We present a generalized POVM formalism for the calculation of quantum optics networks of arbitrary complexity and apply it in a detailed analysis of the Franson-Bell experiment. Our analysis suggests that when it comes to observations of quantum processes, laboratory apparatus cannot always be used according to classical principles.
Quantum experiments are described in terms of time-dependent networks of quantum bits, each qubit representing an elementary information gateway. The emphasis is on the signal properties of apparatus rather than on systems under observation (SUOs), with the quantum states of the theory (the labstates) representing the observer's information about the state of their apparatus, rather than of any SUO. The formalism gives an efficient quantum register description related to the formalism of quantum computation. Experiments conventionally described by the PVM and POVM formalisms are treated in identical terms, the formalism providing an efficient modular approach to quantum optics experiments of arbitrary complexity.
Introduction As the decades of the twentieth century rolled on, quantum mechanics (QM) became more and more sophisticated and mathematical. Understanding what this theory means intuitively was confounded not only by nonclassical concepts such as wave–particle duality and quantum interference, but also by issues to do with observation. The Newtonian classical mechanics (CM) paradigm of reality, wherein reductionist laws of physics describe observer-independent dynamics of systems under observation (SUOs) with observer-independent properties, was found to be inadequate. Quantum theorists were confronted with the measurement problem , which attempts to understand, explain, and rationalize the laws of QM that underpin the processes of observation that go on in the laboratory. They are not the same as those of CM in several puzzling respects. Historically, the first sign of the measurement problem was Planck's quantization of energy (Planck, 1900b) and the second was Bohr's veto on radiation damping in hydrogen (Bohr, 1913). These occurred in the first quarter of the twentieth century, a period in physics often referred to as Old Quantum Mechanics . Another indicator that intuition was inadequate was Born's interpretation of the squared modulus of the Schrödinger wave function as a probability density (Born, 1926). That interpretation has everything to do with observers and observation, because probability without an observer is a vacuous concept. Eventually, the projection-valued measure (PVM) formalism emerged, championed by von Neumann in an influential book on the mathematical formulation of QM (von Neumann, 1955). Subsequently, pioneers such as Ludwig (Ludwig, 1983a,b) and Kraus (Kraus, 1974, 1983) refined the theory into the general positive operator-valued measure (POVM) formalism that we shall discuss and use. The quantized detector network (QDN) approach to quantum experiments is most naturally expressed in the PVM formalism, as QDN focuses on the individual detectors in the laboratory. However, the POVM formalism is more general than the PVM formalism, giving a description of multiple detector processes similar to QDN. This raises the question of how the two approaches, QDN and POVM, are related. The aim of this chapter is to explore this relationship. Before we review the PVM and POVM formalisms, we review some essential mathematical concepts.
QDN (quantized detector networks) is a description of quantum processes in which the principal focus is on observers and their apparatus, rather than on states of SUOs (systems under observation). It is a realization of Heisenberg's original instrumentalist approach to quantum physics, and can deal with time dependent apparatus, multiple observers and inter-frame physics. QDN is most naturally expressed in the mathematical language of quantum computation, a language ideally suited to describe quantum experiments as processes of information exchange between observers and their apparatus. Examples in quantum optics are given, showing how the formalism deals with quantum interference, non-locality and entanglement. Particle decays, relativity and non-linearity in quantum mechanics are also discussed.
Scientists have been debating the meaning of quantum mechanics for over a century. This book for graduate students and researchers gets to the root of the problem; the contextual nature of empirical truth, the laws of observation and how these impact on our understanding of quantum physics. Bridging the gap between non-relativistic quantum mechanics and quantum field theory, this novel approach to quantum mechanics extends the standard formalism to cover the observer and their apparatus. The author demystifies some of the aspects of quantum mechanics that have traditionally been regarded as extraordinary, such as wave-particle duality and quantum superposition, by emphasizing the scientific principles rather than the mathematical modelling involved. Including key experiments and worked examples throughout to encourage the reader to focus on empirically sound concepts, this book avoids metaphysical speculation and also alerts the reader to the use of computer algebra to explore quantum experiments of virtually limitless complexity.
Introduction The quantized detector network (QDN) formalism was designed from the outset to deal with detector networks of arbitrary complexity and rank. However, there is a price to be paid. A qubit register of rank r is a Hilbert space of dimension 2 r , so the dimensionality of quantized detector networks grows exponentially with rank. For example, a relatively small system of, say, 10 detectors involves a Hilbert space of dimension 2 10 = 1 , 024. Even such a relatively small system cannot be dealt with easily by manual calculation, because quantum mechanics (QM) involves entangled states. Labstates in QDN are far more complex (in the sense of having far more mathematical structure) than the corresponding states in a classical register of the same rank. Some of the mathematical entanglement and separability properties of quantum labstates are discussed in Chapters 22 and 23. The quantum entanglement structure of labstates poses a ubiquitous and serious problem, for both theorists and experimentalists. At this time, there is significant interest in quantum entanglement, particularly regarding its use in quantum computing, but theoretical understanding of entanglement is still surprisingly limited. On the experimental side, quantum computers with 2000 qubits are currently being developed (D-Wave Systems, 2016). There is scope here for the application of QDN to networks of rank going into the many thousands. For those sorts of systems under observation (SUOs), quantum entanglement makes calculations by hand far too laborious to be of any practical use. Fortunately, three factors come to our aid here, making the application of QDN to large-rank networks a potentially viable proposition. Modularization We saw in previous chapters that QDN deals with discrete aspects of observation, despite the fact that standard QM and relativistic quantum field theory (RQFT) deal with continuous degrees of freedom. This discreteness comes in three forms, referred to us as stages, nodes, and modules, and all of that is due to three inescapable physical facts. First, all real observations take time and involve discrete signals in finite numbers of detectors. Second, real apparatus is constructed from atoms, not continua. Third, these atoms form finite numbers of well-characterized modules, as discussed in the previous chapter. Contextuality A critically helpful fact here is that in any particular experiment, the contextual subspaces that the observer needs to deal with will usually be of significantly lower dimensions than that of the full quantum registers involved.