Entanglement and Quantum Teleportation

Transcription

1 Entanglement and Quantum Teleportation Stephen Bartlett Centre for Advanced Computing Algorithms and Cryptography Australian Centre of Excellence in Quantum Computer Technology Macquarie University, Sydney, Australia Lecture 2 on Quantum Computing NITP Summer School 2003 Adelaide, Australia 28-3 January 2003

2 Outline - Entanglement What is entanglement? Coupled quantum systems Classical and quantum correlations Using entanglement Superdense coding Quantum teleportation Entanglement as a resource Creating entanglement Quantum teleportation in the lab! Entanglement and Teleportation - NITP

3 An abstract quantum system We describe it using: a Hilbert space, with dimension basis of states State of the system could be a superposition Projective measurements in this basis give result with prob. Example: a qubit (a two-level system) basis e.g., spin ½ particle, photon with pol. Entanglement and Teleportation - NITP

4 Composite quantum systems A A+B B How do we describe two quantum systems together? A: Hilbert space dimension B: Hilbert space dimension A+B: Hilbert space basis: dimension: Let s look at the form of states for A+B Product, not sum Entanglement and Teleportation - NITP

5 Product and entangled states Product state: has the form System A is in the state regardless of B Measurements on A and B will be uncorrelated Entanglement: take a superposition of product states, e.g., Leads to correlated measurements between A and B Entanglement and Teleportation - NITP

6 Example: coupled qubits Let A and B be two-level (qubit) systems Product state: e.g., Basis for : Many qubits: e.g., Basis for : Computational basis Use binary notation to label product state basis Entanglement and Teleportation - NITP

7 Example: the Bell states Entangled state: e.g., for two qubits Bell states: a basis of entangled states for two qubits Check: they are orthogonal and cannot be expressed as product states for A and B Entanglement and Teleportation - NITP

8 Different bases Consider the Bell state What if we changed bases for each qubit? Rewrite the Bell state: Entangled in any basis Entanglement and Teleportation - NITP

9 Using entanglement Take an entangled system (e.g., in a Bell state) Give system A to Alice and B to Bob Bell state Alice Bob Alice and Bob can: transform their systems (quantum evolution) perform measurements on their systems Entanglement and Teleportation - NITP

10 Measurements by Alice or Bob What happens if Alice (or Bob) performs projective measurements on their system? Random result in any basis Basis: Basis: 0 + Random results Measurements on an ensemble of the same Bell state Random results Entanglement and Teleportation - NITP

11 Entanglement and Teleportation - NITP 2003 Measurements by Alice or Bob What happens if Alice and Bob both perform projective measurements and compare? Correlated results if in the same basis B A B A Basis: Basis: B A Different bases

12 Entangled states can generate classical correlations Classical correlations can be useful for secret communication Example: private key cryptography (one-time pad) Alice has a message m (0000) to send Bob Alice and Bob have a channel A If Alice and Bob share a private key of random numbers (00): 0000 A B No transmitted information! Entanglement and Teleportation - NITP B

13 Power of quantum correlations Quantum correlations (from entangled states) can be useful for communication Quantum correlations can lead to classical correlations (one-time pads) which are powerful Without converting to classical correlations, the entangled states have even more power: Tests of local realism (Bell) Superdense coding Quantum teleportation Entanglement swapping Quantum cryptography Quantum computing (?) Other applications??? (field is still growing) Entanglement and Teleportation - NITP

14 Classical communication on a quantum channel Alice and Bob share a quantum channel Quantum channel sends qubits instead of bits Alice wants to send classical messages to Bob They agree on a basis, say 0, Alice wants to communicate a bit b, so sends a qubit b Bob measures in the agreed basis gets the result b with certainty Entanglement and Teleportation - NITP

15 Classical communication on a quantum channel Can we do better? Qubits seem to store two complex numbers: Number of distinguishable states is limited to the dimension of the Hilbert space (for a qubit, it's 2) Only one bit of information can be measured from a qubit One qubit must be sent for every bit, right? NO! Quantum (non-classical) correlations can be used to send two bits with every qubit superdense coding Entanglement and Teleportation - NITP

16 Superdense coding Let Alice and Bob share a Bell state Bell state Alice wants to send two bits b and b 2 to Bob Alice performs a unitary operation on her qubit Check: X and Z are unitary operators If b =, then flip the qubit: If b 2 =, then change the relative phase by π: Entanglement and Teleportation - NITP

17 Superdense coding Result of Alice s operations: bits apply bits result 00 0 I X Resulting effect on total state of both parties 00 0 Ψ + Ψ 0 Z 0 Φ + XZ Φ Alice then sends her qubit to Bob Bob performs a measurement in the Bell basis: with both qubits, Bob can perform a 4-outcome measurement and obtain two bits of information Entanglement and Teleportation - NITP

18 Results from superdense coding Superdense coding transfers two bits of info per qubit b,b 2 b,b 2 The qubit transferred from Alice to Bob is half of one of the four Bell states: Contains no information on its own All the information is in the quantum correlations This coding has the properties of the classical one-time pad, plus the remarkable advantage of sending two classical bits with every qubit! Entanglement and Teleportation - NITP

19 Interpreting superdense coding Alice has managed to communicate two bits of information to Bob by sending only one qubit, provided they shared a Bell state to start To create and share a Bell state, they must have (at some point) transmitted a qubit, although this transmission could be in either direction The important point: the act of sharing the quantum correlation (Bell state) could be long prior to the protocol, and does not involve the transmission of information All the information about the two bits is transmitted with a single qubit... yet somehow this qubit doesn't contain any information either! Quantum correlations (entanglement) are a resource Entanglement and Teleportation - NITP

20 Sending quantum information Say Alice wants to send Bob a qubit (i.e., quantum information rather than classical) Quantum channels are hard to make and maintain! Can Alice send the qubit over a classical channel (i.e., the telephone)? Option : -measure the qubit -send the measurement results to Bob Entanglement and Teleportation - NITP

21 Sending quantum information If Alice has complete information about the qubit: Alice tells Bob all of this information Bob performs a preparation to create this state If Alice has NO information about the qubit: for instance, the qubit is prepared by a third party Could perform a measurement, e.g., in basis If the qubit were in, no information is gained and the qubit is destroyed in the process Without knowledge of the preparation procedure of a qubit, no measurement can determine its state Entanglement and Teleportation - NITP

22 Quantum teleportation Again, entanglement provides a solution! Let Alice and Bob share a Bell state 2 Bell state 3 Alice takes the qubit to send () and the qubit from the Bell state (2) and measures them in the Bell basis One of four possible outcomes two bits of information Send these bits to Bob, who operates on his qubit (3) Entanglement and Teleportation - NITP

23 Quantum teleportation Result of Alice s measurements: result bits Ψ + Ψ Φ + Φ 00 Send bits to Bob, who must apply 0 0 b,b 2 bits apply I X Z XZ Looks like the opposite of superdense coding! Result: any measurement predictions involving the original qubit () now apply to Bob s qubit (3) The qubit has been quantum teleported to Bob Entanglement and Teleportation - NITP

24 Interpreting quantum teleportation The quantum system has not been teleported, only the state of the system The two bits contain no information about the qubit If qubit () was entangled with another system before quantum teleportation, qubit (3) is entangled after After teleportation, qubit () contains no information Entanglement and Teleportation - NITP

25 Quantum teleportation: reality Quantum teleportation has been performed in the lab! 997: Innsbrook, Austria Qubit: polarization state of a single photon Bell state: generated through parametric down conversion 998: Caltech, USA Qubit : coherent state of electromagnetic field mode Bell state : generated through two-mode squeezing Entanglement and Teleportation - NITP

26 Quantum teleportation in Oz 2002: Ping-Koy Lam s group at ANU Similar to Caltech exp. Hi-Fi QT Demonstrates: Entanglement was used Alice gains no info about the system Entanglement and Teleportation - NITP

27 Summary Entanglement is a resource Quantum correlations (from entangled states) can be useful for communication Tests of local realism (Bell) Superdense coding Quantum teleportation Entanglement swapping Quantum cryptography Quantum computing (?) Bell state Other applications??? (field is still growing) Next lecture: quantum algorithms... Entanglement and Teleportation - NITP