Unitary and Anti-Unitary Quantum Descriptions of the Classical Not Gate
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On Quaslfree States of CAR and Bogoliubov Automorphisms
Publ. RIMS Kyoto Univ. Vol. 6 (1970/71), 385-442 On Quaslfree States of CAR and Bogoliubov Automorphisms By Huzihiro ARAKI Abstract A necessary and sufficient condition for two quasifree states of CAR to be quasiequivalent is obtained. Quasif ree states is characterized as the unique KMS state of a Bogoliubov automorphism of CAR. The structure of the group of all inner Bogoliubov automorphisms of CAR is clarified. §1. Introduction A classification of gauge invariant quasifree states of the canonical anticommutation relations (CAR) up to quasi and unitary equivalence is recently obtained by Powers and St0rmer [12]. We shall generalize their result to arbitrary quasifree states. We use the formalism developped earlier [2] and study quasifree state <ps of a self dual CAR algebra. It is then shown that <pSi and <pS2 are quasiequivalent if and only if Sllz — Sllz is in the Hiibert Schmidt class. For a gauge invariant quasifree state <pA in the paper of Powers and St0rmer, S = A@(1 — A) and hence our result is a direct generali- zation of Powers and St0rmer. The quasifree primary state <ps for which 5 does not have eigen- value 1 is shown to be the unique KMS state for the one parameter group r(£/GO) of Bogoiiubov * automorphisms of CAR, where r(t/GO) corresponds to a unitary transformation U(£)=ex.piAH of the direct sum of testing function spaces of creation and annihilation operators and H is related to 5 by S= (l + e"*)"1. This is used to simplify some of arguments. A quasifree state q>s is primary unless 1/2 is an isolated point spectrum of 5, has an odd multiplicity and S(l — S1) is in the Received May 6, 1970 and in revised form November 17, 1970. -
Linear Operators
C H A P T E R 10 Linear Operators Recall that a linear transformation T ∞ L(V) of a vector space into itself is called a (linear) operator. In this chapter we shall elaborate somewhat on the theory of operators. In so doing, we will define several important types of operators, and we will also prove some important diagonalization theorems. Much of this material is directly useful in physics and engineering as well as in mathematics. While some of this chapter overlaps with Chapter 8, we assume that the reader has studied at least Section 8.1. 10.1 LINEAR FUNCTIONALS AND ADJOINTS Recall that in Theorem 9.3 we showed that for a finite-dimensional real inner product space V, the mapping u ’ Lu = Óu, Ô was an isomorphism of V onto V*. This mapping had the property that Lauv = Óau, vÔ = aÓu, vÔ = aLuv, and hence Lau = aLu for all u ∞ V and a ∞ ®. However, if V is a complex space with a Hermitian inner product, then Lauv = Óau, vÔ = a*Óu, vÔ = a*Luv, and hence Lau = a*Lu which is not even linear (this was the definition of an anti- linear (or conjugate linear) transformation given in Section 9.2). Fortunately, there is a closely related result that holds even for complex vector spaces. Let V be finite-dimensional over ç, and assume that V has an inner prod- uct Ó , Ô defined on it (this is just a positive definite Hermitian form on V). Thus for any X, Y ∞ V we have ÓX, YÔ ∞ ç. -
Basic Concepts in Quantum Computation
Basic concepts in quantum computation Artur Ekert, Patrick Hayden and Hitoshi Inamori Centre for Quantum Computation, University of Oxford, Oxford OX1 3PU, United Kingdom February 1, 2008 1 Qubits, gates and networks Consider the two binary strings, 011, (1) 111. (2) The first one can represent, for example, the number 3 (in binary) and the second one the number 7. In general three physical bits can be prepared in 23 = 8 different configurations that can represent, for example, the integers from 0 to 7. However, a register composed of three classical bits can store only one number at a given moment of time. Enter qubits and quantum registers: A qubit is a quantum system in which the Boolean states 0 and 1 are rep- resented by a prescribed pair of normalised and mutually orthogonal quantum states labeled as 0 , 1 [1]. The two states form a ‘computational basis’ and any other (pure) state{| i | ofi} the qubit can be written as a superposition α 0 + β 1 for some α and β such that α 2 + β 2 = 1. A qubit is typically a microscopic| i | i system, such as an atom, a nuclear| | | spin,| or a polarised photon. A collection of n qubits is called a quantum register of size n. We shall assume that information is stored in the registers in binary form. arXiv:quant-ph/0011013v1 2 Nov 2000 For example, the number 6 is represented by a register in state 1 1 | i ⊗ | i⊗ 0 . In more compact notation: a stands for the tensor product an 1 | i | i | − i⊗ an 2 . -
1 Complex Numbers in Quantum Mechanics
1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic fields are all real quantities, and the equations describing them, Newton's laws, Maxwell's equations,etc. are all differential equations involving strictly real quantities. p Quantum mechanics is different. Factors of i = −1 are everywhere, for example from Heisenberg we have his famous commutation relation, QP − PQ = ih;¯ and from Schr¨odingerhis equally famous equation h¯ @ Ψ = HΨ: i t The wave function Ψ is complex and so is practically every other quantity needed to formulate quantum mechanics. Granted that classical mechanics fails at distances of nanometer or less, why is it suddenly so necessary that everything be complex? Comment 1 An interesting question is the distance scale below which a quantum treatment is essential. A nanometer seems like a reasonable choice. But even more intriguing is the question of what happens on much bigger scales, centimeters, meters, and larger. Does quantum mechanics still govern things, or does it somehow break down? In 2011, it is fair to say no definite answer to this question is known. Returning to the question of why quantum mechanics requires complex quantities, one reason becomes apparent upon trying to describe wavelike phenomena for material particles, electrons, neutrons, etc. To see this point, start with a classical electromagnetic wave. For a plane wave traveling along the z−axis, we can write the following formula for the electric field, ~ E = E0x^ cos(kz − !t): (1) To be more realistic, we should of course superpose nearby frequencies so that the wave has a finite extent. -
Quantum State Transformation
Quantum Computing 1 ✬ ✩ Quantum State Transformation Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 2 ✬ ✩ Quantum State Transition Postulate It seems that nature does not allow arbitrary state transformation. Change of states (over time) of a closeda quantum mechanical system are caused by a specific class of transformations, mathematically known as unitary transformations. aBy a closed system we mean that no measurement is performed on the sys- tem. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 3 ✬ Unitary Transformation ✩ A unitary transformation U : H H is an • → isomorphism, where H an inner product space (Hilbert space). So U is a bijection that preserves inner • product. In our notation U : x y and if • | i 7→ | i x , x′ H, then | i | i ∈ < x , x′ >=< Ux , Ux′ >. | i | i | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 4 ✬ ✩ Unitary Matrix A unitary transformation will be represented by a unitary matrix. We call a complex matrix U as unitary if its conjugate transpose (Hermitian transpose) U † (or U ∗) is its inverse i.e UU † = I. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 5 ✬ ✩ One Qubit Transformation 1 0 We encode 0 = 0 and 1 = 1 . • | i | i So an 1-qubit unitary transformation or • 1-qubit gate is a 2 2 unitary matrix U i.e. × U †U = I. 1 This means U † = U − and U † is also unitary. • Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 6 ✬ ✩ Properties It is clear that any quantum transformation is reversible, as U x = y if and only if | i | i U † y = x . | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 7 ✬ ✩ Properties A unitary operator U is linear by definition. -
Automorphisms and Quasi-Free States of the CAR Algebra N
Commun. math. Phys. 43, 181—197 (1975) © by Springer-Verlag 1975 Automorphisms and Quasi-Free States of the CAR Algebra N. M. Hugenholtz Instituut voor Theoretische Natuurkunde, Rijksuniversiteit, Groningen, The Netherlands R. V. Kadison Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, USA Received September 25, 1974 Abstract. We study automorphisms of the CAR algebra which map the family of gauge-invariant, quasi-free states of the CAR algebra onto itself and show (Theorem 3.1) that they are one-particle automorphisms. 1. Introduction The problem discussed in this paper arose from questions regarding equi- librium states of thermodynamical systems. Equilibrium states have been extensively discussed in the framework of C*-algebras of observables (see, for example [6, 15]). Such states are labeled by a very small number of parameters. For example, in the case of a gas of identical particles, the equilibrium states are labeled by the temperature, the chemical potential, and the average velocity of the particles - quantities which relate directly with the conserved quantities - energy, particle number, and total momentum. Since conserved quantities are in one-one correspondence with one-parameter groups of transformations which leave the Hamiltonian invariant, we can describe the situation in a way which remains meaningful for infinite systems. Equilibrium states of thermodynamical systems are labeled by a very small number of parameters which relate directly with one- parameter, automorphism groups of the observable algebra that commute with the time-evolution automorphisms. The fact that there are so few parameters involved, which is related to the fact that there are only a small number of one-parameter groups of automorphisms that commute with the time-evolution, is an aspect of the ergodic nature of most large physical systems.