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States and channels in quantum mechanics without complex numbers

Jarosław Adam Miszczak

Published Jan 16, 2016• 1 citation
OpenAlex

Abstract

In this work we demonstrate a simplified version of quantum mechanics in which the states are constructed using real numbers only. In the standard for-mulation of quantum mechanics the state is represented by positive semidefinite, normalized linear operators. In the following we focus on linearity and hermic-ity properties of density matrices as they are crucial for the properties of quantum channels. The main advantage of the introduced approach is that the simulation of the n-dimensional quantum system requires O(n2) real numbers, in contrast to the standard case where O(n4) real numbers are required. The main disadvantage is the lack of hermicity in the representation of quantum states. This leads to the occurrence of complex eigenvalues of the real-valued density matrices. During the last few years Mathematica computing system has become very popular in the area of quantum information theory and the foundations of quantum mechanics. The main reason for this is its ability to merge the symbolic and nu-

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In this work we demonstrate a simplified version of quantum mechanics in which the states are constructed using real numbers only. In the standard for-mulation of quantum mechanics the state is represented by positive semidefinite, normalized linear operators. In the following we focus on linearity and hermic-ity properties of density matrices as they are crucial for the properties of quantum channels. The main advantage of the introduced approach is that the simulation of the n-dimensional quantum system requires O(n2) real numbers, in contrast to the standard case where O(n4) real numbers are required. The main disadvantage is the lack of hermicity in the representation of quantum states. This leads to the occurrence of complex eigenvalues of the real-valued density matrices. During the last few years Mathematica computing system has become very popular in the area of quantum information theory and the foundations of quantum mechanics. The main reason for this is its ability to merge the symbolic and nu-
We show that every Clifford algebra can be transformed into a formalism that uses only real numbers. The rules we must follow to achieve this are universal. To illustrate the method we construct the real pendant of SU(2) and we derive a version of the Dirac equation that uses only real numbers. The advantage of the complex formulations is that they reduce the size of the representation matrices by a factor of 2. The reason for the presence of complex numbers in quantum mechanics is that it is formulated in the language of group representation theory.
It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.
Abstract With this chapter, we begin to transit from digital information processing to quantum information processing. However, to be able to master this transition, we first need to familiarize ourselves with quantum mechanical principles and the mathematics used to describe them. Since we already know about abstract state spaces, tensor products, and Hamiltonian operators and their eigenvectors and eigenvalues, this should not be too difficult or too alien. And yet, there remain a few challenges: Mathematical descriptions of quantum states usually involve the Dirac notation, require complex linear algebra in high dimensional spaces, and encode unintuitive probabilistic interpretations. Rising to these challenges, we briefly study the notion of Hilbert spaces, introduce the Dirac notation, and discuss the axioms of quantum mechanics and some of their consequences. All this will admittedly be dense and abstract but prepare us for everything that is still to follow. Our main goal is to emphasize that quantum mechanics and therefore quantum computing is just complex linear algebra in very high dimensional spaces so that quantum computing needs a different kind of thinking than we are used to in digital computing.