d a central role in various aspects of quantum information processing because quantum teleportation, [1] quan


 Gerard Matthews
 1 years ago
 Views:
Transcription
1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society Vol. 49, No. 3, March 5, 008 Calculation of Entanglement Entropy for ContinuousVariable Entangled State Based on General TwoMode Boson Exponential Quadratic Operator in Fock Space DAI FangWen and MA Lei Department of Physics, East China Normal University, Shanghai 0006, China Received February 5, 007 Abstract We obtain an explicit formula to calculate the entanglement entropy of bipartite entangled state of general twomode boson exponential quadratic operator with continuous variables in Fock space. The simplicity and generality of our formula are shown by some examples. PACS numbers: Ud, p, Ta Key words: entanglement entropy, continuousvariable entangled states, linear quantum transformation Introduction 0 Σ B =, Continuous quantum variables have emerged as an alternative to discretelevel systems for performing quan 0 tum information processing tasks. Entanglement plays M M M c M Σ B = c d a central role in various aspects of quantum information processing because quantum teleportation, [] quan Thus, from Eq. one can conveniently obtain the c b c. 3 tum cryptography, [] and quantum dense coding, [3] etc., entanglement entropy of the states in Eq. only by calculating the eigenvalues of negative Hermitian matrix N. are all based on it. Thus, it is interesting to measure the entanglement of continuous variables systems. The von But in general cases, the form of a bipartite continuousvariable entangled state is not written as Eq. explicitly, Neumann entropy [4] of either partial trace of the density operator for any bipartite pure entangled state is considered as a good measurement of quantum entangle complicated to use Eq.. and then N is more implicitly. In certain cases, it is too ment. Based on the Schmidt decompositions, Parker et In this paper, we shall derive an explicit formula al. [5] have developed an elegant method to calculate the to calculate the entanglement entropy of any bipartite entanglement of the bipartite pure entangled states with continuousvariable entangled states in Fock space. This continuous variables by means of the integral eigenvalue paper is arranged as follows. In Sec., we shall obtain equations in the coordinatemomentum space. But in certain cases, it is somewhat harder to find such a Schmidt the general form of the bipartite continuousvariable entangled states, and then derive the explicit formula to calculate the entanglement entropy of the states. In Sec. 3, basis for the continuous variable systems. By virtue of the linear quantum transformation theory LQTT, [6] Lu et al. [3,4] have provided an alter by virtue of the explicit formula, we shall calculate the degree of entanglement of some bipartite continuousvariable entangled states. A brief conclusion will close the paper native method to calculate the entanglement entropy for in the last section. any bipartite pure entangled Gaussian state with continuous variables in Fock space. If the density operator of Explicit Formula for Calculating Entanglement Entropy of Bipartite Continuous the bipartite continuousvariable entangled state can be written as follows: { a ρ = A 0 : exp [a, a a Variable Entangled States M + a a, a M. General Form of Entangled State of Two a Mode Boson Exponential Quadratic Operator a ]} + a, a M : a where A 0 is a normalization factor, M i i =, are Hermitian matrices, and M = e f f e with e and f being two arbitrary complex numbers. Then the entanglement entropy of any bipartite pure entangled Gaussian state with continuous variables in Fock space can be calculated by the following formula, E = { ln det e N + tr [ N e N ]}, ln where a d N = lnm, M =, ac bd =, b c Let us consider the entangled state of a twomode boson operator with continuous variables as follows: ψ = U 00, 4 where U is a twomode boson exponential quadratic operator. If we denote Λ = a, ã a = a, a ã = a, a, 5 a i and a i i =, are twomode boson creation and annihilation operators, respectively. Thus, without any loss of generality, the ordinary form of U can be written as U = exp ΛNΣ Λ B, 6 where Σ B, N C 4 4, Σ B = 0 I I 0, I = 0 0, and N satisfies NΣ B = NΣ B. The project supported by the National Fundamental Research Program under Grant No. 006CB904 and National Natural Science Foundation of China under Grant No
2 No. 3 Calculation of Entanglement Entropy for ContinuousVariable Entangled State Based on General TwoMode 59 Denoting M = e N = A D B C, A, B, C, and D are four complex matrices, U and M thus satisfy [6] By using Eq. 8, we immediately have UΛU = ΛM, 7 MΣ B M = Σ B. 8 Ã B BA = 0, Ã C BD =, 9 If C exists, by using Eq. 9, we can rewrite matrix M as I C M = e N D C 0 I 0 = 0 I 0 C BC. 0 I Therefore, from LQTT, the action of U is equivalent to the product of three LQT operators, i.e., U = U + U 0 U, where U +, U 0, and U satisfy the following expressions, I C U + ΛU+ D = ΛM + = Λ, 0 I C U 0 ΛU0 0 = ΛM 0 = Λ, 0 C I 0 U ΛU = ΛM = Λ BC, I By virtue of LQTT, we can easily obtain 0 C D N + = lnm + =, 0 0 lnc 0 N 0 = lnm 0 =, N = lnm = 0 ln C, BC 0 U + = exp ΛN +Σ B Λ = exp a C Dã, U 0 = exp ΛN 0Σ B Λ = detc / : exp [ a C a ] :, U = exp ΛN Σ B Λ = exp ãbc a. 4 U +, U 0, and U only contain the terms of a i a j, a i a j, and a i a j, respectively, where i, j =,. By noticing expa i a j 00 = exp a i a j 00 = 00, 5 the bipartite continuousvariable entangled state 4 can be rewritten as ψ = U 00 = U + U 0 U 00 = det C / exp a C Dã From the above discussions, we obtain the general form of the entangled state of any twomode boson exponential quadratic operator 4 as the following form, ψ = det C / expαa + βa + γa a 00, 7 where α, β, and γ are determined by C D.. Explicit Formula of Calculating Entanglement Entropy It is easy to see that the entanglement entropy, i.e., the degree of entanglement of the entangled state 7 between a and a, is determined by α, β, and γ, but which parameter is more predominant? Are α and β really symmetrical? If γ = 0, obviously, the state 7 is a separable state, and its entanglement entropy should be zero. When γ 0, the density operator of Eq. 7 can be written as ρ =detc expαa + βa + γa a exp α a + β a + γ a a. 8 By noticing the normal ordering form of the twomode vacuum state projector = : e a a a a : 9 after substituting Eq. 9 into Eq. 8, we have, { α ρ =det C : exp a [, a α so we get α M = α a a β + a, a β a a γ 0 + a, a 0 γ a ]} : 0 β γ 0, M = β, M = 0 γ. By using Eq. 3, we find the explicit form of M as follows: M = γ 4 4 α 4 β 4 αβγ + α β γ α 4α β + β γ γ. α 4α β + βγ 4 β Then, substituting it into Eq., we obtain the explicit formula for calculating the entanglement entropy for the continuous variables entangled states 7 as follows: L 4 L + L E = log 4 L L 4L log 4 L + L log 4 L L 4 + log 4 + log L, 3 where the logarithm is defined in the complex domain and L = γ + γ 4 α 4 β 4 αβγ γ + α β γ. 4 From Eq. 4, we can see that α and β are symmetrical and exchangeable for calculating the entanglement entropy. Obviously, it is true because the modes a and a are symmetrical in Fock space. a
3 59 DAI FangWen and MA Lei Vol. 49 When 0 L <, equation 3 can be simplified as E = + L L + i 4 L L arctan log L. 5 When L >, equation 3 can be simplified as E = L + L log L + L 4 log L. 6 With Eqs. 5 and 6, one can easily have E L and E L +, so we have E L=. 7 Equation 7 shows that if, the entangled state 7 will be maximally entangled. The entanglement entropy is plotted as a function of L in Fig., and figure shows the entanglement entropy as a function of α and γ β is fixed to be 0.5. Fig. Entanglement entropy as a function of L E when L. Fig. Entanglement entropy as a function of α and γ β is fixed to be Some Applications 3. Calculating Entanglement Entropy of Common TwoMode Squeezed Vacuum State Now, let us consider twomode squeezed vacuum state ξ = exp ξ a a ξa a 00, where ξ = r e iθ is the squeezing parameter, by virtue of LQTT, [6] we can obtain ξ = sech rexp a a eiθ tanhr 00, where α = 0, β = 0, γ = e iθ tanhr, according to Eq. 4, we have By using Eq. 6, we can easily obtain L = tanh r + coth r >. E = cosh r log cosh r sinh r log sinh r. This result has been obtained by other authors through different methods, [5,68] but our formula is more explicit. figure 3a shows that the amount of entanglement entropy of twomode squeezed vacuum state is approximately linear against the amount of squeezing parameter r. 3. Calculating Entanglement Entropy of TwoMode OneSided Squeezed Vacuum State Reference [5] has constructed twomode onesided squeezed vacuum state { λ S = exp a + a 4[ a + ]} [ a 00 = sech / λ exp where α = β = tanhλ/4, γ = tanhλ/, so tanhλ 4 a + ] a 00, L = 4 coth λ > and then by Eq. 6 we obtain E = coshλ log + cosh λ cosh λ log sinh λ log cosh λ.
4 No. 3 Calculation of Entanglement Entropy for ContinuousVariable Entangled State Based on General TwoMode 593 Figure 3b shows that the amount of entanglement entropy of twomode onesided squeezed vacuum state is also approximately linear against the amount of squeezing parameter λ, but is about only half of amount of entanglement entropy of the twomode squeezed vacuum state with the squeezing parameter λ = r. Fig. 3 a Entanglement entropy of twomode squeezed vacuum state as a function of squeezing parameter r. b The entanglement entropy of twomode onesided squeezed vacuum state as a function of squeezing parameter λ. 3.3 Calculating the Entanglement Entropy of Entangled States Produced by a Beam Splitter with Squeezed States Inputs When the two input fields are squeezed, the output state from a beam splitter is [6] B θ, φs ζ S ζ 00, 8 where B θ, φ is the beam splitter operator, [ θ B θ, φ = exp a a e iφ a a e iφ]. t = cosθ/ and r = sinθ/ are the amplitude reflection and transmission coefficients. The beam splitter gives the phase difference between the reflected and transmitted fields. S ζ is the singlemode squeezed operator S ζ = exp ζ a ζa, ζ = s e iϕ is the squeezing parameter. By virtue of LQTT, [6] we can obtain B θ, φs ζ S ζ 00 = coshs coshs exp αa + βa + γa a 00, where α = [ e iϕ cos θ tanhs + e iϕ+φ sin θ ] tanhs, β = [ e iϕ φ sin θ tanhs + e iϕ cos θ ] tanhs, γ = sinθ e iϕ φ tanhs e iϕ+φ tanhs. 9 Thus, substituting Eq. 9 into Eq. 4, the entanglement entropy of the state 8 can be calculated explicitly. The entanglement entropy of the state 8 is plotted in Fig. 4 against the squeezing parameter s and reflection coefficient for s = 0.5. The relative phase φ = 0 in Fig. 4a and φ = π/ in Fig. 4b. These two figures have been obtained in Ref. [6], but it did not give the explicit form of the entanglement entropy. It is necessary to point out that in Ref. [6], natural logarithm is used to calculate the entanglement entropy, but in this paper, the logarithm with base is used. 3.4 Calculate the Entanglement Entropy of the Thermal Vacuum State of a Free Boson Now, we investigate the entanglement entropy of the thermal vacuum state of a free boson, [8], 0 β = e βω / exp e βω/ a ã 00. Substituting α = β = 0, γ = e βω/ into Eq. 4, we have and then, by using Eq. 6, we can easily get L = e βω + e βω >, E = log e βω e βω e βω log e βω.
5 594 DAI FangWen and MA Lei Vol. 49 Reference 8] obtained the same result by virtue of the technique of IWOP, [5] but we get the result more simply. Fig. 4 Entanglement entropy of the beamsplitter output field. The squeezing parameter for one squeezed input is fixed to S = 0.5 while the squeezing parameter for other squeezed state is varied from S = 0 to. The transitivity is R R r. The beam splitter gives phase difference φ = 0 a and φ = π/ b between reflected and transmitted fields. 4 Conclusions In Fock space, by virtue of LQTT, we show that the entangled state of any twomode boson exponential quadratic operator ψ = U 00 can be rewritten as ψ = detc / expαa + βa + γa a 00, and we obtain the explicit formula to calculate the entanglement entropy of the state. The simplicity and generality of our formula are shown by several entangled states of continuous variables. These states include the common twomode squeezed vacuum state, the twomode onesided squeezed vacuum state, the output states produced by a beam splitter with squeezed states inputs and the thermal vacuum state of a free single boson. It is shown that LQTT and the explicit formula we obtained are power tools for investigating the degree of entanglement for entangled states of twomode boson system with exponential quadratic operator. Acknowledgments We thank Dr. JinMing Liu for beneficial discussions. References [] C.H. Bennett, G. Brassard, et al., Phys. Rev. Lett [] K. Ekert, Phys. Rev. Lett [3] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett [4] C.H. Bennett, H.J. Herbert, S. Popescu, and B. Schumacher, Phys. Rev. A [5] S. Parker, S. Bose, and M.B. Plenio, Phys. Rev. A [6] Y.D. Zhang and Z. Tang, J. Math. Phys [7] Y.D. Zhang and Z. Tang, Nuovo Cimento B [8] L. Ma and Y.D. Zhang, Nuovo Cimento B [9] Y.D. Zhang and Z. Tang, Commun. Theor. Phys. Beijing, China [0] Y.D. Zhang, L. Ma, X.B. Wang, et al., Commun. Theor. Phys. Beijing, China [] S.X. Yu and Y.D. Zhang, Commun. Theor. Phys. Beijing, China [] L. Ma, Doctoral Dissertation, University of Science and Technology of China 995. [3] H.X. Lu, Z.B. Chen, J.W. Pan, and Y.D. Zhang, LANL eprint quantph/ [4] H.X. Lu, Doctoral Dissertation, University of Science and Technology of China 003. [5] H.Y. Fan, Int. J. Mod. Phys. B [6] M.S. Kim, W. Son, V. Buzek, and P.L. Knight, Phys. Rev. A [7] S.J. van Enk, Phys. Rev. A [8] X.T. Liang, Commun. Theor. Phys. Beijing, China
Phillips Policy Rules 3.1 A simple textbook macrodynamic model
1 2 3 ( ) ( ) = ( ) + ( ) + ( ) ( ) ( ) ( ) [ ] &( ) = α ( ) ( ) α > 0 4 ( ) ( ) = ( ) 0 < < 1 ( ) = [ ] &( ) = α ( 1) ( ) + + ( ) 0 < < 1 + = 1 5 ( ) ( ) &( ) = β ( ( ) ( )) β > 0 ( ) ( ) ( ) β ( ) =
More informationQuantum Discord in TwoQubit System Constructed from the Yang Baxter Equation
Commun. Theor. Phys. 61 (014) 349 353 Vol. 61, No. 3, March 1, 014 Quantum Discord in TwoQubit System Constructed from the Yang Baxter Equation GOU LiDan ( ), 1 WANG XiaoQian ( ¼), 1, XU YuMei (Å Ö),
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More informationManybody entanglement in decoherence processes
PHYSICAL REVIEW A 68, 63814 3 Manybody entanglement in decoherence processes Helen McAneney, Jinhyoung Lee, and M. S. Kim School of Mathematics and Physics, Queen s University, Belfast BT7 1NN, United
More informationclassical vs. quantum statistics, quasiprobability distributions
Lecture 6: Quantum states in phase space classical vs. quantum statistics, quasiprobability distributions operator expansion in phase space Classical vs. quantum statistics, quasiprobability distributions:
More informationExact Jacobian Elliptic Function Solutions to sinegordon Equation
Commun. Theor. Phys. (Beijing, China 44 (005 pp. 3 30 c International Academic Publishers Vol. 44, No., July 5, 005 Exact Jacobian Elliptic Function Solutions to sinegordon Equation FU ZunTao,,, YAO
More informationPhysics 505 Fall 2007 Homework Assignment #2 Solutions. Textbook problems: Ch. 2: 2.2, 2.8, 2.10, 2.11
Physics 55 Fall 27 Homework Assignment #2 Solutions Textbook problems: Ch. 2: 2.2, 2.8, 2., 2. 2.2 Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting
More informationTeleportation of a twoparticle entangled state via W class states
Teleportation of a twoparticle entangled state via W class states ZhuoLiang Cao Wei Song Department of Physics nhui University Hefei 339 P. R. of China bstract scheme for teleporting an unknown twoparticle
More informationQuantum key distribution via quantum encryption
Quantum key distribution via quantum encryption YongSheng Zhang, ChuanFeng Li, GuangCan Guo Laboratory of Quantum Communication and Quantum Computation and Department of Physics, University of Science
More informationQuantification of entanglement via uncertainties
Quantification of entanglement via uncertainties Barış Öztop Bilkent University Department of Physics September 2007 In blessed memory of Alexander Stanislaw Shumovsky Outline Introduction what can we
More informationPh 219a/CS 219a. Exercises Due: Friday 2 December 2005
1 Ph 219a/CS 219a Exercises Due: Friday 2 December 2005 3.1 Two inequivalent types of tripartitite entangled pure states Alice, Bob, and Charlie share a GHZ state of three qubits: GHZ = 1 2 ( 000 + 111
More informationarxiv:quantph/9607009v1 11 Jul 1996
Distillability of Inseparable Quantum Systems Micha l Horodecki Department of Mathematics and Physics University of Gdańsk, 80 952 Gdańsk, Poland arxiv:quantph/9607009v1 11 Jul 1996 Pawe l Horodecki Faculty
More informationMath Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, f(z) = 3x + y + i (3y x)
Math 311  Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, 2014 Question 1. [p 77, #1 (a)] Apply the theorem in Sec. 22 to verify that the function is entire. f(z) = 3x + y + i
More informationTeleportation improvement by inconclusive photon subtraction
PHYSICAL REVIEW A 67, 03314 003 Teleportation improvement by inconclusive photon subtraction Stefano Olivares, 1 Matteo G. A. Paris, and Rodolfo Bonifacio 1 1 Dipartimento di Fisica and Unità INFM, Università
More informationSmall oscillations with many degrees of freedom
Analytical Dynamics  Graduate Center CUNY  Fall 007 Professor Dmitry Garanin Small oscillations with many degrees of freedom General formalism Consider a dynamical system with N degrees of freedom near
More informationQuantum Squeezing of Dark Solitons in Optical Fibers
Commun. Theor. Phys. 56 (11) 3 36 Vol. 56, No., August 15, 11 Quantum Squeezing of Dark Solitons in Optical Fibers G.R. Honarasa, 1,, M. Hatami, 1 and M.K. Tavassoly 1 1 Atomic and Molecular Group, Faculty
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationCONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK NOVIKOV VESSELOV EQUATION
International Journal of Modern Physics C Vol. 15, No. 4 (004 595 606 c World Scientific Publishing Company CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC
More informationON THE FIBONACCI NUMBERS
ON THE FIBONACCI NUMBERS Prepared by Kei Nakamura The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationIs bound entanglement for continuous variables a rare phenomenon?
Is bound entanglement for continuous variables a rare phenomenon? Pawe l Horodecki 1 and Maciej Lewenstein 2 1 Faculty of Applied Physics and Mathematics Technical University of Gdańsk, 80 952 Gdańsk,
More information(x) = lim. x 0 x. (2.1)
Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationA note on the symmetry properties of Löwdin s orthogonalization schemes. Abstract
A note on the symmetry properties of Löwdin s orthogonalization schemes T. A. Rokob Chemical Research Center, ungarian Academy of Sciences, 155 Budapest, POB 17, ungary Á. Szabados and P. R. Surján Eötvös
More informationPaper I ( ALGEBRA AND TRIGNOMETRY )
Paper I ( ALGEBRA AND TRIGNOMETRY ) Dr. J. N. Chaudhari Prof. P. N. Tayade Prof. Miss. R. N. Mahajan Prof. P. N. Bhirud Prof. J. D. Patil M. J. College, Jalgaon Dr. A. G. D. Bendale Mahila Mahavidyalaya,
More informationModule 3F2: Systems and Control EXAMPLES PAPER 1  STATESPACE MODELS
Cambridge University Engineering Dept. Third year Module 3F2: Systems and Control EXAMPLES PAPER  STATESPACE MODELS. A feedback arrangement for control of the angular position of an inertial load is
More informationRutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006
Rutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J
More informationLinear Algebra In Dirac Notation
Chapter 3 Linear Algebra In Dirac Notation 3.1 Hilbert Space and Inner Product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex
More information1 Ordinary Differential Equations Separation of Variables
Ordinary Differential Equations Separation of Variables. Introduction Calculus is fundamentally important for the simple reason that almost everything we stu is subject to change. In many if not most such
More informationLinear systems of ordinary differential equations
Linear systems of ordinary differential equations (This is a draft and preliminary version of the lectures given by Prof. Colin Atkinson FRS on 2st, 22nd and 25th April 2008 at Tecnun Introduction. This
More information9 Matrices, determinants, inverse matrix, Cramer s Rule
AAC  Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:
More informationQuantum implementation of elementary arithmetic operations
Quantum implementation of elementary arithmetic operations G. Florio and D. Picca INFN (Sezione di Bari and Dipartimento di Fisica, Università di Bari, Via Orabona 4, 706, Bari, Italy Abstract Quantum
More informationDefinition of entanglement for pure and mixed states
Definition of entanglement for pure and mixed states seminar talk given by Marius Krumm in the master studies seminar course Selected Topics in Mathematical Physics: Quantum Information Theory at the University
More informationLecture Quantitative Finance Spring Term 2015
(1,1) Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas : May 21, 2015 1 / 58 Outline (1,1) 1 (1,1) 2 / 58 Outline of the Presentation (1,1) 1 (1,1) 3 / 58 (1,1) The goal of this
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationTHE LAWS OF COSINES FOR NONEUCLIDEAN TETRAHEDRA
THE LAWS OF COSINES FOR NONEUCLIDEAN TETRAHEDRA B.D.S. MCCONNELL Darko Veljan s article The 500YearOld Pythagorean Theorem 1 discusses the history and lore of probably the only nontrivial theorem in
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationth ey are.. r.. ve.. an t.. bo th for th e.. st ru c.. tur alan d.. t he.. quait.. ta ive un de \ centerline {. Inrodu \quad c t i o na r s
   I  ˆ    q I q I ˆ I q R R q I q q I R R R R    ˆ @ & q k 7 q k O q k 8 & q & k P S q k q k ˆ q k 3 q k ˆ A & [ 7 O [8 P & S & [ [ 3 ˆ A @ q ˆ U q  : U [ U φ : U D φ φ Dφ A ψ A : I SS N :
More informationBit error rate in multipath wireless channels with several specular paths
Bit error rate in multipath wireless channels with several specular paths C. Chen and A. Adi In this letter a recursive and computationally efficient new formula for it error rate (BER) in multipath channels
More informationMITES 2010: Physics III Survey of Modern Physics Final Exam Solutions
MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in onedimension. Its position at time t is x(t. As a function of
More informationFourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27
24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about
More informationMathematical Background
Appendix A Mathematical Background A.1 Joint, Marginal and Conditional Probability Let the n (discrete or continuous) random variables y 1,..., y n have a joint joint probability probability p(y 1,...,
More informationLecture Quantitative Finance
Lecture Quantitative Finance Spring 2011 Prof. Dr. Erich Walter Farkas Lecture 12: May 19, 2011 Chapter 8: Estimating volatility and correlations Prof. Dr. Erich Walter Farkas Quantitative Finance 11:
More informationFrom the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties:
3. Exponential and trigonometric functions From the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties: 1. f(z) is
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationModule 3: SecondOrder Partial Differential Equations
Module 3: SecondOrder Partial Differential Equations In Module 3, we shall discuss some general concepts associated with secondorder linear PDEs. These types of PDEs arise in connection with various
More information3.12 SL(2, C) and Its Subgroups: SL(2, R), SU(2), SU(1, 1) and SO(1, 2)
The Ricci curvature tensor of a biinvariant metric for semisimple group is proportional to the metric and, therefore, the scalar curvature is Ric = 1 4 g, (3.211) R = n 4. (3.212) 3.12 SL(2, ) and Its
More informationIntroduction II. The Fieller Method
ρ ψ = φ ρ = β φ = β = β + ε ε σ 2 β = σ = 㭗匷 㭗匷 ( ( ) ) β β σ ψ ψ ρ = φ ρ = β φ = β % ψ ψ + ψ + = = β σ = σ β β = β σ ± > ( ψ ψ ) = β = β = > > < é ( )% { } ( ) ( ) = β β ψ {( β ) ( β ) ψ } ± { ( ) ( σ
More informationOn a Problem of LargeAmplitude Oscillation of a Nonlinear Conservative System with Inertia and Static Nonlinearity
Applied Mathematics & Information Sciences 1(2)(2007), 173183 An International Journal c 2007 Dixie W Publishing Corporation, U. S. A. On a Problem of LargeAmplitude Oscillation of a Nonlinear Conservative
More informationAnalysis of the Probability Model of Wind Load on the Offshore Wind Turbine
Open Journal of Civil Engineering, 015, 5, 6167 Published Online June 015 in SciRes. http://www.scirp.org/journal/ojce http://dx.doi.org/10.436/ojce.015.506 Analysis of the Probability Model of Wind Load
More informationAREAS OF SPHERICAL AND HYPERBOLIC TRIANGLES IN TERMS OF THEIR MIDPOINTS. G.M. Tuynman
ARAS OF SPHRICA AND HYPRBOIC TRIANGS IN TRMS OF THIR MIDPOINTS GM Tuynman Abstract et M be either the 2sphere S 2 R 3 or the hyperbolic plane H 2 R 3 If abc is a geodesic triangle on M with corners at
More information5. Möbius Transformations
5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a onetoone mapping
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More informationarxiv:condmat/ v3 11 Feb 1998
MaxwellSchrödinger Equation for Polarized Light and Evolution of the Stokes Parameters Hiroshi Kuratsuji and Shouhei Kakigi Department of Physics, Ritsumeikan UniversityBKC, Kusatsu City 558577, Japan
More informationMATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations
MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations 19/24 April 2012 Lagrange s work on general solution formulae for polynomial equations The formulae for the cubic and
More informationIntroduction to Quantum Information Theory. Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT)
Introduction to Quantum Information Theory Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT) carlospalazuelos@icmat.es Madrid, Spain March 013 Contents Chapter 1. A comment on these notes 3
More informationVI. Transcendental Functions. x = ln y. In general, two functions f, g are said to be inverse to each other when the
VI Transcendental Functions 6 Inverse Functions The functions e x and ln x are inverses to each other in the sense that the two statements y = e x, x = ln y are equivalent statements In general, two functions
More informationProject description a) What is quantum information theory with continuous variables?
Project description a) What is quantum information theory with continuous variables? According to prequantum physics, any theory should display the property of locality defined in three almost equivalent
More informationPreCalculus Review Lesson 1 Polynomials and Rational Functions
If a and b are real numbers and a < b, then PreCalculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b
More informationQuantum Mechanics I: Basic Principles
Quantum Mechanics I: Basic Principles Michael A. Nielsen University of Queensland I ain t no physicist but I know what matters  Popeye the Sailor Goal of this and the next lecture: to introduce all the
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTIAL METHODS FOR ENGINEERS Unit code: A/6/ QF Level: redit value: 5 OUTOME  ALULUS TUTORIAL INTEGRATION The calculus: the concept of the limit and continuity; definition of the derivative;
More informationMethods  Solutions to Sheet B
Methods  Solutions to Sheet B 1. Equation is u t + u = 0 with, t > 0 and u(0, t) = sin t and u(, 0) = sin. First parametrise PDE: ds = 1...(1); d ds = 1...(2); du ds = ds u t + d ds u = 0...(3) Net, parametrise
More informationMISO Capacity with PerAntenna Power Constraint
1 MISO Capacity with PerAntenna Power Constraint Mai Vu Department of Electrical and Computer Engineering, McGill University, Montreal, HAA7 Email: mai.h.vu@mcgill.ca arxiv:100.178v [cs.it] 18 Jan 011
More informationENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE. Ivette Fuentes University of Nottingham
ENTANGLEMENT OF BOSONIC AND FERMIONIC FIELDS IN AN EXPANDING UNIVERSE Ivette Fuentes University of Nottingham with: Robert B. Mann (U Waterloo) Shahpoor Moradi (U Razi) Eduardo Martin Martinez (CISC Madrid)
More informationMATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET
MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET 06 Copyright School Curriculum and Standards Authority, 06 This document apart from any third party copyright material contained in it may be freely copied,
More informationModern Geometry Homework.
Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z
More informationarxiv: v1 [grqc] 21 Jan 2010
Notes Concerning On the Origin of Gravity and the Laws of Newton by E. Verlinde Jarmo Mäkelä Vaasa University of Applied Sciences, Wolffintie 30, 65200 Vaasa, Finland We point out that certain equations
More informationRADIO PROPAGATION MODELS
RADIO PROPAGATION MODELS 1 Radio Propagation Models 1 Path Loss Free Space Loss Ground Reflections Surface Waves Diffraction Channelization Shadowing 3 Multipath Reception and Scattering Dispersion Time
More informationMathematical Formulation of the Superposition Principle
Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.
More informationReport Assignment 1: A Cancer Model
Report Assignment 1: A Cancer Model The first report assignment concerns a mathematical model of cancer. It is based on an article by John D. Nagy [Competition and natural selection in a mathematical model
More informationarxiv:hepph/ v1 4 Mar 1996
LU TP 968 February 996 arxiv:hepph/96039v 4 Mar 996 Colour: A Computer Program for QCD Colour Factor Calculations Jari Häkkinen, Hamid Kharraziha Department of Theoretical Physics, Lund University, Sölvegatan
More informationDamped neutrino oscillations
Damped neutrino oscillations Erroneous measurements of neutrino oscillation parameters? Based on hepph/0502147 Mattias Blennow Division of Mathematical Physics Department of Physics Royal Institute of
More informationM J M j {1,..., J M } i j u ij = X j β αp j + ξ j + ζ ig + (1 σ)ϵ ij, X j ξ j j ϵ ij ζ ig g i ζ + (1 σ)ϵ σ δ j = X j β α i p j + ξ j j D g = ( ) k g δk 1 σ s j = ( ) δj 1 σ Dg σ (1 + Dg 1 σ ), l ij > 0
More informationFullwave synthetic acoustic logs in porous media
Global Geology 15 2 151155 2012 doi 10. 3969 /j. issn. 16739736. 2012. 02. 11 Article ID 16739736 2012 02015105 Fullwave synthetic acoustic logs in porous media LI Hanqing WANG Zhuwen and ZHANG Xueang
More informationEIGENVALUES AND EIGENVECTORS
Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called
More informationQuantitative Methods for Economics Tutorial 3. Katherine Eyal
Quantitative Methods for Economics Tutorial 3 Katherine Eyal TUTORIAL 3 9th13th August 2010 ECO3021S PART 1 1. Let the national income model be: Y = C + I + G C = a + b(y T ) a > 0, 0 < b < 1 T = d +
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationEE 321 Analog Electronics, Fall 2013 Homework #5 solution
EE 321 Analog Electronics, Fall 2013 Homework #5 solution 3.26. For the circuit shown in Fig. P3.26, both diodes are identical, conducting 10mA at 0.7V, and 100mA at 0.8V. Find the value of for which V
More information3. A LITTLE ABOUT GROUP THEORY
3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of
More information, < x < Using separation of variables, u(x, t) = Φ(x)h(t) (4) we obtain the differential equations. d 2 Φ = λφ (6) Φ(± ) < (7)
Chapter1: Fourier Transform Solutions of PDEs In this chapter we show how the method of separation of variables may be extended to solve PDEs defined on an infinite or semiinfinite spatial domain. Several
More informationConvex Optimization 3. Convex Functions
Convex Optimization 3. Convex Functions Goele Pipeleers KU Leuven, Feb. 913, 2015 Overview definition properties of convex functions Jensen s inequality restriction to line firstorder condition secondorder
More information2.2 Second quantization
2.2 Second quantization We introduced a compact notation for Slater determinants Ψ S = ψ 1... ψ N. This notation hides the fact that ψ 1... ψ N really stands for a full antisymmetric state, which can
More informationEntangling rates and the quantum holographic butterfly
Entangling rates and the quantum holographic butterfly David Berenstein DAMTP/UCSB. C. Asplund, D.B. arxiv:1503.04857 Work in progress with C. Asplund, A. Garcia Garcia Questions If a black hole is in
More informationInverse Problems for Selfadjoint Matrix Polynomials
Peter Lancaster, University of Calgary, Canada Castro Urdiales, SPAIN, 2013. Castro Urdiales, SPAIN, 2013. 1 / Preliminaries Given A 0, A 1,, A l C n n (or possibly in R n n ), L(λ) := l A j λ j, λ C,
More informationLesson A  Natural Exponential Function and Natural Logarithm Functions
A Lesson A  Natural Exponential Function and Natural Logarithm Functions Natural Exponential Function In Lesson 2, we explored the world of logarithms in base 0. The natural logarithm has a base of e.
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationA numerical solution of Nagumo telegraph equation by Adomian decomposition method
Mathematics Scientific Journal Vol. 6, No. 2, S. N. 13, (2011), 7381 A numerical solution of Nagumo telegraph equation by Adomian decomposition method H. Rouhparvar a,1 a Department of Mathematics, Islamic
More informationQuantum Fields in Curved Spacetime
Quantum Fields in Curved Spacetime Lecture 1: Introduction Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 20, 2016. Intro: Fields Setting: many microscopic degrees of freedom interacting
More informationProjectile Motion. Newton s Laws of Motion Click to edit Master title style. Mechanical Phenomena. Physics and Astronomy Department
Projectile Motion PHY 5200 Mechanical Phenomena PHY 5200 Mechanical Phenomena Newton s Laws of Motion Click to edit Master title style Claude A Pruneau Physics and Astronomy Department Wayne State University
More informationStrategy. Theorem Fermat s Last Theorem: If n > 2, then there are no nontrivial integer solutions to x n + y n = z n.
1. Rewrite equation as α n + β n + γ 3 = 0. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β, γ is divisible by 3. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β,
More informationVONNEUMANN STABILITY ANALYSIS
universitylogo VONNEUMANN STABILITY ANALYSIS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 EXPONENTIAL GROWTH Stability under exponential growth 3 VONNEUMANN STABILITY ANALYSIS 4 SUMMARY
More informationHomework One Solutions. Keith Fratus
Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product
More informationNotes on spherical geometry
Notes on spherical geometry Math 130 Course web site: www.courses.fas.harvard.edu/5811 This handout covers some spherical trigonometry (following yan s exposition) and the topic of dual triangles. 1. Some
More informationLecture 10: Characteristic Functions
Lecture 0: Characteristic Functions. Definition of characteristic functions. Complex random variables.2 Definition and basic properties of characteristic functions.3 Examples.4 Inversion formulas 2. Applications
More informationHow is a vector rotated?
How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 6168 (1999) Introduction In an earlier series
More informationQUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS
QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...
More informationQuadratic Polynomials
Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs
More informationModule M6.3 Solving second order differential equations
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M6. Solving second order differential equations Opening items. Module introduction.2 Fast track questions. Rea to stu? 2 Methods
More informationRAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER
Volume 9 (008), Issue 3, Article 89, pp. RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER MARK B. VILLARINO DEPTO. DE MATEMÁTICA, UNIVERSIDAD DE COSTA RICA, 060 SAN JOSÉ,
More informationDensity Evolution. Telecommunications Laboratory. Alex BalatsoukasStimming. Technical University of Crete. October 21, 2009
Density Evolution Telecommunications Laboratory Alex BalatsoukasStimming Technical University of Crete October 21, 2009 Telecommunications Laboratory (TUC) Density Evolution October 21, 2009 1 / 27 Outline
More information