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Introductory Topics in Quantum Paradigm Introductory Topics in Quantum Paradigm Subhamoy Maitra Indian Statistical Institute [[email protected]] 21 May, 2016 Subhamoy Maitra Quantum Information Outline of the talk Preliminaries of Quantum Paradigm What is a Qubit? Entanglement Quantum Gates No Cloning Indistinguishability of quantum states Details of BB84 Quantum Key Distribution Algorithm Description Eavesdropping State of the art: News and Industries Walsh Transform in Deutsch-Jozsa (DJ) Algorithm Deutsch-Jozsa Algorithm Walsh Transform Relating the above two Some implications Subhamoy Maitra Quantum Information Qubit and Measurement A qubit: αj0i + βj1i; 2 2 α; β 2 C, jαj + jβj = 1. Measurement in fj0i; j1ig basis: we will get j0i with probability jαj2, j1i with probability jβj2. The original state gets destroyed. Example: 1 + i 1 j0i + p j1i: 2 2 After measurement: we will get 1 j0i with probability 2 , 1 j1i with probability 2 . Subhamoy Maitra Quantum Information Basic Algebra Basic algebra: (α1j0i + β1j1i) ⊗ (α2j0i + β2j1i) = α1α2j00i + α1β2j01i + β1α2j10i + β1β2j11i, can be seen as tensor product. Any 2-qubit state may not be decomposed as above. Consider the state γ1j00i + γ2j11i with γ1 6= 0; γ2 6= 0. This cannot be written as (α1j0i + β1j1i) ⊗ (α2j0i + β2j1i). This is called entanglement. Known as Bell states or EPR pairs. An example of maximally entangled state is j00i + j11i p : 2 Subhamoy Maitra Quantum Information Quantum Gates n inputs, n outputs. Can be seen as 2n × 2n unitary matrices where the elements are complex numbers. Single input single output quantum gates. Quantum input Quantum gate Quantum Output αj0i + βj1i X βj0i + αj1i αj0i + βj1i Z αj0i − βj1i j0i+j1i j0i−|1i αj0i + βj1i H α p + β p 2 2 Subhamoy Maitra Quantum Information Quantum Gates (contd.) 1 input, 1 output. Can be seen as 21 × 21 unitary matrices where the elements are complex numbers. 0 1 α β The X gate: = 1 0 β α " 1 1 # " α+β # p p α p The H gate: 2 2 = 2 p1 − p1 β αp−β 2 2 2 Subhamoy Maitra Quantum Information A True Random Number Generator j0i+j1i j0i−|1i f p ; p g 2 2 H Start with any qubit Random bit stream M Measurement at fj0i; j1ig basis Subhamoy Maitra Quantum Information Quantum Gates (contd.) 2-input 2-output quantum gates. Can be seen as 22 × 22 unitary matrices where the elements are complex numbers. These are basically 4 × 4 unitary matrices. An example is the CNOT gate. j00i ! j00i, j01i ! j01i, j10i ! j11i, j11i ! j10i. The matrix is as follows: 2 1 0 0 0 3 6 0 1 0 0 7 6 7 4 0 0 0 1 5 0 0 1 0 Subhamoy Maitra Quantum Information Preliminaries (circuit for entangled state) x H . jβxy i y ⊕ Figure: Quantum circuit for creating entangled state Bell State Description j00i+j11i jβ i p 00 2 j01i+j10i jβ i p 01 2 j00i−|11i jβ i p 10 2 j01i−|10i jβ i p 11 2 Subhamoy Maitra Quantum Information Teleportation j i A M1 G M2 jβxy i Alice * f Bob + j i B C """"" j 0i j 1i j 2i j 3i j 4i Figure: Quantum circuit for teleporting a qubit Subhamoy Maitra Quantum Information Teleportation (Example) 1 jβxy i = jβ00i, 2 G is CNOT, i.e., j00i ! j00i, j01i ! j01i, j10i ! j11i, j11i ! j10i. M M 3 A = H; B = X 2 ; C = Z 1 An extension: 1 Use any jβxy i, 2 G is CNOT, M ⊕x M ⊕y 3 A = H; B = X 2 ; C = Z 1 We are studying the effect of other gates apart from CNOT. Subhamoy Maitra Quantum Information Teleportation (step by step) (j00i+j11i) j i = j ijβ i = (αj0i + βj1i) p 0 00 2 (j00i+j11i) (j10i+j01i) j i = αj0i p + βj1i p 1 2 2 j0i+j1i (j00i+j11i) j0i−|1i (j10i+j01i) j i = α p p + β p p = 2 2 2 2 2 1 2 (j00i(αj0i + βj1i) + j01i(βj0i + αj1i)+ j10i(αj0i − βj1i) − j11i(βj0i − αj1i)) Observe 00, nothing to do. Observe 01, apply X . Observe 10, apply Z. Observe 11, apply both X ; Z. Subhamoy Maitra Quantum Information Remote state preparation In teleportation the state j i is unknown. In this case the θ θ iφ state is known: j i = cos 2 j0i + sin 2 e j1i: j01i−|10i Entangled state p is shared between two parties. 2 Transformed to different basis j i, j ?i. The entangled state j i−| i on the new basis can be seen as ? p ? . 2 Alice measures on j i; j ?i basis. If Alice gets j ?i, then Bob obtains j i, so no change is required. Alice sends cbit 0 in this case. If Alice gets j i, then Bob obtains j ?i, so he needs to apply a transformation. Alice sends cbit 1 in this case. Note that only one cbit communication is required for remote state preparation. Subhamoy Maitra Quantum Information Multi Party Pseudo Telepathy For any n ≥ 3, the game Gn consists of n players. The bit string x1 ::: xn contains even number of 1's. Each player Ai receives a single input bit xi and is requested to produce an output bit yi . x1 ::: xn is the question and y1 ::: yn is the answer. The game Gn will be won by this team of n players if n n X 1 X y ≡ x (mod2): i 2 i i=1 i=1 Subhamoy Maitra Quantum Information Multi Party Pseudo Telepathy (Contd.) No communication is allowed among the n participants after receiving the inputs and before producing the outputs. It has been proved that no classical strategy for the game Gn 1 −dn=2e can be successful with a probability better than 2 + 2 . Quantum entanglement serves to eliminate the classical need to communicate and it is shown that there exists a perfect quantum protocol where the n parties will always win the game. Subhamoy Maitra Quantum Information Pseudo Telepathy (the set up) Define 1 1 jΦ+i = p j0ni + p j1ni n 2 2 and 1 1 jΦ−i = p j0ni − p j1ni: n 2 2 H denotes Hadamard transform. S denotes the unitary transformation Sj0i 7! j0i; Sj1i 7! ij1i. + If S is applied to any two qubits of jΦn i leaving the other − qubits undisturbes then the resulting state is jΦn i and vice versa. Subhamoy Maitra Quantum Information Pseudo Telepathy (the set up, contd.) + If jΦn i is distributed among n players and if exactly m of them apply S to their qubit, then the resulting global state + − will be jΦn i if m ≡ 0 mod 4 and jΦn i if m ≡ 2 mod 4. Note that 1 X (H⊗n)jΦ+i = p jyi n n−1 2 wt(y)≡0 mod 2 and 1 X (H⊗n)jΦ−i = p jyi: n n−1 2 wt(y)≡1 mod 2 Subhamoy Maitra Quantum Information Pseudo Telepathy (the quantum algorithm) The players are allowed to share a prior entanglement, the state + jΦn i. 1 If xi = 1, Ai applies transformation S to his qubit; otherwise he does nothing. 2 He applies H to his qubit. 3 He measures his qubit in order to obtain y. 4 He produces yi as his output. The game Gn is always won by the n distributed parties without any communication among themselves. Open question: In this algorithm n ≥ 3. It is open to find a two party pseudo telepathy problem that admits a perfect quantum solution, yet any classical protocol would have probability of success close to half. Subhamoy Maitra Quantum Information More on CNOT jµi: control qubit · may be entangled L g j0i: target qubit jµi is either j0i or j1i. Then it will be copied perfectly without creating any disturbance to jµi. j0i+j1i Say jµi = p . Then at the output we will get entangled 2 j00i+j11i state p . Thus copying is not successful here. 2 Subhamoy Maitra Quantum Information Cloning: Possible in classical domain, not in quantum Possible to copy a classical bit Not possible for an unknown quantum bit Subhamoy Maitra Quantum Information No cloning A result of quantum mechanics Not possible to create identical copies of an arbitrary unknown quantum state It was stated by Wootters, Zurek, and Dieks in 1982 W. K. Wootters and W. H. Zurek. A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802803. D. Dieks. Communication by EPR devices, Physics Letters A, vol. 92(6) (1982), pp. 271272. Huge implications in quantum computing, quantum information, quantum cryptography and related fields. Subhamoy Maitra Quantum Information No cloning (contd.) It is not possible to copy an unknown Quantum state. Consider a quantum slot machine with two slots labeled A and B. A is the data slot set in a pure unknown quantum state j i whereas B is target slot set in a pure state jsi where A will be copied. Subhamoy Maitra Quantum Information No cloning (contd.) Let there exist a unitary operator which does the copying procedure. Mathematically it is written as U(j ijsi) = j ij i. U: unitary operator, UUy = I . y (U )ij = Uji , transpose and scalar complex conjugate. Let this copying procedure works for two particular pure states, j i and jφi. Then we have U(j ijsi) = j ij i; U(jφijsi) = jφijφi Take the inner product: hsjh jUyUjφijsi = h jh jjφijφi. This implies h jφi = (h jφi)2. Subhamoy Maitra Quantum Information No Cloning (contd.) x = x2 has only two solutions: x = 0 and x = 1. Thus we get either j i = jφi or inner product of them equals to zero, i.e., j i and jφi are orthogonal to each other.
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