Abstract Moving completely into the quantum realm, the key concept of quantum state vector is described in three ways: (1) arrows in state space, (2) Dirac notation symbols that are important for the formal mathematical manipulation of quantum states, and (3) collections of 1s and 0s grouped in columns or rows. A qubit (two-level system) has two basis states. These basis state vectors can be added to give superposition states and multiplied using scalar products to change the state coefficients. These abstract concepts are given a physical meaning in a system of polarized light, a familiar phenomenon because of its use in polarized sunglasses. Light polarization has been widely used to demonstrate many fundamental aspects of quantum computing and quantum information science.
Abstract Moving completely into the quantum realm, the key concept of quantum state vector is described in three ways: (1) arrows in state space, (2) Dirac notation symbols that are important for the formal mathematical manipulation of quantum states, and (3) collections of 1s and 0s grouped in columns or rows. A qubit (two-level system) has two basis states. These basis state vectors can be added to give superposition states and multiplied using scalar products to change the state coefficients. These abstract concepts are given a physical meaning in a system of polarized light, a familiar phenomenon because of its use in polarized sunglasses. Light polarization has been widely used to demonstrate many fundamental aspects of quantum computing and quantum information science.
Abstract After a quick review of Chapter 2’s bits and gates, Bob and Alice introduce Cardy to several concepts and symbolic representations that will be of crucial importance in understanding quantum computing: state vectors, pictorial representations of state vectors, and simple mathematical forms that help us manipulate and reason about state vectors. State vectors may be represented by arrows in a state space diagram. States may also be represented more abstractly using Dirac notation symbols |⟩. Logic gates, critically important for carrying out computational work, can be represented by mathematical objects called matrices. States are represented by column vectors. Everything you need to know about matrices is presented over just a few pages.
Mathematical representations of the state of a quantum system are provided by vectors. The hold information about the probability that the system will be in a particular state once a measurement has been made. A directed line segment is just a segment of a straight line with an arrow to indicate its direction. Vectors having the same number of components, n say, can be added or subtracted in an element-by-element fashion to yield another vector of the same size. Any vector can be multiplied by a scalar by multiplying each element of the vector by that number, again yielding another vector of the same size. A qubit or quantum bit is a mathematical representation of the state of the simplest quantum system. A word of warning is necessary. Conventions differ among computer scientists, physicists, engineers and mathematicians as to how the inner product is defined.
The aim of this chapter is to present the vector nature of quantum states. We shall also become acquainted with the concepts of projectors and the mathematical tool of matrix calculation. In this chapter we shall elaborate the concepts presented in the previous chapter. To this end, let us consider another important property of the photon: polarization. Light is an electromagnetic wave and ordinarily it oscillates in all directions orthogonal to its propagation direction, as it happens for the light coming from the sun or spreading out of a normal electric bulb. However, when passing through a polarization filter (e.g., polarized sunglasses) light may acquire a privileged oscillation direction, and in this casewe say that it is polarized in that direction (in this way polarized sunglasses are able to eliminate reflected glare).
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.