Question

Quantum entanglement is one of the fascinating phenomena in quantum mechanics. Consider a system of two entangled particles that are prepared in the singlet state. The spin measurement of one of these particles, let's say Particle A, always reveals the opposite spin state of the other, Particle B, regardless of the measurement direction.

1) Explain why this phenomenon does not violate the principles of special relativity, specifically the no-signalling theorem.

2) If an observer measures the spin of Particle A along the z-axis and finds it to be "up," what can be said about the state of Particle B? If a second observer, unaware of the first measurement, measures the spin of Particle B along the x-axis, what are the possible outcomes and their respective probabilities?

3) Discuss how this type of entanglement can be used in the quantum teleportation protocol and provide a step-by-step explanation of the process.

Answer 1

1) The phenomenon of quantum entanglement, where two particles become correlated in such a way that the state of one particle is instantaneously influenced by the measurement of the other particle, might seem to violate the principles of special relativity, which state that no information can be transmitted faster than the speed of light. However, this is not the case, and quantum entanglement does not violate the no-signalling theorem.

The no-signalling theorem ensures that no information can be communicated between two distant entangled particles through their entanglement faster than the speed of light. While the measurement of one entangled particle can instantaneously determine the state of the other, this does not enable any form of communication or information transfer. The outcomes of the measurements on the entangled particles are inherently random and unpredictable before the measurements take place, so there is no way to use entangled particles to transmit information in a controlled manner.

2) If an observer measures the spin of Particle A along the z-axis and finds it to be "up," the state of Particle B will be instantaneously determined to be "down" in the z-axis direction. This is a consequence of their entanglement, which guarantees opposite spin outcomes.

If a second observer, unaware of the first measurement, measures the spin of Particle B along the x-axis, there are two possible outcomes:

- The probability of finding Particle B's spin to be "up" along the x-axis is 50%.

- The probability of finding Particle B's spin to be "down" along the x-axis is also 50%.

The previous measurement of Particle A along the z-axis does not affect the outcome probabilities of the x-axis measurement on Particle B, as the outcomes are statistically independent.

3) Quantum teleportation is a remarkable quantum communication protocol that allows the transfer of an unknown quantum state from one location (sender) to another distant location (receiver) using entanglement as a resource. Here's a step-by-step explanation of the process:

- Entanglement Preparation: Alice and Bob share an entangled pair of particles, typically photons, in a Bell state. Alice keeps one particle, and Bob keeps the other.

- Initial State Preparation: Alice possesses a quantum state |ψ⟩ that she wants to teleport to Bob.

- Bell Measurement: Alice performs a joint measurement of her unknown state |ψ⟩ and her part of the entangled pair.

- Communication of Classical Information: Alice obtains two classical bits of information as a result of the measurement and communicates these to Bob through classical channels.

- Conditional Operations: Based on the classical information, Bob performs specific quantum operations on his particle.

- Teleportation Complete: After Bob performs the operations, his particle now holds the exact quantum state |ψ⟩ that Alice wanted to teleport.

Quantum teleportation is a fundamental process in quantum information science and enables secure quantum communication and quantum computing.

Answer 2

Quantum entanglement is a physical phenomenon that occurs when two or more particles are linked together in such a way that they share the same quantum state, even when they are separated by a large distance

Step-by-step explanation:

1.

The no-signaling theorem states that it is impossible to transmit information faster than the speed of light. Quantum entanglement does not violate this theorem because the entangled particles do not actually communicate with each other.

Instead, they are described by a single quantum state, and when one particle is measured, the other particle is instantaneously forced into the opposite state.

2.

If an observer measures the spin of Particle A along the z-axis and finds it to be "up," then Particle B must be "down" along the z-axis. This is because the two particles are entangled, and they are described by a single quantum state. The state of Particle B is therefore fixed by the measurement of Particle A.

If a second observer, unaware of the first measurement, measures the spin of Particle B along the x-axis, then there are two possible outcomes: "up" or "down." The probability of each outcome is 50%.

This is because the two states "up" and "down" are equally likely along the x-axis.

3.

Quantum teleportation is a protocol that allows for the transmission of quantum information between two parties.

The protocol works by entangling two particles, one of which is held by each party. The parties then measure their respective particles, and the results of the measurements are used to reconstruct the quantum state of the particle that was initially held by the first party.

The steps of the quantum teleportation protocol are as follows:

• The two parties, Alice and Bob, prepare two entangled particles.
• Alice measures her particle along a chosen axis.
• Bob uses the results of Alice's measurement to reconstruct the quantum state of his particle.

The quantum teleportation protocol is a remarkable feat of quantum mechanics. It allows for the transmission of quantum information between two parties without actually transmitting the particles themselves.

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