Ph 219a/CS 219a. Exercises Due: Wednesday 23 October 2013

Size: px
Start display at page:

Download "Ph 219a/CS 219a. Exercises Due: Wednesday 23 October 2013"

Transcription

1 1 Ph 219a/CS 219a Exercses Due: Wednesday 23 October How far apart are two quantum states? Consder two quantum states descrbed by densty operators ρ and ρ n an N-dmensonal Hlbert space, and consder the complete orthogonal measurement {E a, a = 1, 2, 3,...N}, where the E a s are onedmensonal projectors satsfyng N E a = I. (1) a=1 When the measurement s performed, outcome a occurs wth probablty p a = tr ρe a f the state s ρ and wth probablty p a = tr ρe a f the state s ρ. The L 1 dstance between the two probablty dstrbutons s defned as d(p, p) p p 1 1 N p a p a ; (2) 2 ths dstance s zero f the two dstrbutons are dentcal, and attans ts maxmum value one f the two dstrbutons have support on dsjont sets. a) Show that d(p, p) 1 2 a=1 N λ (3) where the λ s are the egenvalues of the Hermtan operator ρ ρ. Hnt: Workng n the bass n whch ρ ρ s dagonal, fnd an expresson for p a p a, and then fnd an upper bound on p a p a. Fnally, use the completeness property eq. (1) to bound d(p, p). b) Fnd a choce for the orthogonal projector {E a } that saturates the upper bound eq. (3). =1

2 2 Defne a dstance d(ρ, ρ) between densty operators as the maxmal L 1 dstance between the correspondng probablty dstrbutons that can be acheved by any orthogonal measurement. From the results of (a) and (b), we have found that d(ρ, ρ) = 1 2 N λ. (4) =1 c) The L 1 norm A 1 of an operator A s defned as [ A 1 tr (AA ) 1/2]. (5) How can the dstance d(ρ, ρ) be expressed as the L 1 norm of an operator? Now suppose that the states ρ and ρ are pure states ρ = ψ ψ and ρ = ψ ψ. If we adopt a sutable bass n the space spanned by the two vectors, and approprate phase conventons, then these vectors can be expressed as ψ = where ψ ψ = sn θ. ( ) cosθ/2 sn θ/2, ψ = ( ) sn θ/2 cos θ/2 d) Express the dstance d(ρ, ρ) n terms of the angle θ., (6) e) Express ψ ψ 2 (where denotes the Hlbert space norm) n terms of θ, and by comparng wth the result of (d), derve the bound d( ψ ψ, ψ ψ ) ψ ψ. (7) f) Bob thnks that the norm ψ ψ should be a good measure of the dstngushablty of the pure quantum states ρ and ρ. Explan why Bob s wrong. Hnt: Remember that quantum states are rays. 1.2 Whch state dd Alce make? Consder a game n whch Alce prepares one of two possble states: ether ρ 1 wth a pror probablty p 1, or ρ 2 wth a pror probablty

3 3 p 2 = 1 p 1. Bob s to perform a measurement and on the bass of the outcome to guess whch state Alce prepared. If Bob s guess s rght, he wns; f he guesses wrong, Alce wns. In ths exercse you wll fnd Bob s best strategy, and determne hs optmal probablty of error. Let s suppose (for now) that Bob performs a POVM wth two possble outcomes, correspondng to the two nonnegatve Hermtan operators E 1 and E 2 = I E 1. If Bob s outcome s E 1, he guesses that Alce s state was ρ 1, and f t s E 2, he guesses ρ 2. Then the probablty that Bob guesses wrong s p error = p 1 tr (ρ 1 E 2 ) + p 2 tr (ρ 2 E 1 ). (8) a) Show that p error = p 1 + λ E 1, (9) where { } denotes the orthonormal bass of egenstates of the Hermtan operator p 2 ρ 2 p 1 ρ 1, and the λ s are the correspondng egenvalues. b) Bob s best strategy s to perform the two-outcome POVM that mnmzes ths error probablty. Fnd the nonnegatve operator E 1 that mnmzes p error, and show that error probablty when Bob performs ths optmal two-outcome POVM s (p error ) optmal = p 1 + neg λ. (10) where neg denotes the sum over all of the negatve egenvalues of p 2 ρ 2 p 1 ρ 1. c) It s convenent to express ths optmal error probablty n terms of the L 1 norm of the operator p 2 ρ 2 p 1 ρ 1, p 2 ρ 2 p 1 ρ 1 1 = tr p 2 ρ 2 p 1 ρ 1 = pos λ λ, (11) neg the dfference between the sum of postve egenvalues and the sum of negatve egenvalues. Use the property tr (p 2 ρ 2 p 1 ρ 1 ) = p 2 p 1 to show that (p error ) optmal = p 2ρ 2 p 1 ρ 1 1. (12)

4 4 Check whether the answer makes sense n the case where ρ 1 = ρ 2 and n the case where ρ 1 and ρ 2 have support on orthogonal subspaces. d) Now suppose that Alce decdes at random (wth p 1 = p 2 = 1/2) to prepare one of two pure states ψ 1, ψ 2 of a sngle qubt, wth ψ 1 ψ 2 = sn(2α), 0 α π/4. (13) Wth a sutable choce of bass, the two states can be expressed as ( ) ( ) cos α snα ψ 1 =, ψ snα 2 =. (14) cos α Fnd Bob s optmal two-outcome measurement, and compute the optmal error probablty. e) Bob wonders whether he can fnd a better strategy f hs POVM {E } has more than two possble outcomes. Let p(a ) denote the probablty that state a was prepared, gven that the measurement outcome was ; t can be computed usng the relatons p p(1 ) = p 1 p( 1) = p 1 tr ρ 1 E, p p(2 ) = p 2 p( 2) = p 2 tr ρ 2 E ; (15) here p( a) denotes the probablty that Bob fnds measurement outcome f Alce prepared the state ρ a, and p denotes the probablty that Bob fnds measurement outcome, averaged over Alce s choce of state. For each outcome, Bob wll make hs decson accordng to whch of the two quanttes p(1 ), p(2 ) (16) s the larger; the probablty that he makes a mstake s the smaller of these two quanttes. Ths probablty of error, gven that Bob obtans outcome, can be wrtten as p error () = mn(p(1 ), p(2 )) = p(2 ) p(1 ). (17) 2 Show that the probablty of error, averaged over the measurement outcomes, s p error = p p error () = tr(p 2 ρ 2 p 1 ρ 1 )E. (18) 2

5 5 f) By expandng n terms of the bass of egenstates of p 2 ρ 2 p 1 ρ 1, show that p error p 2ρ 2 p 1 ρ 1 1. (19) (Hnt: Use the completeness property E = I.) Snce we have already shown that ths bound can be saturated wth a twooutcome POVM, the POVM wth many outcomes s no better. 1.3 Eavesdroppng and dsturbance Alce wants to send a message to Bob. Alce s equpped to prepare ether one of the two states u or v. These two states, n a sutable bass, can be expressed as u = ( cosα sn α ), v = ( ) snα cos α, (20) where 0 < α < π/4. Suppose that Alce decdes at random to send ether u or v to Bob, and Bob s to make a measurement to determne what she sent. Snce the two states are not orthogonal, Bob cannot dstngush the states perfectly. a) Bob realzes that he can t expect to be able to dentfy Alce s qubt every tme, so he settles for a procedure that s successful only some of the tme. He performs a POVM wth three possble outcomes: u, v, or DON T KNOW. If he obtans the result u, he s certan that v was sent, and f he obtans v, he s certan that u was sent. If the result s DON T KNOW, then hs measurement s nconclusve. Ths POVM s defned by the operators E u = A(I u u ), E v = A(I v v ), E DK = (1 2A)I + A ( u u + v v ), (21) where A s a postve real number. How should Bob choose A to mnmze the probablty of the outcome DK, and what s ths mnmal DK probablty (assumng that Alce chooses from { u, v } equprobably)? Hnt: If A s too large, E DK wll have negatve egenvalues, and Eq.(21) wll not be a POVM. b) Eve also wants to know what Alce s sendng to Bob. Hopng that Alce and Bob won t notce, she ntercepts each qubt that Alce

6 6 sends, by performng an orthogonal measurement that projects onto the bass {( ( 1 0), 0 ( 1)}. If she obtans the outcome 1 0), she sends the state u on to Bob, and f she obtans the outcome ( 0 1), she sends v on to Bob. Therefore each tme Bob s POVM has a conclusve outcome, Eve knows wth certanty what that outcome s. But Eve s tamperng causes detectable errors; sometmes Bob obtans a conclusve outcome that actually dffers from what Alce sent. What s the probablty of such an error, when Bob s outcome s conclusve? 1.4 What probablty dstrbutons are consstent wth a mxed state? A densty operator ρ, expressed n the orthonormal bass { α } that dagonalzes t, s ρ = p α α. (22) We would lke to realze ths densty operator as an ensemble of pure states { ϕ µ }, where ϕ µ s prepared wth a specfed probablty q µ. Ths preparaton s possble f the ϕ µ s can be chosen so that ρ = µ q µ ϕ µ ϕ µ. (23) We say that a probablty vector q (a vector whose components are nonnegatve real numbers that sum to 1) s majorzed by a probablty vector p (denoted q p), f there exsts a doubly stochastc matrx D such that q µ = D µ p. (24) A matrx s doubly stochastc f ts entres are nonnegatve real numbers such that µ D µ = D µ = 1. That the columns sum to one assures that D maps probablty vectors to probablty vectors (.e., s stochastc). That the rows sum to one assures that D maps the unform dstrbuton to tself. Appled repeatedly, D takes any nput dstrbuton closer and closer to the unform dstrbuton (unless D s a permutaton, wth one nonzero entry n each row and column). Thus we can vew majorzaton as a partal order on probablty vectors such that q p means that q s more nearly unform than p (or equally close to unform, n the case where D s a permutaton).

7 7 Show that normalzed pure states { ϕ µ } exst such that eq. (23) s satsfed f and only f q p, where p s the vector of egenvalues of ρ. Hnt: Recall that, accordng to the Hughston-Jozsa-Wootters Theorem, f eq. (22) and eq. (23) are both satsfed then there s a untary matrx V µ such that qµ ϕ µ = p V µ α. (25) You may also use (but need not prove) Horn s Lemma: f q p, then there exsts a untary (n fact, orthogonal) matrx U µ such that q = Dp and D µ = U µ 2.

Ph 219a/CS 219a. Exercises Due: Wednesday 12 November 2008

Ph 219a/CS 219a. Exercises Due: Wednesday 12 November 2008 1 Ph 19a/CS 19a Exercses Due: Wednesday 1 November 008.1 Whch state dd Alce make? Consder a game n whch Alce prepares one of two possble states: ether ρ 1 wth a pror probablty p 1, or ρ wth a pror probablty

More information

Solution 1 for USTC class Physics of Quantum Information

Solution 1 for USTC class Physics of Quantum Information Soluton 1 for 017 018 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.

763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1. 7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)

More information

PHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions

PHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions PHYS 5C: Quantum Mechancs Sprng 07 Problem Set 3 Solutons Prof. Matthew Fsher Solutons prepared by: Chatanya Murthy and James Sully June 4, 07 Please let me know f you encounter any typos n the solutons.

More information

Solution 1 for USTC class Physics of Quantum Information

Solution 1 for USTC class Physics of Quantum Information Soluton 1 for 018 019 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and

More information

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac

More information

Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems

Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems Explct constructons of all separable two-qubts densty matrces and related problems for three-qubts systems Y. en-ryeh and. Mann Physcs Department, Technon-Israel Insttute of Technology, Hafa 2000, Israel

More information

ψ = i c i u i c i a i b i u i = i b 0 0 b 0 0

ψ = i c i u i c i a i b i u i = i b 0 0 b 0 0 Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by  and ˆB, the set of egenstates by { u }, and the egenvalues as  u = a u and ˆB u = b u. Snce the

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2

Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2 P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons

More information

Quantum Mechanics I - Session 4

Quantum Mechanics I - Session 4 Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Lecture 4: Constant Time SVD Approximation

Lecture 4: Constant Time SVD Approximation Spectral Algorthms and Representatons eb. 17, Mar. 3 and 8, 005 Lecture 4: Constant Tme SVD Approxmaton Lecturer: Santosh Vempala Scrbe: Jangzhuo Chen Ths topc conssts of three lectures 0/17, 03/03, 03/08),

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

1 Vectors over the complex numbers

1 Vectors over the complex numbers Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What

More information

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

Composite Hypotheses testing

Composite Hypotheses testing Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

Lecture 3. Ax x i a i. i i

Lecture 3. Ax x i a i. i i 18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Singular Value Decomposition: Theory and Applications

Singular Value Decomposition: Theory and Applications Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real

More information

Grover s Algorithm + Quantum Zeno Effect + Vaidman

Grover s Algorithm + Quantum Zeno Effect + Vaidman Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the

More information

Dynamic Systems on Graphs

Dynamic Systems on Graphs Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,

More information

THEOREMS OF QUANTUM MECHANICS

THEOREMS OF QUANTUM MECHANICS THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn

More information

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 217 Fall 2013 Homework 2 Solutions Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

p 1 c 2 + p 2 c 2 + p 3 c p m c 2

p 1 c 2 + p 2 c 2 + p 3 c p m c 2 Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance

More information

CSE 599d - Quantum Computing Introduction to Quantum Error Correction

CSE 599d - Quantum Computing Introduction to Quantum Error Correction CSE 599d - Quantum Computng Introducton to Quantum Error Correcton Dave Bacon Department of Computer Scence & Engneerng, Unversty of Washngton In the last lecture we saw that open quantum systems could

More information

A how to guide to second quantization method.

A how to guide to second quantization method. Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

Feb 14: Spatial analysis of data fields

Feb 14: Spatial analysis of data fields Feb 4: Spatal analyss of data felds Mappng rregularly sampled data onto a regular grd Many analyss technques for geophyscal data requre the data be located at regular ntervals n space and/or tme. hs s

More information

Solutions to Chapter 1 exercises

Solutions to Chapter 1 exercises Appendx S Solutons to Chapter exercses Soluton to Exercse.. Usng the result of Ex. A.5, we wrte don t forget the complex conjugaton where approprate! ψ ψ N alve ψ dead ψ N 4alve alve + alve dead dead alve

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016 U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

Random Walks on Digraphs

Random Walks on Digraphs Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

arxiv:quant-ph/ Jul 2002

arxiv:quant-ph/ Jul 2002 Lnear optcs mplementaton of general two-photon proectve measurement Andrze Grudka* and Anton Wóck** Faculty of Physcs, Adam Mckewcz Unversty, arxv:quant-ph/ 9 Jul PXOWRZVNDR]QDRODQG Abstract We wll present

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16 STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

für Mathematik in den Naturwissenschaften Leipzig

für Mathematik in den Naturwissenschaften Leipzig ŠܹÈÐ Ò ¹ÁÒ Ø ØÙØ für Mathematk n den Naturwssenschaften Lepzg Asymptotcally optmal dscrmnaton between multple pure quantum states by Mchael Nussbaum, and Arleta Szkola Preprnt no.: 1 2010 Asymptotcally

More information

2. Postulates of Quantum Mechanics. [Last revised: Friday 7 th December, 2018, 21:27]

2. Postulates of Quantum Mechanics. [Last revised: Friday 7 th December, 2018, 21:27] 2. Postulates of Quantum Mechancs [Last revsed: Frday 7 th December, 2018, 21:27] 24 States and physcal systems In the prevous chapter, wth the help of the Stern-Gerlach experment, we have shown the falure

More information

The Second Anti-Mathima on Game Theory

The Second Anti-Mathima on Game Theory The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

More information

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order: 68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse

More information

Asymptotically Optimal Discrimination between Pure Quantum States

Asymptotically Optimal Discrimination between Pure Quantum States Asymptotcally Optmal Dscrmnaton between Pure Quantum States Mchael Nussbaum 1, and Arleta Szko la 2 1 Department of Mathematcs, Cornell Unversty, Ithaca NY, USA 2 Max Planck Insttute for Mathematcs n the

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Density matrix. c α (t)φ α (q)

Density matrix. c α (t)φ α (q) Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n

More information

Lecture 4: November 17, Part 1 Single Buffer Management

Lecture 4: November 17, Part 1 Single Buffer Management Lecturer: Ad Rosén Algorthms for the anagement of Networs Fall 2003-2004 Lecture 4: November 7, 2003 Scrbe: Guy Grebla Part Sngle Buffer anagement In the prevous lecture we taled about the Combned Input

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Winter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan

Winter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments

More information

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 17. a ij x (k) b i. a ij x (k+1) (D + L)x (k+1) = b Ux (k)

STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 17. a ij x (k) b i. a ij x (k+1) (D + L)x (k+1) = b Ux (k) STAT 309: MATHEMATICAL COMPUTATIONS I FALL 08 LECTURE 7. sor method remnder: n coordnatewse form, Jacob method s = [ b a x (k) a and Gauss Sedel method s = [ b a = = remnder: n matrx form, Jacob method

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,

More information

332600_08_1.qxp 4/17/08 11:29 AM Page 481

332600_08_1.qxp 4/17/08 11:29 AM Page 481 336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5

More information

and problem sheet 2

and problem sheet 2 -8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,

More information

Ph 219/CS 219. Exercises Due: Friday 20 October 2006

Ph 219/CS 219. Exercises Due: Friday 20 October 2006 1 Ph 219/CS 219 Exercises Due: Friday 20 October 2006 1.1 How far apart are two quantum states? Consider two quantum states described by density operators ρ and ρ in an N-dimensional Hilbert space, and

More information

Errata to Invariant Theory with Applications January 28, 2017

Errata to Invariant Theory with Applications January 28, 2017 Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng

More information

Chapter 8 Indicator Variables

Chapter 8 Indicator Variables Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n

More information

Error Probability for M Signals

Error Probability for M Signals Chapter 3 rror Probablty for M Sgnals In ths chapter we dscuss the error probablty n decdng whch of M sgnals was transmtted over an arbtrary channel. We assume the sgnals are represented by a set of orthonormal

More information

P R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /

P R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering / Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons

More information

Ph 219/CS 219. Exercises Due: Friday 3 November 2006

Ph 219/CS 219. Exercises Due: Friday 3 November 2006 Ph 9/CS 9 Exercises Due: Friday 3 November 006. Fidelity We saw in Exercise. that the trace norm ρ ρ tr provides a useful measure of the distinguishability of the states ρ and ρ. Another useful measure

More information

7. Products and matrix elements

7. Products and matrix elements 7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also

More information

A crash course in real-world quantum mechanics

A crash course in real-world quantum mechanics A crash course n real-world quantum mechancs Basc postulates for an solated quantum system Pure states (mnmum-uncertanty states) of a physcal system are represented by vectors n a complex Hlbert space.

More information

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all

More information

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS) Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998

More information

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,

More information

14 The Postulates of Quantum mechanics

14 The Postulates of Quantum mechanics 14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable

More information

Matrix Mechanics Exercises Using Polarized Light

Matrix Mechanics Exercises Using Polarized Light Matrx Mechancs Exercses Usng Polarzed Lght Frank Roux Egenstates and operators are provded for a seres of matrx mechancs exercses nvolvng polarzed lght. Egenstate for a -polarzed lght: Θ( θ) ( ) smplfy

More information

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty

Additional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,

More information

10-701/ Machine Learning, Fall 2005 Homework 3

10-701/ Machine Learning, Fall 2005 Homework 3 10-701/15-781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons-10701@autonlaborg for queston Problem 1 Regresson and Cross-valdaton [40

More information

w ). Then use the Cauchy-Schwartz inequality ( v w v w ).] = in R 4. Can you find a vector u 4 in R 4 such that the

w ). Then use the Cauchy-Schwartz inequality ( v w v w ).] = in R 4. Can you find a vector u 4 in R 4 such that the Math S-b Summer 8 Homework #5 Problems due Wed, July 8: Secton 5: Gve an algebrac proof for the trangle nequalty v+ w v + w Draw a sketch [Hnt: Expand v+ w ( v+ w) ( v+ w ) hen use the Cauchy-Schwartz

More information

Multilayer Perceptron (MLP)

Multilayer Perceptron (MLP) Multlayer Perceptron (MLP) Seungjn Cho Department of Computer Scence and Engneerng Pohang Unversty of Scence and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjn@postech.ac.kr 1 / 20 Outlne

More information

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Module 9. Lecture 6. Duality in Assignment Problems

Module 9. Lecture 6. Duality in Assignment Problems Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept

More information

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t 8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes

More information

Eigenvalues of Random Graphs

Eigenvalues of Random Graphs Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the

More information

Supplementary Information

Supplementary Information Supplementary Informaton Quantum correlatons wth no causal order Ognyan Oreshkov 2 Fabo Costa Časlav Brukner 3 Faculty of Physcs Unversty of Venna Boltzmanngasse 5 A-090 Venna Austra. 2 QuIC Ecole Polytechnque

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

Google PageRank with Stochastic Matrix

Google PageRank with Stochastic Matrix Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate

More information

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013 Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned

More information

A 2D Bounded Linear Program (H,c) 2D Linear Programming

A 2D Bounded Linear Program (H,c) 2D Linear Programming A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

Norms, Condition Numbers, Eigenvalues and Eigenvectors

Norms, Condition Numbers, Eigenvalues and Eigenvectors Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b

More information