Quantum Mechanics Basic Principles - cl.cam.ac.uk

1 1QM slides by Michael A. Nielsen, University of QueenslandQuantum Mechanics Basic PrinciplesWhat is Quantum Mechanics ?It is a frameworkfor the development of physical is nota complete physical theory in its own electrodynamics (QED)Operating systemApplications softwareQuantum mechanicsSpecific rulesNewton s laws of motionNewtonian gravitationQM consists of four mathematical postulateswhich lay theground rules for our description of the successful is Quantum Mechanics ?It is deviations from Quantum Mechanics are knownMost physicists believe that any theory of everything will be a Quantum mechanical theoryNot just for the small stuff!QM crucial to explain why stars shine, how the Universeformed, and the stability of conceptual issue, the so-called measurement problem , remains to be clarified. Attempts to describe gravitation in the framework of Quantum Mechanics have (so far) structure of Quantum mechanicslinear algebraDirac notation4 postulates ofquantum mechanics1.

2 How to describe Quantum states of a closed How to describe Quantum How to describe measurements of a Quantum How to describe Quantum state of a composite system. state vectors and state space unitary evolution projective measurements tensor products ,,A Example: qubits(two-level Quantum systems)01 +01 +=22|| || 1 Normalization 0 and 1 are thecomputational basis statesphotonselectron spinnuclear spinetcetera All we do is draw little arrows on a piece of paper - that's all. -Richard FeynmanPostulate 1: Rough FormQuantum Mechanics does not prescribe the state spacesof specific systems, such as electrons. That s the job ofa physical theory like Quantum to any Quantum system is a complex vector space known as state : we ll work mainly with qubits, which have statespace + The state of a closed Quantum system is a unit vector instate space. 2A few conventionsThis is the write vectors in state space as: We always assume that our physical systems have finite-dimensional state :ddd =++++ = QuditdC(= ) rnearlyvQuantum not gate:01; qubitOutput qubit0 101 0110X = + 01?

3 + +01 10 Matrix representation:General dynamics of a closed Quantum system(including logic gates) can be represented as aunitary : Quantum logic gatesabAcd = Hermitian conjugation; taking the adjointUnitary matrices() *TAA=**acbd = Ais said to be unitary if AAA A I==We usually write unitary matrices as U. Example: 01 01 10XX1010 01I === Nomenclature tipsmatrix=(linear) operator=(linear) transformation=(linear) map= Quantum gate (modulo unitarity)Postulate 2 The of a is describedevolutionclosed Quantum systemunitary transforma by a tion.'U =Why unitaries?Unitary maps are the only linear maps that preservenormalization.'U =implies '1U ===Exercise: prove that unitary evolution preserves ; X 10 ; X10X === XYZP auli gates1 gate (AKA or )xX 2Y gate (AKA or )y 0 Notation: I 001 ; Y 10 ; Y0iYiii == = 1000 ; Z 11 ; Z01Z == = 3Z gate (AKA or )z Exercise: prove that XY=iZExercise: prove that X2=Y2=Z2=IMeasuring a qubit: a rough and ready prescription01 =+ Quantum Mechanics DOES NOT allow us to determine and.

4 We can, however, read out limited information about and . Measuring in the computational basis 22(0); (1)PP ==Measurement the system, leaving it in a state 0 or 1 determiunavoidably disned by the a qubit01110122 1(0)(1)2PP==More general measurements1 Let ,.., be an orthonormal basis for .ddeeC12A gives re"measuremenst of in the basis ,..,"(ult with probability .) djjeePje = **Reminder: + Measurement the system, leaving it in a state deterunavoidablmined by ty dhe example01 =+0101 Introduce orthonormal basis 22+ += =()211Pr + =12 22 +=2=2 +2Pr( )2 =4 Inner products and dualsThe inner product is used to define the of a vectodualr . ()If lives in then the of is a function defined bduya: l ddCCC Young man, in mathematics you don t understandthings, you just get used to them. - John von NeumannExample:()100 1=0 += Simplified notation: ()()()()** Properties: , since ,, , since ,,abbaa bb aAbbAAb cb A cbA c== ==Duals as row vectors()*Suppose = and =.

5 Then ,jjjjjjjaaj bbjaba bab== 1**122baab = KM**12identificatThis suggests the very useful of withthe row vector .ionaaa KPostulate 3: rough form12If we measure in an orthonormal basis ,..,, then we obtain the result with probability ).( jdPjejee =The measurement the system, leaving it in a state determined by thedis measurement problemQuantum systemMeasuring apparatusRest of the UniversePostulate 3 Postulates 1 and 2 Research problem: solve the measurement of global phase 12 Suppose we measure in the orthonormal basis ,..,.Pr(Then ).djeeje =212 Suppose we measure in the orthonormal basis ,..,.Then Pr ).( idijjjeeeeee ==global phase factor unobservabThe is thus , and we mayidentify the states and .leiiee Revised postulate 1 Associated to any Quantum system is a complex inner product space known as state space. The state of a closed Quantum system is a unit vectorin state space. Note: These inner product spaces are often calledHilbert spaces.

6 5 Multiple-qubit systems0001101100011011 +++ =2(,)| |xyPxyMeasurement in the computational basis:General state of nqubits:{} 0,1nxxx()Classically, requires 2 bits to describe the Hilbert space is a big place - Carlton Caves Perhaps [..] we need a mathematical theory of quantumautomata. [..] the Quantum state space has far greatercapacity than the classical one: [..] in the Quantum casewe get the exponential growth [..] the Quantum behaviorof the system might be much more complex than itsclassical simulation. Yu Manin (1980)Postulate 4 The state space of a composite physical system is thetensor product of the state spaces of the component :224 Two-qubit state space is CCC =Computational basis states: 00 ; 01 ; 10 ; 11 Alternative notations: 0 0 ; 0, 0 ; 00 .Properties()()( )zv wzvw vzw = = 1212()vv wv wv w+ = + 1212()vww vwvw + = + Some conventions implicit in Postulate 4If Alice prepares her system in state , and Bob prepareshis in state , then the joint state is.

7 Abab AliceBobConversely, if the joint state is then we say thatAlice's system is , and Bob's system iin the state inthe stats e .abab () means "Alice thatapplies is applied to the joint systethe gate to her system" ()ABv w Av Bw = ()()=iiabea eb Suppose a two-qubit system is in thestate 11 . A NOT gate is applied to thesecond qubit, and a measurement performed in thecomputational basis. What are the Worked exeprobabilitrciseo:ies f+r thepossible measurement outcomes?Suppose a NOT gate is applied to the second qubit of the 11 . +++()()The resulting state is 11IX +++ 10 .=+++ExamplesQuantum entanglementAliceBob00112 +=ab Schroedinger (1935): I would not call[entanglement] onebut rather the characteristic trait of Quantum Mechanics , the one that enforces its entire departure from classical lines of thought. ()()0101 =+ +00100111 =+++0 or 0. = =SummaryPostulate 1: A closed Quantum system is described by a unit vector in a complex inner product space known as state 2: The evolution of a closed Quantum system is described by a unitary transformation.

8 'U =12If we measure in an orthonormal basis,..,, then we obtain the result with probability ( ).Postulate 3 :djeejPje =The measurement disturbs the system, leaving it in a state determined by the 4: The state space of a composite physical systemis the tensor product of the state spaces of the componentsystems.