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Progress in Electromagnetic Research Symposium

Direct measurement of an one-million-dimensional photonic state

Zhimin Shi, Mohammad Mirhosseini, Jessica Margiewicz, Mehul Malik, Freida Rivera, Ziyi Zhu, Robert W. Boyd

Published Aug 1, 2016• 3 citations
Publisher

Abstract

The state vector of a pure quantum system is a set of complex probability amplitudes used for describing the system in each state of a given Hilbert space. Characterizing the state of a quantum system is crucial for fundamental studies in quantum mechanics as well as for manipulating and utilizing quantum systems for practical applications.

DOI
10.1109/piers.2016.7734289

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The state vector of a pure quantum system is a set of complex probability amplitudes used for describing the system in each state of a given Hilbert space. Characterizing the state of a quantum system is crucial for fundamental studies in quantum mechanics as well as for manipulating and utilizing quantum systems for practical applications.
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The state of a quantum mechanical system is represented by a nonzero vector in a separable Hilbert space. The word ‘represented’ means that knowledge of the state vector gives complete information on the properties of the system. In quantum mechanics, the state vectors are denoted using Dirac's notation. Several quantum systems possess a property called spin. Spin is a purely quantum mechanical feature which has no direct classical counterpart. The set of density matrices is a convex subset of the space of Hermitian operators whose ‘extreme’ points represent pure states. In quantum mechanics, observed quantities are called observables. A continuous spectrum is such that it contains no isolated points. The chapter discusses the case of measurement of observables with discrete spectrum.
The of a quantum state poses a unique challenge for experimentalists. Recently, the technique of measurement was proposed for characterizing a quantum state in-situ through sequential weak and strong measurements. While this method has been used for measuring polarization states, its real potential lies in the of states with a large dimensionality. Here we show the practical direct of a high-dimensional state vector in the discrete basis of orbital-angular momentum. Through weak measurements of orbital-angular momentum and strong measurements of angular position, we measure the complex probability amplitudes of a pure state with a dimensionality, d=27. Further, we use our method to directly observe the relationship between rotations of a state vector and the relative phase between its orbital-angular-momentum components. Our technique has important applications in high-dimensional classical and quantum information systems, and can be extended to characterize other types of large quantum states.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.