I discuss the process of measurement in the context of a communication system. The setting is a transmitter which encodes some physical object and sends it off, a receiver which measures some property of the transmitted physical object (TPO) in order to get some information, and some path between the transmitter and the receiver over which the TPO is sent. The object of the game here is to characterize the TPO, either as an end in itself (research), or to examine its potential for information transmittal (communication). If the TPO is a 'small' object then quantum mechanics is needed to play this game. In the course of this work, I hope to broaden the concept of measurement in quantum mechanics to include noisy measurements and incomplete measurements. I suggest that quantum density operators are logically associated with the transmitter and some portion of the path, and that quantum measurement operators are logically associated with the rest of the path and the receiver. Communication involves the overlap of a density operator and a measurement operator. That is, the fundamental conditional probability of the communication process is given by the trace of the product of a density operator and a measurement operator.
I discuss the process of measurement in the context of a communication system. The setting is a transmitter which encodes some physical object and sends it off, a receiver which measures some property of the transmitted physical object (TPO) in order to get some information, and some path between the transmitter and the receiver over which the TPO is sent. The object of the game here is to characterize the TPO, either as an end in itself (research), or to examine its potential for information transmittal (communication). If the TPO is a 'small' object then quantum mechanics is needed to play this game. In the course of this work, I hope to broaden the concept of measurement in quantum mechanics to include noisy measurements and incomplete measurements. I suggest that quantum density operators are logically associated with the transmitter and some portion of the path, and that quantum measurement operators are logically associated with the rest of the path and the receiver. Communication involves the overlap of a density operator and a measurement operator. That is, the fundamental conditional probability of the communication process is given by the trace of the product of a density operator and a measurement operator.
The term "measurement" in quantum theory (as well as in other physical theories) is ambiguous: It is used to describe both an experience - e.g., an observation in an experiment - and an interaction with the system under scrutiny. If doing physics is regarded as a creative activity to develop a meaningful description of the world, then one has to carefully discriminate between the two notions: An observer's account of experience - consitutive to meaning - is hardly expressed exhaustively by the formal framework of an interaction within one particular theory. We develop a corresponding perspective onto central terms in quantum mechanics in general, and onto the measurement problem in particular.
The field of quantum mechanics has revolutionised physics as a subject. Areas such as information theory, computer science and physical sensing have all been affected by the tremendous successes of various quantum protocols. In this thesis I present my contribution to the development of such non-classical protocols. In classical communication theory a message is always carried by physical particles that interact with a transmitter, after which they travel to a receiver. In this thesis I outline a quantum protocol which allows a receiver to obtain a message without receiving any physical object or particles that have interacted with the transmitter—that is, counterfactually. I build my protocol for counterfactual communication on the principles of interaction-free measurements, ensuring that information always propagates in the opposite direction to the protocol particles. The protocol shows how quantum mechanics breaks the previous premise of communication theory. From the perspective of local observers, it is a beautiful manifestation of the non-locality of interaction-free measurements. Furthermore, it is highly robust against experimental errors and external disturbances. The majority of this part of the thesis is based on my published article ‘Quantum counterfactual communication without a weak trace’ [Phys. Rev. A 94, 062303 (2016)]. Previous to my work, Salih et al. attempted to design a counterfactual communication protocol [Phys. Rev. Lett. 110, 170502 (2013)]. This protocol has been highly controversial. As counterfactual phenomena impose restrictions on the inter-measurement paths of quantum particles, and the physical reality of such paths lacks description in the Copenhagen interpretation of quantum mechanics, an extension of current quantum theory is required to facilitate a discussion. In this thesis I present an operational and interpretation-independent methodology, enabling the discussion of inter-measurement paths of quantum particles. I start by considering the interferometers of counterfactual protocols, making the basic assumption that any quantum evolution naturally involves uncontrolled weak interactions. I then show how the Fisher information of these weak interactions, available at the output of counterfactual experiments, can be used to discuss the pre-measurement past of the particles. Based on this analysis, the protocol developed by Salih et al. is found to strongly violate counterfactuality. However, my protocol is more flexible in that it allows particles to propagate in the opposite direction to the message. This leads to counterfactuality being satisfied—even in the presence of large experimental errors. These results are observed both analytically and numerically. This part of the thesis is based on my article ‘Evaluation of counterfactuality in counterfactual communication protocols’ [Phys. Rev. A 96, 062316 (2017)]. The numerical methods are inspired by another of my publications: ‘Protocol for fermionic positive-operator-valued measures’ [Phys. Rev. A 96, 052305 (2017)]. Finally, as the Fisher information measure is found to be useful in evaluating counterfactual protocols, I extend my work by investigating the quantum Fisher information in experiments with general discrete quantum circuits. I prove that the quantum Fisher information of a two-level interaction in a quantum circuit can be expressed by a simple formula. Under certain phase-relations, the formula provides a straightforward connection between the abstract concept of the inter-measurement wavefunction and the quantum Fisher information at the output. With regard to how the information obtained from a certain volume of space influences our perception of classical objects, I argue that the quantum Fisher information measure is highly useful in describing quantum objects. If this measure is applied to observers with a limited set of the experimental measurement outcomes, a quantum object can appear to follow non-classical discontinuous paths. This supports the remarkable conclusion that our perception of the past of a quantum object is subjectively dependent on the measurement we conduct on it.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.
We consider the communication systems sending the classical (Shannon) information through the quantum channel. The classical information source X is represented with symbols and their a priori probabilities At a transmitter, each symbol is modulated to a quantum state and detected at a receiver by the quantum mechanically generalized measurement, so-called the probability operator-valued measure Here we call a detection operator since it represents a detection process deciding a received signal to be ‘j’. The detection operator is no-negative, and resolution of identity, where is the identity operator on the signal Hilbert space spanned by signal quantum states. This detection process can be regarded as a quantum channel characterized by the conditional probability defined as follows:
Information and computation theory have undergone a spurt of new growth, and a renewal of their historic connection to basic physics, as they have expanded to treat the intact transmission and processing of quantum states, and the interaction of such 'quantum information' with traditional forms of information. We may wonder why this did not happen earlier, as quantum principles have long been accepted as fundamental to all of physics. Perhaps the founders of information and computation theory, such as Shannon, Turing and von Neumann, were too accustomed to thinking of information processing in macroscopic terms, not yet having before them the powerful examples of the genetic code and ever-shrinking microelectronics. Be that as it may, information until recently has largely been thought of in classical terms, with quantum mechanics playing a supporting role in the design of the equipment to process it, and setting limits on the rate at which it could be sent through certain channels. Now we know that a fully quantum theory of information and information processing offers, among other benefits, a brand of cryptography whose security rests on fundamental physics, and a reasonable hope of constructing quan- tum computers that could dramatically speed up the solution of certain mathematical problems. These benefits depend on distinc- tively quantum properties such as uncertainty, interference and entanglement. At a more fundamental level, it has become clear that an information theory based on quantum principles extends and completes classical information theory, just as complex numbers extend and complete the reals. Besides quantum generalizations of classical notions such as sources, channels and codes, the new theory includes two complementary, quantifiable kinds of information— classical information and quantum entanglement. Classical infor- mation can be copied at will, but can only be transmitted forward in time, to a receiver in the sender's forward light cone. Entanglement in contrast, cannot be copied, but can connect any two points in space-time. Conventional data processing operations destroy entanglement, but quantum operations can create it and use it for various purposes, such as speeding up certain classical computa- tions and assisting in the transmission of classical information or intact quantum states. Part of the new quantum information theory is the qualitative and quantitative study of entanglement, and its interactions with classical information. Any means, such as an optical fibre, of delivering quantum systems more or less intact from one place to another, may be viewed as a quantum channel. Unlike classical channels, which are well characterized by a single capacity, quantum channels have several distinct capacities, depending on what one is trying to use them for, and what auxiliary resources are brought into play. New effects involving quantum information continue to be discovered, not only in the traditional areas of computation, channel capacity, and cryptography, but in areas such as commu- nication complexity and game theory.
As openly stated by Willis Lamb, the field of quantum measurements has not been helped by the ample participation of theorists in it (although Lamb himself has been described as a theorist turned experimentalist). The point is, as validly exposed by Lamb, that when it comes to quantum measurements there is a plethora of misunderstandings that mystify and confuse those who describe them and those who listen. Here, the so called ‘measurement problem’ comes to mind. The very first concept in explaining what is a measurement, and not just a quantum measurement, is that no measurement is exact . In physics no one can measure https://www.w3.org/1998/Math/MathML" display="inline"> x with https://www.w3.org/1998/Math/MathML" display="inline"> Δ x = 0 . When we measure https://www.w3.org/1998/Math/MathML" display="inline"> x we always measure it as https://www.w3.org/1998/Math/MathML" display="inline"> x ± Δ x where https://www.w3.org/1998/Math/MathML" display="inline"> Δ x is a nonzero quantity. This observation alone immediately brings in stochasticity into the measurement arena. In measurements there is no escape from stochasticity . In this chapter the measurement process is described from an interferometric perspective.