The Weird World of Quantum Mechanics

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1 The Weird World of Quantum Mechanics Libor Nentvich: QC 16 April 2007: The Weird World of QM 1/30

2 How to fing a good bomb (Elitzur-Vaidman) In 1993 Avshalom Elitzur and Lev Vaidman proposed the following problem. 1 We have bombs whose detonator is extremely sensitive. Even a single photon set it on. 2 One half of bombs are good. One half are bad. Quetion: Can we pick up a good one? Libor Nentvich: QC 16 April 2007: The Weird World of QM 2/30

3 Classical answer Yes, we can. But the bomb is destroyed. Libor Nentvich: QC 16 April 2007: The Weird World of QM 3/30

4 Classical answer Yes, we can. But the bomb is destroyed. Libor Nentvich: QC 16 April 2007: The Weird World of QM 4/30

5 Quantum Mechanical answer Yes, we can. Without destroying the bomb. Libor Nentvich: QC 16 April 2007: The Weird World of QM 5/30

6 Quantum Mechanical answer. Why does it work? Dud bomb 1 Before beam-splitter we are in state P. 2 After beam-splitter the state evolves into R + i S. 3 After reflections of two mirrors the state evolves into i T V 4 Finally the two parts of the photon state interfere in a such way that the entire state emerges in state W. The detector at B receives no photon. Libor Nentvich: QC 16 April 2007: The Weird World of QM 6/30

7 Quantum Mechanical answer. Why does it work? Dud bomb Libor Nentvich: QC 16 April 2007: The Weird World of QM 7/30

8 Quantum Mechanical answer. Why does it work? Good bomb 1 Before beam-splitter we are in state P. 2 After beam-splitter the state evolves into R or i S each whith equal probability (property of quantum meaurement). R The photon hits the good detector and the bomb is lost. S The photon chooses the other way. Finally the photon reaches the final beam-splitter and the detector at B has 50% chances to detect it. Libor Nentvich: QC 16 April 2007: The Weird World of QM 8/30

9 Quantum Mechanical answer. Why does it work? Good bomb Libor Nentvich: QC 16 April 2007: The Weird World of QM 9/30

10 Quantum Mechanical answer. Why does it work? Good bomb Libor Nentvich: QC 16 April 2007: The Weird World of QM 10/30

11 When I hear about Schrödinger s cat, I reach for my gun. Stephen Hawking Schrödinger s cat is a thought experiment introduced in 1935 by Erwin Schrödinger. Inside an isolated container is a cat and a device that can be triggered by some quantum event. If that event takes place, then the device smashes a phial containing cyanide and the cat is killed, otherwise the cat lives on. There is an equal chance that the event occurs during one hour. After one hour in what state the cat is? Libor Nentvich: QC 16 April 2007: The Weird World of QM 11/30

12 It s not a lamb, it s the Schrödinger s cat. Libor Nentvich: QC 16 April 2007: The Weird World of QM 12/30

13 (at the beginning) Libor Nentvich: QC 16 April 2007: The Weird World of QM 13/30

14 (some time elapse) Libor Nentvich: QC 16 April 2007: The Weird World of QM 14/30

15 (more time elapse) Libor Nentvich: QC 16 April 2007: The Weird World of QM 15/30

16 (more time elapse) Libor Nentvich: QC 16 April 2007: The Weird World of QM 16/30

17 (after measurement) First possibility Libor Nentvich: QC 16 April 2007: The Weird World of QM 17/30

18 (after measurement) Second possibility Libor Nentvich: QC 16 April 2007: The Weird World of QM 18/30

19 Please, don t measure! Libor Nentvich: QC 16 April 2007: The Weird World of QM 19/30

20 Short history S. Wiesner. Counterfeit proof money R. P. Feynman. Simulating physics by computers C. H. Bennett G. Brassard. Key distribution algorithm D. Deutsch. Deutsch algorithm cm key ditribution m key ditribution P. Shor. Factoring algorithm L. Grover. Database searching algorithm km key ditribution (IBM) Quantum programming languages. 2001?? - 15 = 3 5. Really? Comercially available quantum key distribution devices (id Quantique, MagiQ). Libor Nentvich: QC 16 April 2007: The Weird World of QM 20/30

21 1. postulate 2. postulate 3. postulate 4. postulate Conventions We are using Dirac notations for denoting vectors, its duals, inner products, etc. For example: Dirac ordinary math vector ϑ ϑ dual vector ϑ ϑ inner product ϑ ϕ ϑ ϕ Libor Nentvich: QC 16 April 2007: The Weird World of QM 21/30

22 1. postulate 2. postulate 3. postulate 4. postulate Conventions All state spaces are finite dimensional. All vectors in the ket notations are nonzero unit vectors! For example: 0, 1, 3, 15 are sixteen mutually orthonormal vectors. The null (zero) vector will be denoted by 0. Warning: 0 0. The tensor product of vetors ϕ, ψ ϕ ψ will be denoted also by ϕψ Libor Nentvich: QC 16 April 2007: The Weird World of QM 22/30

23 1. postulate 2. postulate 3. postulate 4. postulate Ask one, get two Deutsch s algorithm Let f : {0, 1} {0, 1} be a function. We are allowed to evaluate f only once. Can we determine if f (0) = f (1)? Libor Nentvich: QC 16 April 2007: The Weird World of QM 23/30

24 1. postulate 2. postulate 3. postulate 4. postulate Ask one, get two Clasical answer. It s not possible Libor Nentvich: QC 16 April 2007: The Weird World of QM 24/30

25 1. postulate 2. postulate 3. postulate 4. postulate Ask one, get two Quantum mechanical answer. ψ 0 The input state is 01. ψ 1 The state is [ ] [ 0 1 ]. ψ 2 The state is ±[ ] [ 0 1 ] if f (0) = f (1) ψ 3 The state is ±[ 0 [ 0 1 ] if f (0) = f (1) 0 H x x H ) 1 ψ 0 H ψ 1 y U f y f (x) ψ 2 ψ 3 Libor Nentvich: QC 16 April 2007: The Weird World of QM 25/30

26 1. postulate 2. postulate 3. postulate 4. postulate The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative... Richard Feynman Libor Nentvich: QC 16 April 2007: The Weird World of QM 26/30

27 1. postulate 2. postulate 3. postulate 4. postulate Postulate 1. Associated to any quantum sytem is a complex vector space state space. The state of a closed quantum space is a unit vector in the state space. Example The simplest quantum mechanical sytem is the qubit. A qubit has a two-dimensional state space. 0, 1 classical bits 0, 1 0, 1 quantum mechanical qubits 0, 1 Libor Nentvich: QC 16 April 2007: The Weird World of QM 27/30

28 1. postulate 2. postulate 3. postulate 4. postulate Postulate 2. The evolution of a closed quantum sytem is described by unitary transformation. φ = U ψ Example ψ ψ X H φ φ X 0 = 1, X 1 = 0 H 0 = 1 2 ( ), H 1 = 1 2 ( 0 1 ). Libor Nentvich: QC 16 April 2007: The Weird World of QM 28/30

29 1. postulate 2. postulate 3. postulate 4. postulate Postulate 3. If we measure ψ in an orthonormal basis φ 1,... φ n, then we obtain the result i with probability P(i) = φ i ψ 2. The measurements disturbs the system, leaving it in the state φ i. Example Let the sytem be in the state ψ = After meaurement in the basis 0, 1 we obtain 0 with probability 1/4 or 1 with probability 3/4. The meaurement leaves the system in the state 0 in the first case and leaves the system in the state 1 in the second one. Libor Nentvich: QC 16 April 2007: The Weird World of QM 29/30

30 1. postulate 2. postulate 3. postulate 4. postulate Postulate 4. The state space of a composite physical system is the tensor product of the state spaces of the component systems. Example Composite sytem of three components H H H (H H H ) = H 0 H 1 H 0 = (0 + 1)(0 1)(0 + 1) Libor Nentvich: QC 16 April 2007: The Weird World of QM 30/30

Quantum Measurements: some technical background

Quantum Measurements: some technical background [From the projection postulate to density matrices & (introduction to) von Neumann measurements] (AKA: the boring lecture) First: One more example I wanted