Introduction to Quantum Information Processing

Transcription

1 Introdction to Qantm Information Processing Lectre 5 Richard Cleve

2 Overview of Lectre 5 Review of some introdctory material: qantm states, operations, and simple qantm circits Commnication tasks: one qbit conveys at most one bit sperdense coding teleportation 2

3 review of introdctory material 3

4 Classical & qantm states Probabilistic states: x, p x p x x = p p p p p p p p Qantm states: x x, x x 2 = C Dirac notation: ψ = x {, } n x x 4

5 Dirac bra-c-ket notation Ket: ψ always denotes a colmn vector, e.g. M Bra: ψ always denotes a row vector that is the conjgate transpose of ψ, e.g. [ * * * ] Bracket: φ ψ denotes φ ψ, the inner prodct of φ and ψ 5

6 Some basic qantm operations Initialize: set a qbit to state or Unitary operations: x x a U x x x x (U U = I ) Measrements: x x x a M with prob with prob M with prob M and the qantm state collapses 6

7 Examples: one-qbit operations Rotation: cos sin θ θ sin cos θ θ NOT (bit flip): σ x = X = Phase flip: σ z = Z = Hadamard: H = 2 7

8 Example: measring a qbit ψ β 2 ψ With respect to the comptational basis {, } 2 There exist other qantm operations, bt they can all be simlated by the aforementioned types Example: measrement with respect to a different orthonormal basis { ψ, ψ } 8

9 Distingishing between two states + ( ) = ( ) Let be in state = + or 2 2 Qestion : can we distingish between the two cases? Distingishing procedre:. apply H 2. measre This works becase H + = and H = Qestion 2: can we distingish between and +? It trns ot that, since they re not orthogonal, they cannot be perfectly distingished 9

10 Strctre among sbsystems qbits: time # U W #2 V #3 #4 nitary operations measrements

11 Qantm comptations Qantm circits:

12 Examples of two-qbit systems (do nothing) U U = Maps basis states as: U U U U The reslting 4x4 matrix is I U = 2

13 Two-qbit gates U Maps basis states as: U U U = Reslting 4x4 matrix is controlled-u = C-U = 3

14 Controlled-NOT (CNOT ( CNOT) X a a b a b Note: control qbit may change on some inpt states + 4

15 Universal sets of gates Theorem: any nitary operation U acting on k qbits can be decomposed into O(4 k ) CNOT and one-qbit gates Therefore, CNOT and all one-qbit gates are niversal (classical analoge: AND and NOT gates) Example: Toffoli gate controlled-controlled-not a b a b c c (a b) Can be simlated by CNOT, H, and = W e i π / 4 5

16 commnication tasks 6

17 How mch classical information in n qbits? 2 n complex nmbers are needed to describe an arbitrary n-qbit pre qantm state: Does this mean that an exponential amont of classical information is stored in n qbits? No! Holevo s Theorem [973] implies: cannot convey more than n bits of information in n qbits How mch information does Natre have to store to maintain an n-qbit qantm state? 7

18 Holevo s Theorem Easy case: Hard case (the general case): ψ n qbits U b b 2... b n cannot convey more than n bits! b b 2 b 3 b n ψ n qbits m qbits U b b 2 b 3 b n b n+ b n+2 b n+3 b n+4 b n+m (proof is omitted here) 8

19 Sperdense coding (prelde) Sppose that Alice wants to convey two classical bits to Bob sending jst one qbit ab Alice Bob ab By Holevo s Theorem, this is impossible 9

20 Sperdense coding In sperdense coding, Bob can send a qbit to Alice first ab Alice Bob ab How can this help? 2

21 How sperdense coding works. Bob creates the state + and sends the first qbit to Alice 2. Alice: if a =then apply X to qbit if b =then apply Z to qbit send the qbit back to Bob X = Z = ab state + + Bell basis 3. Bob measres the two qbits in the Bell basis 2

22 Measrement in the Bell basis Specifically, Bob applies H to his two qbits... and then measres them, yielding ab inpt otpt + + This concldes sperdense coding 22

23 Review of partial measrements Sppose one measres jst the first qbit of the state 2 + i 3 + What is the reslt? = + i , 3 7 +i 4 7 with prob. 7/2, with prob. 5/2 23

24 Teleportation (prelde) Sppose Alice wishes to convey a qbit to Bob by sending jst classical bits + β + β If Alice knows and β, she can send approximations of them bt this reqires infinitely many bits for perfect precision Moreover, if Alice does not know or β, she can at best acqire one bit abot them by a measrement 24

25 Teleportation scenario In teleportation, Alice and Bob also start with a Bell state + β (/ 2)( + ) and Alice can send two classical bits to Bob Note that the initial state of the three qbit system is: (/ 2)( + β )( + ) = (/ 2)( + + β + β ) 25

26 To be contined 26

27 Corse annoncements Send an to to be pt on the corse mailing list. We will normally only send information relevant to the corse here (e.g. comments abot the assignments). Office hors: Carlos Perez (CS): Wednesdays 2:3-4:3pm Donny Cheng (C&O): Thrsdays 2:3-4:3pm Marcs Silva (Physics): Fridays :am-noon or :-3:pm is there a preference? 27