Two State Quantum System

11 Key excerpts on "Two State Quantum System"

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  Thinking about Physics

  • Roger G. Newton(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  It can be applied, however, only when we can imagine a long (in principle, infinitely long) sequence of repetitions of the same experiment, or an ensemble of identical systems subject to similar conditions. An immediate consequence 51 CHAPTER 2 of this interpretation is that the quantum-mechanical state vector never refers to, or completely describes, an individual physical sys-tem, but always refers to an ensemble (which is the reason why some physicists resist this interpretation). We should therefore say more accurately than before that in quantum mechanics, the state of a system is defined as the ensemble to which the system belongs, and this ensemble is represented by a vector in Hilbert space or by a density operator, depending upon the precision with which the ensemble is defined. The quantum state of a system is not a direct description of the individual system. ENTANGLEMENT The issue of entanglement of distant particles, another characteris-tic of quantum states, opens up a separate set of questions. When specifying a classical state of several particles that do not interact with one another, their individual dynamical variables can be inde-pendently assigned, even if in the past they interacted. In quantum mechanics this is not necessarily so; there are the notorious phase correlations, which are difficult to grasp intuitively. Recall Bohm's version of the EPR experiment. 8 Two spin 1/2 particles emerge as the decay products of a spin 0 parent, fly-ing off in opposite directions. Since their total angular momentum must be zero, measurement of any spin projection of one of them allows us to infer that of the other, no matter how far away: they are entangled. If the vertical projection of the spin of particle 1 is measured and found to be up, a measurement of the vertical pro-jection of the spin of particle 2 must yield down. But if it was a horizontal projection of the spin of particle 1 that was measured 8 A.

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  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  2.1 Basic Theoretical Concepts 47 these will be developed later. The wording of this postulate is rather verbose because I have deliberately kept separate the physical concepts from the mathematical objects that represent them. When no confusion is likely to occur from a failure to make such explicit distinctions, we may say, The average of the observable R in the state p is - (2.1). [[ The concept of state is one of the most subtle and controversial concepts in quantum mechanics. In classical mechanics the word state is used to refer to the coordinates and momenta of an individual system, and so early on it was supposed that the quantum state description would also refer to attributes of an individual system. Since it has always been the goal of physics to give an objective realistic description of the world, it might seem that this goal is most easily achieved by interpreting the quantum state function (state operator, state vector, or wave function) as an element of reality in the same sense that the electromagnetic field is an element of reality. Such ideas are very common in the literature, more often appear-ing as implicit unanalyzed assumptions than as explicitly formulated arguments. However, such assumptions lead to contradictions (see Ch. 9), and must be abandoned. The quantum state description may be taken to refer to an ensemble of similarly prepared systems. One of the earliest, and surely the most prominent advocate of the ensemble interpretation, was A. Einstein. His view is concisely expressed as follows [Einstein (1949), quoted here without the supporting argument]: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

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  Quantum Robotics

  A Primer on Current Science and Future Perspectives

  • Prateek Tandon, Stanley Lam, Ben Shih, Tanay Mehta, Alex Mitev, Zhiyang Ong(Authors)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  e electron is measured to be in the ground state j0i with probability j˛j 2 and in the excited state j1i with probability jˇj 2 such that j˛j 2 C jˇj 2 D 1. e notation can be generalized for describing k-level quantum systems. In a k-level quan- tum system, the electron can be in one of k orbitals as opposed to just one of two states. e state of the k-level quantum system j i (when in superposition) can be expressed as: j i D k X i D1 ˛ i ji i s.t. k X i D1 j˛ i j 2 D 1: (2.2) Upon measurement of the system, the superposition collapses to state ji i with probability j˛ i j 2 . e ˛ i are complex numbers, potentially having both real and imaginary parts. 2.2 QUANTUM STATES AND ENTANGLEMENT Previously, we illustrated how a simple electron-orbital system could be represented with bra-ket notation. e ket is a mathematical abstraction, a notation representing a physical state that exists ¹Often, for us, just C N , the space of complex numbered vectors. ²e conjugate transpose of a matrix A D A T . To form the conjugate transpose of A, one takes the transpose of A and then computes the complex conjugate of each entry. e complex conjugate is simply the negation of the imaginary part (but not the real part). 2.2. QUANTUM STATES AND ENTANGLEMENT 7 in the real world. e beauty of this abstraction is that a variety of quantum systems, although implemented differently, can be described by the same underlying theory. For a particular quantum system being studied, a physicist using the bra-ket notation will specify some of the system’s elementary physical states as “pure states.” Pure states are defined as fundamental states of a quantum system that cannot be created from other quantum states. A pure state j i can be described via a density matrix: D j i h j : (2.3) In general, each quantum state (pure or not) has an associated density matrix. Not all states are pure; many are mixtures of pure states.

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  Time, Emergences and Communications

  • Bernard Dugué(Author)
  • 2018(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Bernard Dugué. © ISTE Ltd 2018. Published by ISTE Ltd and John Wiley & Sons, Inc. 2 Time, Emergences and Communications principles of these new mechanics early on and to expose them with undeniable elegance, together with a very platonic aesthetic concern. Dirac exposed the basic quantum mechanics, including the relativistic equation of the electron. In 1930, the great enigma of quantum entanglement was not known and specialists were satisfied with the orthodox interpretation acquired after bitter controversy during the Solvay Congress of 1927. A presentation of quantum mechanics should also include the so-called orthodox interpretation. Quantum mechanics are integrated by two blocks. In the first place, the preparation of the experiment described in the form of a superposition of states, each described by a complex function, ψ 1 , ψ 2 , ψ N , etc. The notation system uses the column vector (bra) or the line vector (ket), symbolized by a | bar, respectively followed or preceded by > or <; each state corresponds to an observable 1 . Let us imagine an experiment with two observable colors, red and blue. The system is described by an addition of the two vectors; each vector is assigned a coefficient whose square corresponds to the probability that the observable associated with its state may actually appear in the experiment. One of the requirements is that the sum of the squares equals one. State vector = √1/3 | ψ-red > + √2/3 | ψ-blue > – This state vector indicates that there is a one in three chance for red to appear and two chances out of three chances for blue to appear. When there is no observation, the wave function is supposed to obey a deterministic equation established by Schrödinger. Evolution is continuous in time and deterministic. When a measurement is made, there is a collapse or a reduction in the state vector.

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  Quantum Processes Systems, and Information

  • Benjamin Schumacher, Michael Westmoreland(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  Any other state of the qubit is a superposition of the states in the standard basis: | ψ = α 0 | 0 + α 1 | 1 . (2.69) The coefficients α k = k | ψ , and so they are probability amplitudes for the two possible outcomes of the standard measurement. Probability amplitudes for other measurements are given by brackets of | ψ with other basis states. If a qubit system is informa-tionally isolated, the change of its state with time will be described by an evolution operator U ( t ) . This same template applies to each of our three examples, though we have used the examples to emphasize different aspects of it. For instance, in the example of the spin-1/2 particle, it is clear that the choice of basis states was arbitrary. Fixing our standard 44 Qubits Fig. 2.15 Measurement apparatus for the |± basis states in the two-beam interferometer. measurement to be a measurement of S z , we have | 0 = | z + , | 1 = | z − . (2.70) On the other hand, there are many other possible measurements on the system. For example, the basis states |+ = 1 √ 2 | 0 + | 1 , |− = 1 √ 2 | 0 − | 1 , (2.71) are states of definite S x . (Note that |± = | x ± .) The choice of S z as the standard measurement, rather than S x or some other spin component, is merely a matter of convention. For the two-beam interferometer, our remarks about basis independence seem at first to be a bit strained. We let | 0 represent the state in which the photon is in the upper beam and | 1 the state in which the photon is in the lower beam. This choice of standard basis seems to be especially natural, since we can easily perform the corresponding measurement with a pair of photon detectors. In fact, we might be tempted to say that this is the only sort of measurement that makes sense. If the “which beam” measurement corresponds to S z , what could possibly correspond to a measurement of S x ? In fact, it is possible to measure the photon in any basis we choose.

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  An Introduction to the Mathematical Structure of Quantum Mechanics

  A Short Course for Mathematicians

  • F Strocchi(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2 Mathematical description of a quantum system 2.1 Heisenberg uncertainty relations and non-abelianess The puzzling situation, briefly discussed in Sect. 1.1, which character- izes the conflict between classical physics and the experimental results on microscopic (atomic) systems, was brilliantly clarified by Heisenberg, who identified the basic crucial point which marks the deep philosophical dif- ference between classical physics and the physics of microscopic (atomic) systems, briefly called quantum systems. In the mathematical description of classical systems outlined in Sect. 1.2, it was recognized that dispersion free states are an unrealistic ideal- ization (which would require infinite precision of measurements) and more correctly a state identified by realistic measurements defines a probability distribution on the random variables which describe the observables. Nev- ertheless, from the experience with classical macroscopic systems it was taken for granted that the ideal limit of dispersion free states could be approximated as closely as one likes, by refining the apparatus involved in the preparation and the identification of the state. This means that, e.g. even if one cannot prepare a state in which the q’s and p’s take a sharp value, one can prepare states win which the mean square deviations or variances (A, q ) 2 , (AWP)~ are as small as one likes. This agrees with the assumption that the C*-algebra of observables of a classical system is abelian (see also the discussion below). This philosophical prejudice seems strongly supported by experiments on macroscopic systems, but it is not so obvious when one deals with microscopic or atomic systems. As a matter of fact, sharper and sharper measurements of the position 37 38 Mathematical descrbtion of a quantum system of a particle, require instruments capable of distinguishing points at smaller and smaller scales.

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  Quantum Computing

  From Linear Algebra to Physical Realizations

  • Mikio Nakahara, Tetsuo Ohmi(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  2 Framework of Quantum Mechanics Quantum mechanics is founded on several postulates, which cannot be proven theoretically. They are justified only through an empirical fact that they are consistent with all the known experimental results. The choice of the postu-lates depends heavily on authors’ taste. Here we give one that turns out to be the most convenient in the study of quantum information and computation. For a general introduction to quantum mechanics, we recommend [1, 2, 3, 4], for example. [5] and [6] contain more advanced subjects than those treated in this chapter. 2.1 Fundamental Postulates Quantum mechanics was discovered roughly a century ago. In spite of its long history, the interpretation of the wave function remains an open question. Here we adopt the most popular one, called the Copenhagen interpreta-tion . A 1 A pure state in quantum mechanics is represented in terms of a normal-ized vector | ψ in a Hilbert space H (a complex vector space with an inner product): ψ | ψ = 1. Suppose two states | ψ 1 and | ψ 2 are physi-cal states of the system. Then their linear superposition c 1 | ψ 1 + c 2 | ψ 2 ( c k ∈ C ) is also a possible state of the same system. This is called the superposition principle . A 2 For any physical quantity (i.e., observable ) a , there exists a corre-sponding Hermitian operator A acting on the Hilbert space H . When we make a measurement of a , we obtain one of the eigenvalues λ j of the operator A . Let λ 1 and λ 2 be two eigenvalues of A : A | λ i = λ i | λ i . Suppose the system is in a superposition state c 1 | λ 1 + c 2 | λ 2 . If we measure a in this state, then the state undergoes an abrupt change to one of the eigenstates corresponding to the observed eigenvalue: If the observed eigenvalue is λ 1 ( λ 2 ), the system undergoes a wave function collapse as follows: c 1 | λ 1 + c 2 | λ 2 → | λ 1 ( | λ 2 ), and the state imme-diately after the measurement is | λ 1 ( | λ 2 ). Suppose we prepare many 29

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  Applied Quantum Computers

  Learn about the Concept, Architecture, Tools, and Adoption Strategies for Quantum Computing and Artificial Intelligence (English Edition)

  • Dr. Patanjali Kashyap(Author)
  • 2023(Publication Date)
  • BPB Publications

    (Publisher)

  Visualize if you could toss a coin that could drop not only in a head or tail position but in both positions at the same time. You can name it as a “quantum toss”. Because this is exactly like the quantum behaviour, where one object stays in “up or down” at the same moment. However, the moment you detect this is a quantum coin, it is compulsory to take either heads or tails not ever knowing what position it was in before. This is one reason that one needs to be cautious when measuring qubits since they change as soon as detected.

  The qubit maybe 0 or 1 or have any proportion among them. A qubit (a state of atom or electron or ion) can be assumed as an arrow that can point all the way in three-dimensional space; when it points up, the qubit is said to be in the 1 state, down is the 0 state, and any extra direction is a mixture of both. Superposition is an essential feature of quantum computing. In conservative (or traditional or contemporary or classical = these are used throughout interchangeably in this this book) computing, a state of n bits a whole number is defined using n digits in a sequence of 0 or 1. In quantum computing, it’s more complex, needing 2n-1 complex numbers to define a state of “n” qubits. This means that an exponential number of classical bits would be required to store the state of a quantum computer, even approximately.

  Two qubits in superposition can signify four states, three represent 8, and 4 represents 16 entangled (associated with each other, so that change in one qubit can incestuously reflect in other) states. Likewise, qubits epitomize 216 = 65,536, and 32 would be 4,294,967,296 options. With just a few qubits, a huge volume of information processing will be achieved. Assuming that a byte (8 bits) is the elementary unit used to stock information in any system, then the number of values that can be deposited concurrently in a quantum computer would be 2n where n is the number of qubits. Compare this against the processing capacity of a classic computer and you realize why qubits are powerful.

  A coinage has two states and makes a decent bit. But a deprived qubit does not, since it cannot keep on in superposition of head and tail for very long like a conventional one. A solitary nuclear spin can be a very decent qubit since the superposition of being associated with or contrary to an exterior magnetic field can be preceding for an extended time. But then again it can be problematic to shape a quantum computer - nuclear spin since their connection is so minor that it is tough to measure the orientation of a single nucleus. A truthful execution must strike an equilibrium between these constrictions.

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  Quantum Mechanical Foundations of Molecular Spectroscopy

  • Max Diem(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  Since the position and momentum can never be determined simultaneously at any point in time, the position (or momentum) in the future cannot be precisely predicted, only the probability of either of them. This is manifested in the postulate that all properties, present or future, of a particle are contained in a quantity known as the wavefunction Ψ of a system. This function, in general, depends on spatial coordinates and time; thus, for a one‐dimensional motion (to be discussed first), the wavefunction is written as Ψ(x, t). The probability of finding a quantum mechanical system at any time is given by the integral of the square of this wavefunction: ∫Ψ(x, t) 2 d x. This is, in fact, one of the “postulates” on which quantum mechanics is based to be discussed next. Different authors list these postulates in different orders and include different postulates necessary for the description of quantum mechanical systems [ 1 ]. Quantum mechanics is unusual in that it is based on postulates, whereas science, in general, is axiom‐based. 2.1 Postulates of Quantum Mechanics Postulate 1: The state of a quantum mechanical system is completely defined by a wavefunction Ψ(x, t). The square of this function, or in the case of complex wavefunction, the product Ψ*(x, t) Ψ(x, t), integrated over a volume element d τ (= d x d y d z in Cartesian coordinates or sin 2 θ d θ d φ in spherical polar coordinates) gives the probability of finding a system in the volume element d τ. Here, Ψ*(x, t) is the complex conjugate of the function Ψ(x, t). This postulate contains the transition from a deterministic to probabilistic description of a quantum mechanical system

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  Mesoscopic Physics meets Quantum Engineering

  • Sergey N Shevchenko(Author)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2

  QUANTUM MECHANICS OF QUBITS

  (Dynamical behaviour of a two-level system) “In addition to the well-studied statics of the quantum systems, mesoscopic physics adds the new aspects of dynamics in quantum mechanics.” [Valiev 2005]

  The problem of dynamical behaviour in a two-level system deserves detailed discussion. First, this gives results for fundamental problems. Second, it is very topical for mesoscopic systems, which have parameters tunable in a wide range, and where the regimes of control and interferometry are important. Third, it is a good example of the accurate solution of a realistic problem. Fourth, this will allow us to introduce useful formulas and approaches.

  2.1.Two-level system

  Consider a two-level system driven periodically. A two-level system with the energy bias ε and the tunneling amplitude Δ is described by the pseudospin Hamiltonian

  in terms of the Pauli matrices σx,z (we wrote about this above, when we discussed Eq. (1.9)). Usually, the value Δ is assumed to be constant, while the bias ε is considered to be a time-dependent controlling parameter. The most interesting is the situation with monochromatic time dependence,

  with the amplitude A, frequency ω, and offset ε0 .

  We can split the one-qubit Hamiltonian into the time-independent and time-dependent parts, H = H0 + V(t) with

  First, let us define eigenvectors and eigenfunctions of the operator H0 from the stationary Schrödinger equation H0 |ψ = E|ψ . Then for |ψ = α|0 + β|1 we have

  Equating the determinant to zero, we obtain

  where we defined the distance between the qubit energy levels .

  Further we solve the system of equations (2.4); from the first equation it follows

  and from the normalization condition 1 = α2 + β2 we have

  Then we find α from the expression α2 = 1 − β2 and take into account that from Eq. (2.6) we have sgn α = ∓sgn β. So, α± = γ∓ and β± = ∓γ± , and for the eigenfunctions of H0

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  Quantum Technologies

  For Engineers

  • Rainer Müller, Franziska Greinert(Authors)
  • 2023(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  atomic physics began to flourish. The structure of atoms, their states and bonds could be unraveled. This achievement helped to put chemistry on a firm theoretical footing.

  In spectroscopy – the experimental investigation of line spectra – increasingly precise resolution could be achieved. Ever finer effects were observed in the complicated spectra of atoms and molecules and were explained on the basis of quantum mechanics. This detailed understanding of energy levels and the transitions between them is now an indispensable foundation for quantum technologies. The physical realization of qubits on the basis of atoms and ions is based on the complete control of these degrees of freedom.

  Figure 2.2 Absorption and emission as transition between energy levels.

  The mathematical formalism of quantum mechanics can be regarded as a wave theory of matter with some limitations. Electrons are described by a wave function

  ψ ( x , t )

  , and the Schrödinger equation governing their propagation bears some resemblance to a wave equation. However, an all-too-intuitive notion of “electrons as waves” is already made impossible by the fact that ψ is, in general, complex-valued. In addition, the space in which the propagation takes place is not three-dimensional for multi-particle systems. With an increasing number of particles, it quickly becomes very high-dimensional. As we will see, this high dimensionality is ultimately why quantum computers can solve large problems with a relatively small number of qubits.

  The term quantum mechanics refers to the theory described above, whose basic equation is the Schrödinger equation. It applies to quantum objects with mass, in particular electrons in atoms, molecules, and solids. A more general description is needed for light, where the corresponding quantum objects are called photons. They have no mass and travel at the speed of light. Photons can be created or destroyed in absorption and emission processes. They are described by a quantum field theory (quantum electrodynamics), the basic features of which will be discussed later. The broader term quantum physics

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