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 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. Polarization beam splitter Half wave plate at 45 degrees Examples for photons: Beam splitter:

Example: distinguishing anti-symmetric and symmetric states - Hong–Ou–Mandel effect 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! Beam splitter operator representation for a single photon: OR ?

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 bya 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 Swapping Experiment

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 Nature 423, (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. Resources (number of EPR pairs): Polynomial in resources, logarithmic (n) in time! Repeat the process for the new state until A and B share an entangled pair.

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