Quantum Nonlocality Pt. 2: No-Signaling and Local Hidden Variables May 1, / 16
|
|
- Horace Jones
- 6 years ago
- Views:
Transcription
1 Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
2 Non-Signling Boxes The primry lesson from lst lecture is tht pir of input/output oxes cn e used for communiction if nd only if they violte the non-signling condition. Non-Signling Condition Two oxes with inputs elonging to set X = {1,, X }, outputs elong to set B = {, 1,, B } nd trnsition proilities {p( x)} B,x X re clled non-signling if p( x) = p( x ) B, x, x X. In this lecture we will return to the quntum setting nd see if quntum entnglement cn e used fster-thn-light communiction. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
3 Quntum Boxes Suppose Alice nd Bo shre some iprtite stte ρ AB. They hve red populr science mgzine climing tht quntum entnglement llows for spooky ction t distnce. They get excited nd think this mens their entnglement cn e used for instntneous communiction. Alice: I will perform one of two quntum opertions on my system. This will cuse n instntneous chnge on your system so tht y mesuring your system you will e le to determine which ction I performed. Then if I wnt to send messge 0, I will perform opertion {A (0) 0, A(0) 1 {A (1) 0, A(1) 1 }. } nd if I wnt to send messge 1, I will perform ction Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
4 Quntum Boxes Bo: Gret ide! But wht mesurement should I use to recover your messge? Let me choose etween two of them - {B (0) 0, B(0) 1 } or {B (1) } - nd see which one works etter. 0, B(1) 1 The mesurements {B (0) 0, B(0) 1 } nd {B(1) 0, B(1) 1 } re two decoding opertions for Alice s messge. He mesures his system nd clims tht Alice sent vlue x if he otins mesurement outcome x. Alice nd Bo s strtegy cn e descried within the lck ox frmework. These oxes re more sophisticted since they hve pirs of inputs nd pirs of inputs. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
5 Quntum Boxes The trnsition proilities re given y p(, x, y) = tr[(a (x) Do these oxes llow for communiction? B (y) )ρab (A (x) B (y) ) ],, x, y {0, 1}. There re two possiilities now: communiction from Alice to Bo or communiction from Bo to Alice. For Alice to Bo, we look t Bo s output distriution p( x, y). For ech of his inputs y, this genertes n input/output ox p y ( x) := p( x, y) where p( x, y) = p(, x, y). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
6 Quntum Boxes No signling from Alice to Bo mens tht p y ( x) = p y ( x ); in other words p(, x, y) = p(, x, y ), x, y, y. No signling from Bo to Alice mens tht p(, x, y) = p(, x, y), y, x, x. Do quntum oxes stisfy the non-signling conditions with trnsition proilities given y p(, x, y) = tr[(a (x) B (y) )ρab (A (x) B (y) ) ]? Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
7 Quntum Boxes Check: p(, x, y) = = tr[(a (x) tr[(a (x) ) A (x) B (y) )ρab (A (x) B (y) ) ] (B (y) ) B (y) ρab ] = tr[i (B (y) ) B (y) ρab ] = tr[(b (y) ) B (y) ρb ]. This is just the proility tht Bo otins outcome when he performs mesurement {B (y) } on his reduced stte ρb. It is independent of oth Alice s input nd output p( x, y) = p( x, y) It stisfies the non-signling conditions. Quntum entnglement does not llow instntneous communiction! Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
8 Quntum Boxes The trnsition proilities {p(, x, y)},,x,y generted y quntum oxes (i.e. generted y mesuring n entngled quntum stte) do not llow for communiction etween two sptil positions X 1 nd X 2 in time intervl fster thn c X (i.e. Specil Reltivity is not violted). Does this men tht sptilly seprted clssicl systems cn lso generte these trnsition proilities {p(, x, y)}? To properly nswer this question, let us develop generl frmework of input/output oxes tht will llow us to distinguish etween clssicl nd quntum oxes. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
9 The Theory of Correlted Boxes Definition Let A, B, X, nd Y e ritrry finite sets of integers. A set of correltions is collection of non-negtive numers {p(, x, y)} A, B,x X,y Y such tht A, B p(, x, y) = 1 for ll (x, y) X Y. The correltions re clled non-signling (NS) if p(, x, y) = p(, x, y ), x, y, y p(, x, y) = p(, x, y), y, x, x. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
10 The Theory of Correlted Boxes The correltions re clled quntum correltions (QC) if there exists iprtite quntum stte ρ AB s well s quntum opertions {A (x) } A nd {B (y) } B for ll (x, y) X Y such tht p(, x, y) = tr[(a (x) B (y) )ρab (A (x) B (y) ) ]. Wht re the type of correltions tht cn e generted y clssicl oxes? Clssicl systems re ssumed to stisfy property known s loclity. The following mening of loclity will suffice for our purposes. Two oxes re sid to function loclly if their correltions tke the form p(, x, y) = p( x)p( y). In other words, for every input nd output of one ox, the input/output sttistics of the other ox re not chnged (i.e. p(, x, y) = p(, x, y ) nd p(, x, y) = p(, x, y); compre with NS oxes). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
11 The Theory of Correlted Boxes However, clssicl physics clerly does not require tht every pir of oxes ehve loclly. Clssicl physics llows for interction etween systems, nd this interction estlishes correltion. Consider drwer of full of socks nd ech sock hs two properties: (i) Color {Blck, White} nd (ii) Fric {Cotton, Nylon}. For simplicity lso ssume tht ech sock either elongs to right or left foot. Ech pir in the drwer consists of right nd left sock, ut the color nd fric properties re not mtched! Ech pir is descried y list λ CL F L C R F R = (Color C L, Fric F L, Color C R, Fric F R ), where C L /F L is the color/fric of the left sock nd C R /F R is the color/fric of the right sock. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
12 The Theory of Correlted Boxes Suppose tht the pirs of socks re distriuted in the drwer ccording to proilities p(λ CL F L C R F R ) with C L/R {B, W } nd F L/R {C, N}. I rech into the drwer nd rndomly select pir. I then put the left sock in lck ox nd give tht to Alice, nd I put the right sock in lck ox nd give tht to Bo. Ech oxes re designed so tht it revels only the Color or the Fric of the sock it holds, ut Alice nd Bo cn choose which property is reveled. Let x {C, F } e the property tht property tht Alice chooses to lern, nd y {C, F } the property tht Bo chooses to lern. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
13 The Theory of Correlted Boxes Their oxes re descried y the correltions p(x L, y L x, y). The outputs (x L, y L ) will depend not only on the choices (x, y) ut lso which socks were drwn from the drwer. To compute p(x L, y L x, y) we need to sum over ll the socks in the drwer: p(x L, y L x, y) = C L F L C R F R p(x L, y L, λ CL F L C R F R x, y) = C L F L C R F R p(x L, y L x, y, λ CL F L C R F R )p(λ CL F L C R F R ). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
14 The Theory of Correlted Boxes We oviously hve tht p(x L, y L x, y, λ CL F L C R F R ) = p(x L x, λ CL F L C R F R )p(y L y, λ CL F L C R F R ) since λ CL F L C R F R specifies the vlue of x L nd y L for every choice of x nd y. Thus, Alice nd Bo s function loclly given the vlue of λ CL F L C R F R. Their correltions hve the form p(x L, y L x, y) = p(x L x, λ CL F L C R F R )p(y L y, λ CL F L C R F R )p(λ CL F L C R F R ). C L F L C R F R Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
15 The Theory of Correlted Boxes Alice nd Bo s oxes re not locl since they do not hve the form p(x L, y L x, y) = p(x L x)p(y L y). However their oxes re locl given the vlue of some hidden vrile λ CL F L C R F R. The oxes ecme dependent on the hidden vrile λ CL F L C R F R y some interction in the pst. Specificlly, this interction ws when I distriuted the pir of socks to ech ox. The oxes cme together t sometime in the pst when I distriuted the socks. This mde them correlted for ll future times, regrdless of how fr prt they were seprted. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
16 Locl Hidden Vriles In generl, set of correltions {p(, x, y)} re clled locl hidden vrile (LHV) correltions if they cn e written s p(, x, y) = λ p(λ)p A ( x, λ)p B ( y, λ),, x, y for some vrile λ with distriution p(λ). Here p A ( x, λ)p B ( y, λ) is product of locl proility distriutions. Wht is the reltionship etween the vrious correltions LHV, QC, nd NS? We hve lredy shown tht QC NS. To conclude tody s lecture, we will mke two simple oservtions: (i) LHV CQ nd (ii) The quntum correltions generted y every seprle stte elong to LHV. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, / 16
Extended nonlocal games from quantum-classical games
Extended nonlocl gmes from quntum-clssicl gmes Theory Seminr incent Russo niversity of Wterloo October 17, 2016 Outline Extended nonlocl gmes nd quntum-clssicl gmes Entngled vlues nd the dimension of entnglement
More informationEntanglement Purification
Lecture Note Entnglement Purifiction Jin-Wei Pn 6.5. Introduction( Both long distnce quntum teleporttion or glol quntum key distriution need to distriute certin supply of pirs of prticles in mximlly entngled
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:
Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )
More informationHomework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
More informationC/CS/Phys C191 Bell Inequalities, No Cloning, Teleportation 9/13/07 Fall 2007 Lecture 6
C/CS/Phys C9 Bell Inequlities, o Cloning, Teleporttion 9/3/7 Fll 7 Lecture 6 Redings Benenti, Csti, nd Strini: o Cloning Ch.4. Teleporttion Ch. 4.5 Bell inequlities See lecture notes from H. Muchi, Cltech,
More informationStrong Bisimulation. Overview. References. Actions Labeled transition system Transition semantics Simulation Bisimulation
Strong Bisimultion Overview Actions Lbeled trnsition system Trnsition semntics Simultion Bisimultion References Robin Milner, Communiction nd Concurrency Robin Milner, Communicting nd Mobil Systems 32
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationAn Overview of Integration
An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is
More informationLecture 08: Feb. 08, 2019
4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationLecture Notes PH 411/511 ECE 598 A. La Rosa Portland State University INTRODUCTION TO QUANTUM MECHANICS
Lecture Notes PH 4/5 ECE 598. L Ros Portlnd Stte University INTRODUCTION TO QUNTUM MECHNICS Underlying subject of the PROJECT ssignment: QUNTUM ENTNGLEMENT Fundmentls: EPR s view on the completeness of
More informationBeginning Darboux Integration, Math 317, Intro to Analysis II
Beginning Droux Integrtion, Mth 317, Intro to Anlysis II Lets strt y rememering how to integrte function over n intervl. (you lerned this in Clculus I, ut mye it didn t stick.) This set of lecture notes
More informationBayesian Networks: Approximate Inference
pproches to inference yesin Networks: pproximte Inference xct inference Vrillimintion Join tree lgorithm pproximte inference Simplify the structure of the network to mkxct inferencfficient (vritionl methods,
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationSpacetime and the Quantum World Questions Fall 2010
Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationSpecial Relativity solved examples using an Electrical Analog Circuit
1-1-15 Specil Reltivity solved exmples using n Electricl Anlog Circuit Mourici Shchter mourici@gmil.com mourici@wll.co.il ISRAE, HOON 54-54855 Introduction In this pper, I develop simple nlog electricl
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationexpression simply by forming an OR of the ANDs of all input variables for which the output is
2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationContinuous Random Variables
CPSC 53 Systems Modeling nd Simultion Continuous Rndom Vriles Dr. Anirn Mhnti Deprtment of Computer Science University of Clgry mhnti@cpsc.uclgry.c Definitions A rndom vrile is sid to e continuous if there
More information( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationSection 3.2 Maximum Principle and Uniqueness
Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationEE273 Lecture 15 Asynchronous Design November 16, Today s Assignment
EE273 Lecture 15 Asynchronous Design Novemer 16, 199 Willim J. Dlly Computer Systems Lortory Stnford University illd@csl.stnford.edu 1 Tody s Assignment Term Project see project updte hndout on we checkpoint
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More information5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.
Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationFirst Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationExercises with (Some) Solutions
Exercises with (Some) Solutions Techer: Luc Tesei Mster of Science in Computer Science - University of Cmerino Contents 1 Strong Bisimultion nd HML 2 2 Wek Bisimultion 31 3 Complete Lttices nd Fix Points
More information3 Regular expressions
3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll
More informationAnalyzing Quantum Coin-Flipping Protocols using Optimization Techniques
Anlyzing Quntum Coin-Fliing Protocols using Otimiztion Techniques Jmie Sikor (Université Pris 7, Frnce) joint work with Ashwin Nyk (University of Wterloo, Cnd) Levent Tunçel (University of Wterloo, Cnd)
More informationChapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationDo the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationPH12b 2010 Solutions HW#3
PH 00 Solutions HW#3. The Hmiltonin of this two level system is where E g < E e The experimentlist sis is H E g jgi hgj + E e jei hej j+i p (jgi + jei) j i p (jgi jei) ) At t 0 the stte is j (0)i j+i,
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationFirst Midterm Examination
24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationDensity of Energy Stored in the Electric Field
Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationLocal orthogonality: a multipartite principle for (quantum) correlations
Locl orthogonlity: multiprtite principle for (quntum) correltions Antonio Acín ICREA Professor t ICFO-Institut de Ciencies Fotoniques, Brcelon Cusl Structure in Quntum Theory, Bensque, Spin, June 2013
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationEECS 141 Due 04/19/02, 5pm, in 558 Cory
UIVERSITY OF CALIFORIA College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Lst modified on April 8, 2002 y Tufn Krlr (tufn@eecs.erkeley.edu) Jn M. Rey, Andrei Vldemirescu Homework
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationTorsion in Groups of Integral Triangles
Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,
More informationIntroduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationPhysics 1402: Lecture 7 Today s Agenda
1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationNow, given the derivative, can we find the function back? Can we antidifferenitate it?
Fundmentl Theorem of Clculus. Prt I Connection between integrtion nd differentition. Tody we will discuss reltionship between two mjor concepts of Clculus: integrtion nd differentition. We will show tht
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts
More informationSTRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada
STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the
More informationWhat Is Calculus? 42 CHAPTER 1 Limits and Their Properties
60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More information