representations of the algebra of canonical commutation relations (I): Unexpected properties of Rabi oscillations -- theory and experiment

We discuss the Jaynes-Cummings model in different representations of the algebra of canonical commutation relations. The first conclusion is that all the irreducible representations lead to equivalent physical predictions. However, the reducible representation recently introduced as a candidate for `QED without infinities' leads to new effects. We analyze from this perspective the experiments on Rabi oscillations performed by the Kastler-Brossel Laboratory group from Paris. Surprisingly, the results seem to support the reducible representation approach. We also discuss possibilities of more definitive tests of the new formalism.

quant-ph↗
arXiv2005

On Entanglement with Vacuum

The so-called entanglement with vacuum is not a property of the Fock space, but of some rather pathological representations of CCR/CAR algebras. In some other Fock space representations the notion simply does not exist. We have checked all the main Gedanken experiments where the notion of entanglement with vacuum was used, and found that all the calculations could be performed at a representation-independent level. In particular any such experiment can be formulated in a Fock-space representation where the notion of entanglement with vacuum is meaningless. So, for the moment there is no single experiment where the notion is needed, and probably it is simply unphysical.

quant-ph↗
arXiv2005

Towards a quantum evolutionary scheme: violating Bell's inequalities in language

We show the presence of genuine quantum structures in human language. The neo-Darwinian evolutionary scheme is founded on a probability structure that satisfies the Kolmogorovian axioms, and as a consequence cannot incorporate quantum-like evolutionary change. In earlier research we revealed quantum structures in processes taking place in conceptual space. We argue that the presence of quantum structures in language and the earlier detected quantum structures in conceptual change make the neo-Darwinian evolutionary scheme strictly too limited for Evolutionary Epistemology. We sketch how we believe that evolution in a more general way should be implemented in epistemology and conceptual change, but also in biology, and how this view would lead to another relation between both biology and epistemology.

quant-ph↗
arXiv2004

The fundamental problem of quantum cryptography

It is known that statistical predictions of quantum theory do not depend on its interpretation. In particular, an experiment cannot distinguish between the Copenhagen interpretation (involving no hidden variables) and the de Broglie-Bohm interpretation based on nonlocal hidden variables. Quantum cryptographic protocols, such as BB84 or E91, are secure and mutually equivalent as long as one works within the framework of Copenhagen interpretation. But they are inequivalent and insecure if one considers attacks allowed by the de Broglie-Bohm interpretation. The fundametal problem of quantum cryptography is therefore this: Are all the statements about security of quantum protocols based on our belief in one of the two allowed interpretations of quantum mechanics? We show that this is not the case. Ekert-type protocols can be modified in a way that makes them secure even if the de Broglie-Bohm nonlocal hidden variables exist. Bennett-Brassard-type cryptography does not seem to allow for such a correction.

quant-ph↗