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Quantum Computing MAS 725 Hartmut Klauck NTU 5.3.2012 More about measurements(I) Some linear algebra: Vector space V (dimension d) Subspaces: U µ V and U is also a vector space (dim e < d) There is an orthonormal basis: v1,...,ve,...,vd, span(v1,...,ve)=U Projection onto U: PU = i=1...e | v iih vi | Example: Projector P=|v1ih v1| P |v1i = |v1ih v1| v1i = |v1i P ( |v1i+ |v2i)= |v1i + |v1ih v1| v2i = |v1i Measurements (II) A? B iff v?w for all v2 A, w2 B A © B=V iff A? B and for all v2 V: v = u + w, u2A, w2B, kuk,kvk = 1, ||2 + ||2=kvk2 Measurements (III) Hilbert spaces V, dim k Observable: System l · k of subspaces S0,...,Sl-1, pairwise orthogonal S0© ©Sl-1 = V Probability of measuring i is k Proj(Si) |i k2 The state |i collapses to Proj(Si) |i / kProj(Si) |ik (renormalized) Observables correspond to a measurement device We only consider projections measurements Measurements (IV) A matrix A is Hermitian, if A=Ay Hermitian matrices have only real eigenvalues and are diagonalizable: Uy A U is diagonal for some unitary U The eigenspaces of A decompose Cn Cn=©Vi Let ¸(i) denote an eigenvalue of A, and Pi the projection onto its eigenspace Then A=i ¸(i) Pi Measurements (V) A Hermitian matrix is a concise representation of an observable Eigenvalues correspond to measurement outcomes Eigenspaces decompose Cn Measurements: example Two qubits, living in C4 Observable: S0=span(|00i,|01i) S1=span(|10i,|11i) S0 ? S1 This observable corresponds to measuring the first qubit: S0 indicates that it is 0, S1 indicates 1 Measuring an EPR-Pair The state is 1/21/2 ¢ (|00i+|11i) We measure the first qubit Result: If we measure 0, then the state collapses to |00i If we measure 1 we get |11i Each happens with probability ½ Qubit 2 collapses right after measuring qubit 1 The qubits act like a shared public coin toss. This is even true if the qubits are spatially separated Summary Hilbert space: register holding a quantum state Vectors: states Unitary transformation: evolution of states (computation) Observables: for measuring the computation‘s output The probability distribution on results: output of the computation Projected and normalized vector: the remaining quantum state Measurement is the only way to extract information from a quantum state More about qubits No-Cloning Theorem Bell states Quantum Teleportation No Cloning Suppose we are given a quantum state |i Can we make a copy? Copying classical information is trivial (ask the music industry about that…) I.e., there is a unitary transformation Un: Un|xi|0i = |xi|xi for all x2{0,1}n But then by linearity U1 1/2.5(|0i+|1i) |0i= 1/21/2 (|00i+|11i) 1/21/2(|0i+|1i) times 1/2.5(|0i+|1i) No Cloning Theorem Theorem: There is no unitary U, such that for all quantum states |i on n qubits: U |i |0mi = |i|i|()i for some m und |()i (which is garbage) I.e. there is no universal way to copy unknown quantum states without error! [Dieks, Wootters/Zurek 82] Proof: No Cloning Let a linear U be given (this also fixes m). Then U |0ni |0mi =|0ni |0ni |0i U |1ni |0mi =|1ni |1ni |1i Then also (due to linearity) U 1/21/2 ( |0ni+|1ni ) |0mi = 1/21/2 ( U |0ni|0mi + U |1ni|0mi ) = 1/21/2 ( |02ni |0i + |12ni |1i ) But we wanted (for some |2i) 1/2 (|0ni+|1ni) (|0ni+|1ni) |2i Proof: No Cloning (II) We have 1/21/2 ( |02ni |0i + |12ni |1i ) We want 1/2 (|0ni+|1ni) (|0ni+|1ni) |2i Claim: This is not the same! Proof of the claim: we compute the inner product. Same state ) inner product = 1 h | i = h |i ¢ h|i h + | i=h|i + h|i Proof No Cloning (III) We get 1/21/2 ( |02ni |0i + |12ni |1i ) We expect 1/2 (|0ni+|1ni) (|0ni+|1ni) |2i Inner product: 1/23/2 ¢ ( ( h0n|0ni+h0n|1ni ) ¢ ( h0n|0ni+h0n|1ni ) ¢ h0|2i ) +( h1n|0ni+h1n|1ni) ¢ ( h1n|0ni+h1n|1ni) ¢ h1|2i ) ) = 1/23/2 ¢ (1 ¢ 1 ¢ a0,2 + 1 ¢ 1 ¢ a1,2) . Then |1/23/2 ¢ ( a0,2+a1,2)| · 2/23/2 < 1 No Cloning No unitary can map both a state and another state that are not orthogonal in a way that wakes a copy It is possible to clone quantum states with small success probability Bell States Consider the following basis (John Bell) This is orthonormal in C4 Generating Bell states x H |x,yi y CNOT Gate: CNOT |0, yi =|0, yi; CNOT |1,yi=|1, 1-yi |0,0i = 1/21/2 (|00i +|11i)=|+i Quantum Teleportation We are given a quantum state |i Can we reproduce that state in another location? We cannot just send them over…. And we cannot copy them Classical communication is possible This problem seems hard since a single qubit 0 |0i+1 |1i might contain a lot of information due to possibly irrational amplitudes Quantum Teleportation |i Local operations Classical Communication Local operations 1/21/2 (|00i+|11i) Quantum Teleportation [Bennett et al. 93] Alice and Bob share an EPR pair (Alice has the first qubit, Bob the other) Alice has |i (1 qubit quantum state) Alice applies CNOT to her qubit q0 with |i (control) and her EPR qubit q1 Alice applies H to q0 and measures q0 and q1 Alice sends the result of the measurement (2 bits) Bob applies a unitary transformation depending on the message to his EPR qubit q2 Quantum Teleportation q1, q2: EPR Pair; q0: |i; q0,q1 with Alice, q2 with Bob Measurement Pauli Transformations X: (NOT, Bit Flip) Z: (Phase Flip) Y: Quantum Teleportation Create EPR Pair Quantum Teleportation Then: CNOT on q0 and q1 x=0,1 x |xi y=0,1 1/21/2|y,yi CNOT: x,y x/21/2 |x, x © y, yi Measure q1: Result is a Prob. for 0/1 is 0.5 each: |0|2/2+|1|2/2=1/2 Remaining state: yy©a |y © a , yi on q0,q2 Send a to Bob a=0 ) Bob does nothing a=1 ) Bob applies X-Gate (Bit Flip) Result in both cases: y y©a|y © a, y © ai = y y |y,yi Quantum Teleportation State y y |y,yi Problem: Bob’s qubit q2 is still entangled with q0 y y |y,yi Quantum Teleportation Alice applies H to q0 Alice measures q0, assume the result is b, she sends b to Bob Applying H: y y |y,yi 1/21/2 z,y y (-1)z¢ y |z,yi Measuring q0 (Result is b; 0/1 with Prob. 1/2): y y (-1)b¢ y |yi on q2 Bob corrects by applying a Z-Gate, if necessary Quantum Teleportation y y|yi Remarks The EPR-pair is consumed The original state |i is destroyed [no cloning!] States with many qubits can be teleported one by one (we need to make sure entanglement between the teleported qubits is preserved)