Mixed states and pure states

Save this PDF as:

Size: px
Start display at page:

Transcription

1 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 are indicated by a P. You should try and prove these statements. If you understand the formalism, then these statements should not be hard to prove; they are good tests for your understanding. The homework will contain more difficult questions. 1. A pure state of a quantum system is denoted by a vector (ket) ψ with unit length, i.e. ψ ψ = 1, in a complex Hilbert space H. Previously, we (and the textbook) just called this a state, but now we call it a pure state to distinguish it from a more general type of quantum states ( mixed states, see step 21). 2. We already know from the textbook that we can define dual vectors (bra) φ as linear maps from the Hilbert space H to the field C of complex numbers. Formally, we write φ ( ψ ) = φ ψ. The object on the right-hand side denotes the inner product in H for two vectors φ and ψ. That notation for the inner product used to be just that, notation. Now that we have defined φ as a dual vector it has acquired a second meaning. 3. Given vectors and dual vectors we can define operators (i.e., maps from H to H) of the form Ô = ψ φ. Ô acts on vectors in H and produces as result vectors in H (P). 4. The hermitian conjugate of this operator is Ô φ ψ. This follows (P) straight from the definition of the hermitian conjugate: ( m Ô n ) = n Ô m, for all states n and m in H.

2 5. A special case of such an operator is ˆP ψ = ψ ψ. It is hermitian and it satisfies (P) ˆP 2 ψ = ˆP ψ. That is, it is a projection operator, or projector. 6. Completeness of a basis { n } of H can be expressed as Î = n n n, where Î is the identity operator on H. Inserting the identity is a useful trick. We ll use it several times in the following, and it may be useful for proving some of the P statements. [If n is a continuous variable, we should integrate over n rather than sum. In the following we ll keep using sums to keep notation simple, except when we re discussing position or momentum.] 7. It is useful to define the Trace operation: Tr( ˆK) = n n ˆK n, where ˆK is an arbitrary operator, and the sum is over a set of basis vectors { n }. If we write down a matrix representation for Ô, i.e., a matrix with elements n Ô m, then the Trace is the sum over all diagonal elements (i.e., with m = n). 8. A nice property of the Trace operation is that a basis change leaves it invariant (P): that is, it does not matter which basis we choose in the definition (step 7) of Trace. Indeed, the Trace would be far less useful if it did depend on the basis chosen. 9. Another property of the Trace is the cyclical property. For example, for the Trace of three arbitrary operators (which do not necessarily commute!) we have (P) Tr(Â ˆBĈ) = Tr( ˆBĈÂ) = Tr(ĈÂ ˆB). Generalizations to any number of operators are obvious. In an infinite-dimensional Hilbert space some care has to be taken... see homework!

3 10. On the other hand, in general we have Tr(Â ˆBĈ) Tr(Ĉ ˆBÂ). 11. Expectation values can be expressed in terms of projectors ˆP ψ rather than in terms of state vectors ψ. Namely, by inserting the identity Î = n n n we find that for any operator Ô we can write (P) Ô ψ = ψ Ô ψ = Tr(Ô ˆP ψ ) = Tr( ˆP ψ Ô). 12. Similarly, we can express probabilities and overlaps in terms of projectors (P): φ ψ 2 = Tr( ˆP ψ φ φ ) = Tr( φ φ ˆP ψ ). 13. And so we might as well use the projector ˆP ψ to describe all physical quantities we can derive from the state ψ. We use the symbol ˆρ to indicate (or emphasize) we re talking about a physical state rather than an arbitrary operator ˆρ = ψ ψ, and we call ˆρ the density matrix, or the density operator describing the state ψ. 14. We always have the normalization condition (P) Tr(ˆρ) = For example, if we take the two pure states ψ ± = 0 ± 1 2, then the corresponding ˆρs are 2x2 matrices. Written in the basis { 0, 1 }, they have the form 1/2 ±1/2 ˆρ ± =. ±1/2 1/2 16. The notation in step 15 is borrowed from quantum information theory: there, instead of considering classical bits, which take values 0 and 1, we consider qubits, which can be in any superposition of 0 and 1. You could think of these states as spin up and spin down, and, of a spin-1/2 particle, as an example.

4 17. For another example, consider a particle moving in 1-D. Its state ψ lives in an infinitedimensional Hilbert space. Suppose we, as usual, define the wavefunction ψ(x) by x ψ = ψ(x). Then the corresponding density matrix in the basis { x }, x (, + ), is ˆρ = ψ ψ = dx dx x x ψ ψ x x where we inserted two identities. Rearranging terms gives ˆρ = We can denote the matrix elements of ˆρ as dx dx ψ (x)ψ(x ) x x x ˆρ x ρ(x, x) ρ x x = ψ(x )ψ (x). 18. On the diagonal of the matrix ρ(x, x), where x = x, we get the usual probability density ψ(x) 2. We also have Tr(ˆρ) = dx ψ(x) 2 = 1, which expresses normalization in old and new ways. 19. Similarly, we can use momentum eigenstates and expand the same matrix in the form (P) ˆρ = dp dp ψ (p) ψ(p ) p p where ψ(p) = p ψ. 20. Here is one advantage a density operator has compared to a ket: a given physical state can be described by any ket of the form exp(iθ) ψ with θ an arbitrary phase, but by only one density matrix ˆρ. This is more economical, to say the least. 21. Now let us define a more general type of states, still described by density operators and keeping the advantage of step 20, by introducing mixtures of pure states: N ˆρ = p k ψ k ψ k, k=1

5 where { ψ k } is some set of pure states, not necessarily orthogonal. The number N could be anything, and is not limited by the dimension of the Hilbert space. The N numbers (or weights ) p k are nonzero and satisfy the relations 0 < p k 1; N p k = 1. k=1 The normalization of the weights p k expresses the condition Tr(ˆρ) = 1 (P). 22. We could interpret the weights p k as probabilities, but we have to be careful: we should not think of p k as the probability to find the particle in the state ψ k! You can give one reason for that right now (P), and soon we will see another reason. 23. The quantum state described by ˆρ is called a mixed state whenever ˆρ cannot be written as a density matrix for a pure state (for which N = 1 and p 1 = 1). 24. An example: from statistical physics you may know the following statistical mixture of energy eigenstates ψ n in thermal equilibrium: ˆρ = n p n ψ n ψ n, where p n = exp( E n /kt )/Z with Z = n exp( E n /kt ) the partition function. When the Hamiltonian does not depend on time, this mixture is time-independent (P). 25. Expectation values in mixed states are probabilistic weighted averages of the expectation values of the pure states: Tr(ˆρÔ) = k p k Tr( ψ k ψ k Ô). That is, in the pure state ψ k we would expect an average value of O k Tr( ψ k ψ k Ô) if we measured Ô, and for the mixed state we simply get k p k O k. 26. Since ˆρ is hermitian, we can diagonalize it, such that M ˆρ = λ k φ k φ k, k=1 where the states φ k are orthogonal (unlike those used in step 21). The numbers λ k satisfy 0 λ k 1; M λ k = 1. k=1

6 The numbers λ k are, in fact, nothing but the eigenvalues of ˆρ. They sum to one because of normalization. There are exactly M = d of these numbers, where d is the dimension of the Hilbert space. In contrast, in step 21, N could be any number. 27. Comparing steps 21 and 26 shows that a given mixed (not pure) density matrix can be written in multiple [infinitely many ways, in fact] ways as probabilistic mixtures of pure states. And that is another reason why we have to be careful interpreting coefficients λ k or p k as probabilities. 28. On the other hand, we can certainly prepare a mixed state in a probabilistic way. If we prepare with probability p k a pure state ψ k, and then forget which pure state we prepared, the resulting mixed state is ˆρ = k p k ψ k ψ k. In this case, p k certainly has the meaning of probability. 29. For example, consider the following mixture: with probability 1/2 we prepare 0 and with probability 1/2 we prepare ( )/ 2. The mixture can be represented as a 2x2 matrix in the basis { 0, 1 }: ˆρ = /2 1/2 1/2 1/2 = The eigenvalues and eigenvectors of this matrix are λ ± = 1/2 ± 2/4, 3/4 1/4. 1/4 1/4 and φ ± = λ ± 0 1 λ ± 1. And so we can also view the mixed state as a mixture of the two eigenstates φ ± with weights equal to λ ±. 30. There are two simple tests to determine whether ˆρ describes a pure state or not: pure state : ˆρ 2 = ˆρ; mixed state : ˆρ 2 ˆρ. Or (P): pure state : Tr(ˆρ 2 ) = 1; mixed state : Tr(ˆρ 2 ) < 1. In fact, P Tr(ˆρ 2 ) is called the purity of a state. A state is pure when its purity equals 1, and mixed otherwise.

7 31. Another important quantity is the entropy S(ρ) = Tr(ˆρ log[ˆρ]). You might wonder how to calculate the log of a matrix: just diagonalize it, and take the log of the diagonal elements. That is, d S(ˆρ) = λ k log λ k, k=1 where λ k are the eigenvalues of ˆρ. Note that these eigenvalues are nonnegative, so the entropy can always be defined. Indeed, a zero eigenvalue contributes zero to the entropy, as lim x log x = 0. x We use the log base 2, so that the unit of entropy is the bit. We can then interpret the entropy as the missing information (in bits) about the state. 33. Using the entropy we get another criterion for pure states vs mixed states (P): pure state : S(ˆρ) = 0 mixed state : S(ˆρ) > 0. Thus, there is no missing information for a pure state. A pure quantum state corresponds to maximum information. It doesn t tell us all we could know classically (for example, momentum and position), but it is the maximum knowledge quantum mechanics allows us to have. 34. Example: both 2x2 matrices displayed in step 15 have two eigenvalues, 1 and 0. Thus their entropies are zero, and they both describe pure states; we already knew that, of course. 35. In a Hilbert space of dimension d the entropy can be at most log d bits, namely when all eigenvalues λ k are equal; they must equal 1/d then (P). That state is called the maximally mixed (mm) state in that Hilbert space, and the density matrix is proportional to the identity matrix Î: ˆρmm = Î d.

8 36. A unitary operation Û which satisfies by definition ÛÛ = Û Û = Î acts on kets as ψ Û ψ, and hence (P) it acts on density matrices as ˆρ Û ˆρÛ. 37. The maximally mixed state ˆρ mm is easily seen to be a scalar, invariant under rotations, and in fact invariant under any unitary transformation (P). Don t confuse ˆρ mm with a zero-angular momentum pure state, which is also invariant under rotations, but for a different reason (P)! 38. Now let us finally see how mixed states arise from pure states of multiple quantum systems, if we consider only one of the systems by itself. Consider a pure state for two quantum systems A and B. In general, we can write such a state as a superposition Ψ AB = a nm n A m B, nm in terms of bases { n A } for A and { m B } for B. Because of normalization we have a nm 2 = 1. nm 39. Now consider a measurement just on system A, say an observable ÔA. The expectation value of that observable in terms of ˆρ AB = Ψ AB Ψ is ÔA = Tr(ˆρ AB Ô A ) = n n m ˆρ AB m BOA ˆ n A, m where the ket m B has been moved to the left, as ÔA doesn t do anything to states of system B (that is, the observable is really Ô AB = ÔAÎB). Thus we can define an operator ˆρ A = m ˆρ AB m B Tr B ˆρ AB, m in terms of which ÔA = Tr(ˆρ A Ô A ). That is, ˆρ A as defined above, describes the state of system A by itself. We also call ˆρ A the reduced density matrix of system A.

9 40. The operation indicated by Tr B in the previous step is called a partial trace, and more in particular, a trace over B. We also say, we traced out system B. So, if you want to describe a quantum system by itself, you have to trace out the rest of the universe! 41. For example, take a pure state of the form What is ˆρ A? Answer: First calculate Ψ AB = 0 A 1 B + 1 A 0 B. 2 Ψ AB Ψ = 1 2 [ 0 A 0 1 B A 1 1 B A 0 0 B A 1 0 B 0 ], where for convenience I inserted a sign, to indicate the different projectors before and after the sign act on different Hilbert spaces (namely, those of A and B, respectively). Then take the trace over B. Only the first and fourth term survive this: This is a mixed state: Tr (ˆρ 2 A) = 1/2 < 1. ˆρ A = Tr B Ψ AB Ψ = 1 2 [ 0 A A 1 ]. 42. In matrix form we have (w.r.t to the basis { 0, 1 }) ˆρ A = 1/ /2 Compare this to step 15: this is really a different matrix than an equal superposition. ˆρ A could be interpreted, while exercising lots of caution, as either 0 or 1, each with probability 1/ Another example: Take a pure state of the form Ψ AB = ψ A φ B. Then we have ˆρ A = Tr B ψ A ψ φ B φ = ψ A ψ. This is so because n φ φ n = n n Thus, ˆρ A is pure in this case. φ n n φ = φ φ = 1.

10 44. The pure state Ψ AB = 0 A 1 B + 1 A 0 B 2 is entangled, i.e., it cannot be written as a product of states of A and B: Ψ AB ψ A φ B. How can we be sure that a pure state cannot be written as a product state? Answer: just trace out system B, and check if ˆρ A is pure or not Ψ AB entangled ˆρ A mixed Ψ AB not entangled ˆρ A pure 45. Example: take Ψ AB = ( 0 A 1 B + 1 A 1 B + 0 A 0 B + 1 A 0 B )/2. This state is not entangled, as tracing out B gives a pure state for A (P). Similarly, tracing out A yields a pure state for B. 46. In fact, for pure states of two systems, A and B, a measure of entanglement is the entropy of the reduced density matrix of either A or B (both give the same number!). E( Ψ AB ) = S(ˆρ A ). Entanglement has units of bits, too. We actually call them ebits. (This simple expression for the amount of entanglement does not hold for mixed states of two systems, unfortunately. Quantifying entanglement for mixed states is a hard problem.) 47. We can always consider a mixed state of a system A as entangled with some fictitious auxiliary system, F. Namely, first write it in its diagonal form: ˆρ A = k λ k φ k φ k, then write Ψ AF = k λ k φ k A k F, where { k } is some set of orthogonal states of system F. Tracing out F gives ˆρ A. Note the square root here! Square roots must appear as we will construct ˆρ A from the operator Ψ Ψ, which is bilinear in Ψ.

11 48. One more (long) application: consider the 1-D harmonic oscillator with frequency ω. We know one nice basis to describe this system, { n }, the set of number states, eigenstates of the Hamiltonian Ĥ n = hω(n + 1/2) n. 49. But there is another set of states (not a basis!), which are very useful in practice, the so-called coherent states. For reasons soon to become clear, these states are sometimes called quasi-classical states. They can be defined as eigenstates of the lowering (or annihilation) operator â. Since that operator is not hermitian, its eigenvalues do not have to be real. So let s solve the equation â α = α α, where α is a complex number, the eigenvalue of a. 50. Expanding α in the number states basis as α = a n n, n=0 and substituting this in the eigenvalue equation gives a n n = αan 1. for all n > 0. We can use this recursive relation to express all coefficients in terms of a 0 : a 1 = αa 0 ; a 2 = α 2 a 0 / 2; a 3 = α 3 a 0 / 3!... That is, we find a n = αn n! a 0. Now the state α should be normalized: and so we can choose 1 = n a n 2 = ( α 2 ) n a 0 2 = exp( α 2 ) a 0 2, n n! a 0 = exp( α 2 /2). And thus we finally arrive at α = exp( α 2 /2) n αn n! n

12 51. The coherent state evolves in time as α (t) = exp( α 2 /2) n exp( i(n + 1/2)ωt) αn n! n = exp( iωt/2) α exp( iωt). That is, the coherent state stays a coherent state, but its eigenvalue evolves in time! Note the different meaning of the two phase factors appearing here (P)! 52. We like coherent states because the expectation values of position and momentum evolve in time in a nice, classical way (both oscillate at frequency ω!): x (t) = α(t) ˆx α(t) = A cos(ωt φ) and p (t) = α(t) ˆp α(t) = mωa sin(ωt φ) = md x /dt, where the amplitude of the oscillation is A = 2 h mω α, and α α exp(iφ). All this follows from the expectation values of a and a + : α(t) â α(t) = α(t), and α(t) â + α(t) = (α(t)). 53. Now consider the following mixed state: ˆρ = 1 2π dφ α exp(iφ) α exp(iφ). 2π 0 We can view this as a mixture of coherent states with random phase and fixed amplitude α. We can perform the integration over φ, if we first expand the coherent states in number states. The result is that we can rewrite the same mixed state as ˆρ = exp( α 2 ) n α 2 n! n n. That is, the mixture of coherent states with random phase but fixed amplitude is equivalent to a mixture of number states with a Poissonian distribution, with n = α 2.

13 54. And one more application: Let us see why a mixed state can be written in infinitely many different ways as mixtures of pure states. This will take a few steps that will teach us two things along the way: (i) that tracing out a system can be interpreted in a new way, namely as resulting from a particular measurement on that system; and (ii) that entanglement cannot be used for signaling superluminally (some researchers in the past mistakenly thought one could: by the time we get to the no-cloning theorem in Chapter 12 of the textbook, we ll return to this subject). 55. Given a mixed state for system A written in its diagonal form as ˆρ A = k λ k φ k φ k, we write as in item 47 Ψ AF = λ k φ k A k F, k where { k F } are orthogonal states of (a fictitious) system F. When we calculate the Trace over system F by sandwiching between the orthogonal bras and kets k and k, we get back the diagonal form of ˆρ A. We can interpret this as follows: Suppose we measure some observable Ô on system F that has the set { k } as eigenstates. Then, if we get outcome k, the state of A collapses to φ k, and this occurs with probability λ k. Thus, the diagonal representation of ˆρ A can be interpreted as arising from this particular measurement on F. 56. We can write the same Trace in (infinitely many) other ways. Namely, we can pretend we measure a different observable on F, with different eigenstates, say, ψ m = k U mk k. For the set of states { ψ m } to be orthonormal we require that the complex numbers U mk are matrix elements of a unitary matrix U (P). In terms of these states we can rewrite Ψ AF = k λ k φ k m U mk ψ m = m,k U mk λ k φ k ψ m. since U 1 = U = U ( T ). So, if we introduce the normalized states φ k U m = mk λk φ k k U mk λk φ k, k λ k U mk Umk pm then we can write pm φ m ψ m. Ψ AF = m

14 That is, we can now also write the reduced density matrix as ˆρ A = m p m φ m φ m. Note the states φ m are not orthogonal in general. 57. Exactly because the density matrix stays the same no matter what is measured on system F, as long as the outcome is not known, no information about what observable has been measured is transmitted. 58. If the dimension of system A s Hilbert space is d, then there are at most d terms in the decomposition of ρ A from step 56. However, we can actually also write it as a mixture of a larger number of different terms. Namely, just make the system F larger by adding more orthonormal states (F is a fictitious system after all: we can choose its Hilbert space to be larger than that of A.) Going through the same procedure but with a different (larger) matrix U instead of U leads to a similar expression ˆρ A = m p m φ m φ m, except that now the sum is over more terms. 59. Example: Take ˆρ A = Write 3 3 Ψ AF = Let s say we measure F in the basis ( 0 ± 1 )/ 2. Then we collapse A to the normalized states (P: insert the identity!) ± 3 1 with equal probability (of 1/2). It is easy to check that an equal mixture of these two states indeed gives back the same density operator. But we could also pretend F has a third dimension and measure F in the 3-d basis ( )/ 2, ( )/ 3, ( )/ 6 This collapses A to the normalized (but nonorthogonal!) states 1 2 0, , with probabilities 1/6, 1/3, 1/2, respectively. This gives yet another way of decomposing the mixture ˆρ A in terms of pure states. Check this explicitly!

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

2. Introduction to quantum mechanics

2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian

Linear 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

The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

MITES 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 one-dimension. Its position at time t is x(t. As a function of

For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

We can represent the eigenstates for angular momentum of a spin-1/2 particle along each of the three spatial axes with column vectors: 1 +y =

Chapter 0 Pauli Spin Matrices We can represent the eigenstates for angular momentum of a spin-/ particle along each of the three spatial axes with column vectors: +z z [ ] 0 [ ] 0 +y y [ ] / i/ [ ] i/

Introduces the bra and ket notation and gives some examples of its use.

Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to

Homework 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

Harmonic 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 time-independent Schrödinger equation: d ψx

4. The Infinite Square Well

4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the

A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>.

Quantum Mechanical Operators and Commutation C I. Bra-Ket Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates

Contents Quantum states and observables

Contents 2 Quantum states and observables 3 2.1 Quantum states of spin-1/2 particles................ 3 2.1.1 State vectors of spin degrees of freedom.......... 3 2.1.2 Pauli operators........................

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

Linear Algebra Concepts

University of Central Florida School of Electrical Engineering Computer Science EGN-3420 - Engineering Analysis. Fall 2009 - dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector

Bra-ket notation - Wikipedia, the free encyclopedia

Page 1 Bra-ket notation FromWikipedia,thefreeencyclopedia Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract

Ph 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

Hilbert Space Quantum Mechanics

qitd114 Hilbert Space Quantum Mechanics Robert B. Griffiths Version of 16 January 2014 Contents 1 Introduction 1 1.1 Hilbert space............................................. 1 1.2 Qubit.................................................

0.1 Phase Estimation Technique

Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design

Physics 70007, Fall 2009 Solutions to HW #4

Physics 77, Fall 9 Solutions to HW #4 November 9. Sakurai. Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at t the state vector is given by ipa exp where

The Sixfold Path to understanding bra ket Notation

The Sixfold Path to understanding bra ket Notation Start by seeing the three things that bra ket notation can represent. Then build understanding by looking in detail at each of those three in a progressive

Chapter 2 Basics of Quantum Mechanics

Chapter 2 Basics of Quantum Mechanics Those who are not shocked when they first come across quantum theory cannot possibly have understood it. N. Bohr [1] Abstract Before we can start with the quantum

Lecture 2: Essential quantum mechanics

Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics Jani-Petri Martikainen

MATH 56A SPRING 2008 STOCHASTIC PROCESSES 31

MATH 56A SPRING 2008 STOCHASTIC PROCESSES 3.3. Invariant probability distribution. Definition.4. A probability distribution is a function π : S [0, ] from the set of states S to the closed unit interval

H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the

1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information

Introduction 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

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2

Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

Mathematical 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.

Lecture 18 Time-dependent perturbation theory

Lecture 18 Time-dependent perturbation theory Time-dependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are time-independent. In such cases,

Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

2.5 Complex Eigenvalues

1 25 Complex Eigenvalues Real Canonical Form A semisimple matrix with complex conjugate eigenvalues can be diagonalized using the procedure previously described However, the eigenvectors corresponding

(January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

BOX. The density operator or density matrix for the ensemble or mixture of states with probabilities is given by

2.4 Density operator/matrix Ensemble of pure states gives a mixed state BOX The density operator or density matrix for the ensemble or mixture of states with probabilities is given by Note: Once mixed,

Quantum Computation CMU BB, Fall 2015

Quantum Computation CMU 5-859BB, Fall 205 Homework 3 Due: Tuesday, Oct. 6, :59pm, email the pdf to pgarriso@andrew.cmu.edu Solve any 5 out of 6. [Swap test.] The CSWAP (controlled-swap) linear operator

Definition 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

Notes on wavefunctions III: time dependence and the Schrödinger equation

Notes on wavefunctions III: time dependence and the Schrödinger equation We now understand that the wavefunctions for traveling particles with momentum p look like wavepackets with wavelength λ = h/p,

07. Quantum Mechanics Basics.

07. Quantum Mechanics Basics. Stern-Gerlach Experiment (Stern & Gerlach 1922) magnets splotches sliver atoms detection screen Suggests: Electrons possess 2-valued "spin" properties. (Goudsmit & Uhlenbeck

Rutgers - 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

Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics

Lecture Relevant sections in text: 3.1, 3. Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in

Lecture Notes for Ph219/CS219: Quantum Information and Computation Chapter 2. John Preskill California Institute of Technology

Lecture Notes for Ph19/CS19: Quantum Information and Computation Chapter John Preskill California Institute of Technology Updated July 015 Contents Foundations I: States and Ensembles 3.1 Axioms of quantum

Lecture 2. Observables

Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation

Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

We have seen in the previous Chapter that there is a sense in which the state of a quantum

Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

Till now, almost all attention has been focussed on discussing the state of a quantum system.

Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done

2 Creation and Annihilation Operators

Physics 195 Course Notes Second Quantization 030304 F. Porter 1 Introduction This note is an introduction to the topic of second quantization, and hence to quantum field theory. In the Electromagnetic

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

2.2 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 anti-symmetric state, which can

Problems in Quantum Mechanics

Problems in Quantum Mechanics Daniel F. Styer July 1994 Contents 1 Stern-Gerlach Analyzers 1 Photon Polarization 5 3 Matrix Mathematics 9 4 The Density Matrix 1 5 Neutral K Mesons 13 6 Continuum Systems

The Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933

The Schrödinger Equation Erwin Schrödinger 1887-1961 Nobel Prize in Physics 1933 The Schrödinger Wave Equation The Schrödinger wave equation in its time-dependent form for a particle of energy E moving

Entangling 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

Harmonic Oscillator and Coherent States

Chapter 5 Harmonic Oscillator and Coherent States 5. Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it s the harmonic oscillator potential

Physics 53. Wave Motion 1

Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004

PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

The Schrödinger Equation

The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development

Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

From Einstein to Klein-Gordon Quantum Mechanics and Relativity

From Einstein to Klein-Gordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic

COHERENT STATES IN QUANTUM MECHANICS

COHERENT STATES IN QUANTUM MECHANICS Moniek Verschuren July 8, 2011 Bachelor thesis Radboud University Nijmegen Supervisor: Dr. W.D. van Suijlekom Abstract In this thesis we construct coherent states

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

An introduction to generalized vector spaces and Fourier analysis. by M. Croft

1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 58-63 This lab is u lab on Fourier analysis and consists of VI parts.

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,

Quantum Mechanics: Fundamental Principles and Applications

Quantum Mechanics: Fundamental Principles and Applications John F. Dawson Department of Physics, University of New Hampshire, Durham, NH 0384 October 14, 009, 9:08am EST c 007 John F. Dawson, all rights

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

Rotation in the Space

Rotation in the Space (Com S 477/577 Notes) Yan-Bin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One

Chapter 3. Principles and Spin. There are many axiomatic presentations of quantum mechanics that aim at

Chapter 3 Principles and Spin 3. Four Principles of Quantum Mechanics There are many axiomatic presentations of quantum mechanics that aim at conceptual economy and rigor. However, it would be unwise for

The 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

Primer on Index Notation

Massachusetts Institute of Technology Department of Physics Physics 8.07 Fall 2004 Primer on Index Notation c 2003-2004 Edmund Bertschinger. All rights reserved. 1 Introduction Equations involving vector

MATH36001 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

1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 )

CHEM 352: Examples for chapter 2. 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = 1 14 2ψ 1,, 3ψ 2,, ψ 3,2,2 ) where ψ n,l,m are eigenfunctions of the

Linear Algebra Review. Vectors

Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

1, A brief review about Quantum Mechanics

IPT5340, Spring 08 p. 1/5 1, A brief review about Quantum Mechanics 1. Basic Quantum Theory 2. Time-Dependent Perturbation Theory 3. Simple Harmonic Oscillator 4. Quantization of the field 5. Canonical

Quantum Physics II (8.05) Fall 2013 Assignment 4

Quantum Physics II (8.05) Fall 2013 Assignment 4 Massachusetts Institute of Technology Physics Department Due October 4, 2013 September 27, 2013 3:00 pm Problem Set 4 1. Identitites for commutators (Based

1 The Fourier Transform

Physics 326: Quantum Mechanics I Prof. Michael S. Vogeley The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1.1 Fourier transform of a periodic function A function f(x) that

The Power Method for Eigenvalues and Eigenvectors

Numerical Analysis Massoud Malek The Power Method for Eigenvalues and Eigenvectors The spectrum of a square matrix A, denoted by σ(a) is the set of all eigenvalues of A. The spectral radius of A, denoted

On the general equation of the second degree

On the general equation of the second degree S Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600 113 e-mail:kesh@imscresin Abstract We give a unified treatment of the

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

The Classical Linear Regression Model

The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,

C 1 x(t) = e ta C = e C n. 2! A2 + t3

Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

1 Spherical Kinematics

ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

3. 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

Quantum 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

Witten Deformations. Quinton Westrich University of Wisconsin-Madison. October 29, 2013

Witten Deformations Quinton Westrich University of Wisconsin-Madison October 29, 2013 Abstract These are notes on Witten s 1982 paper Supersymmetry and Morse Theory, in which Witten introduced a deformation

Solutions for Chapter 8

Solutions for Chapter 8 Solutions for exercises in section 8. 2 8.2.1. The eigenvalues are σ (A ={12, 6} with alg mult A (6 = 2, and it s clear that 12 = ρ(a σ (A. The eigenspace N(A 12I is spanned by

CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2