Quantum teleportation: the key to quantum networks

The first algorithm that is typically associated with quantum computing is the Shor algorithm [1], which is seen as a significant potential threat to the current cybersecurity paradigm. However, this emerging technology also offers promising avenues for enhancing our communication systems, as is the case of the well-known quantum key distribution (QKD) algorithm [2]. In this post, we will examine a fundamental protocol, quantum teleportation [3], which is regarded as a cornerstone in the development of future quantum networks.

Quantum teleportation is a phenomenon in quantum mechanics that enables the transfer of a quantum state from one particle to another without the physical movement of the particle itself. The key objective is to utilize this natural property of quantum entanglement to transfer quantum information between two parties. For the purpose of introducing the quantum teleportation protocol, we will name the two parties as Alice and Bob. Alice wishes to transfer a state |ψ⟩, which she has on a qubit Q₀, to Bob. To achieve this, it is essential that they have previously established a pair of entangled qubits (Q_A and Q_B) and a classical communication channel. Once the requisite conditions have been met, Alice can proceed with the quantum protocol by applying a controlled-not gate between Q₀ and Q_A, followed by a Hadamard gate on Q₀. Alice then measures both qubits and transmits the data via the classical channel to Bob. If Bob applies a NOT gate controlled by the classical measurement of Q_A and an Z gate controlled by the measurement of Q₀, he will recover the |ψ⟩ state.

Figure 1: Quantum teleportation protocol circuit.

We will now provide a brief mathematical description of what is going on during the protocol. We first start at the state |ψ00⟩. Once the entangled pair (the Φ⁺ bell state) is created we get . Note that as all requisite conditions are fulfilled, we can start to apply the algorithm. Before the algorithm starts, and considering that the state |ψ⟩ can be written as α|0⟩ + β |1⟩, we find ourselves at the state:

Alice now executes her part of the algorithm, commencing with the C-NOT gate:

Subsequently the Hadamard gate is applied:

The next stage is for Alice to measure her qubits, with four possible outcomes (00,01,10,11). It should be noted that, given the entanglement of the system, the state of Bob is dependent on Alice outcomes, resulting in four potential scenarios:

Figure 2: Four possible outcomes after Alice measured her state.

It is evident that, once Bob has completed the algorithm, he recovers the state α|0⟩ + β |1⟩ which was the decomposition we initially did of |ψ⟩. This means that the quantum teleportation protocol enables Alice to send quantum information to Bob. For those seeking deeper insight close connection can be observed between this experiment and the EPR paradox [4], a phenomenon that has captivated the scientific community, sparking interest and debate within the scientific community for much of the past century.

References

1. P.W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 124–134, November 1994.

2. Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J Cerf, Miloslav Dusek, Norbert Lütkenhaus, and Momtchil Peev. The security of practical quantum key distribution. Reviews of modern physics, 81(3):1301–1350, 2009.

3. Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, and Anton Zeilinger. Experimental quantum teleportation. 390, 1997.

4. A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(10):777–780, May 1935.