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Bell Measurements and Teleportation. Overview. Entanglement Bell states and Bell measurements Limitations on Bell measurements using linear devices Teleportation Dense coding Entanglement swapping Entanglement purification Quantum repeaters. Entanglement.
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Overview • Entanglement • Bell states and Bell measurements • Limitations on Bell measurements using linear devices • Teleportation • Dense coding • Entanglement swapping • Entanglement purification • Quantum repeaters
Entanglement • Two systems described by two separable Hilbert spaces. • States of the two systems can be described by the tensor product of their state spaces. • Schmidt decomposition: • If and the state is said to be separable. If more than one then is said to be entangled. • The state of one system cannot be specified without the other.
Bell States • For two two-state systems denoted each by the Bell states form a basis for the whole system and are maximally entangled: where is anti-symmetric and are symmetric with respect to particle interchanging.
Bell Measurements Distinguishing Bell states using linear elements such as beam splitters, phase shifters, photo-detectors etc. All elements can be described by unitary transformations. In linear ones particle number is conserved. Examples for photons: Beam splitter: Polarization beam splitter Half wave plate at 45 degrees
Example: distinguishing anti-symmetric and symmetric states - Hong–Ou–Mandel effect OR ? Beam splitter operator representation for a single photon: • Double transmission obtains a minus sign relative to double reflection. • symmetric states have zero amplitude for d1-d2 coincidence. • d1 + d2 simultaneous “click” the state has collapsed to • By measuring the Bell operator we have created entanglement!
Distinguishing Bell States • The goal: To create a set of unitary operators that would make a different set of detectors “click” for each Bell state.
Distinguishing Bell states – cont. A scheme to measure 2+ Bell states. • Turns out this is the best we can do with linear elements. • Non-linear devices can achieve a complete measurement but with low efficiency.
Teleportation • Alice wants to send a quantum bit to Bob. • She cannot measure the state and send the results. • If she sends the qubit itself it might deteriorate on the way or take too much time to get there if it is a state of a massive object.
Teleportation – cont. • Alice has a photon-qubit that she wants to teleport. • Alice creates two entangled photons, 2 and 3, and sends photon3 to Bob. • She performs a Bell measurement on photon1 and photon2 and sends Bob the result. • Bob performs a transformation of his photon3 according to Alice’s Bell measurement result and photon3 becomes a replica of photon1.
How does it work? • Before Alice’s Bell measurement the complete state is: which can be expressed as • By performing a Bell measurement on photons 1 and 2 they make photon3 collapse into one of the above states. • By sending the result Alice instructs Bob which transformation to perform – Pauli matrices.
Experimentally • Alice takes two photons (2,3) from a PDC in an anti-symmetric entangled state and sends photon3 to Bob. • Alice creates photon1 at 45 degrees, measures only on photons 1 and 2 and indicates to Bob about it. • In this configuration, Bob’s photon is immediately a replica of photon1. • Photon1 is destroyed in accordance with the no-cloning theorem.
Teleportation with complete BSM Very low efficiency…
Dense Coding • By manipulating one photon entangled in a Bell state we can convert it to another Bell state. • Manipulation of one photon = four Bell states = two bits! • We can measure 2+“1” out of four Bell states. • A “trit”: enhancement of the channel capacity by a factor of
Dense Coding Experiment Phys. Rev. Lett. 76, 4656–4659
Entanglement Swapping • Making photons that have never interacted entangle using mediators. • We want to entangle photons 1 and 4. • We entangle photons 1 with 2 and 3 with 4. The complete state is: • Now, performing a Bell measurement on photons 2+3 results in entanglement of 1+4 into the same state as 2+3. OR
Entanglement Purification - Motivation • Distribution of entangled states between distant locations is essential for quantum communication over large distances. • The quality of entangled states generally decreases exponentially with the channel length. • Error correction in quantum computation.
Entanglement purification Take only “four mode” cases Nature423, 417-422 (22 May 2003)
Quantum Repeaters • Classical repeaters: divide the channel into N segments and enhance the signal at each node. • Qubits cannot be cloned at each node and re-sent. • Quantum repeaters: A teleportation scheme involving entanglement swapping and purification. • Works in logarithmic time and polynomial in resources with respect to the channel length.
The Scheme • Divide the channel between A and B into N segments by N-1 nodes: • Create an EPR pair of fidelity between every two adjacent nodes. Example: • At every Node perform a Bell measurement of one photon on both sides.
Purify the entanglement between using M copies to achieve higher fidelity. • Repeat the process for the new state until A and B share an entangled pair. Resources (number of EPR pairs): Polynomial in resources, logarithmic (n) in time!
