Quantum Entanglement Measurement Without Disturbing the System

Interference is not involved in entanglement. In the case of a two-electron singlet, the two particles cannot interfere, yet they remain entangled. This article explores the methods of measuring quantum entanglement without causing any changes to it, specifically focusing on the principles of quantum tomography and Bell's Theorem.

Understanding Quantum Entanglement

Quantum entanglement is a phenomenon where two or more particles become interconnected in a way that the state of one particle directly influences the state of another, regardless of the distance separating them. This interconnectedness persists even when the particles no longer interact.

In a standard case of a two-electron singlet state, the electrons cannot interfere with each other, yet they are entangled. This entanglement is a fundamental aspect of quantum mechanics and is not directly related to interference, as interference involves particles passing through each other, which is not the case here.

Measurement and Its Impact on Quantum Systems

Classical mechanics might make it seem like one could measure something with minimal disturbance. However, in quantum mechanics, any measurement is an interaction that affects the system being measured. To measure quantum entanglement, you must be affected differently by the states related to the entanglement.

For example, take two electrons with opposite spins. Their spin states are quantum indeterminate, meaning they cannot be known with certainty. Two states of their combined system are where one electron spins counterclockwise while the other spins clockwise. The entangled state is a superposition of these two states.

A complete measurement of the spin of one electron causes the system to lean towards one of those two possibilities. While the system is no longer in a superposition, it still slightly influences the observer and vice versa, introducing a degree of entanglement.

Quantum Tomography: A Non-Disturbing Measurement Technique

Quantum tomography is a method that allows for the measurement of a quantum system without significantly disturbing it. It involves repeating the experiment under the same conditions to infer details about the initial state of the system.

In the case of entangled electrons, quantum tomography requires performing 15 different possible measurements. The results are organized into a 4x4 matrix of complex numbers that reflect the entangled states of the electrons. By analyzing this matrix, it is possible to compute the degree of entanglement between the electrons.

The matrix provides a profile of the states of the system, and the method does not rely on the theory of spin being correct. However, the structure of a 4x4 matrix arises because we believe there are just two independent spin states for each electron, and 2 x 2 4.

Bell's Theorem: A Proof of Quantum Entanglement

Bell's theorem is a fundamental principle that distinguishes classical theories from quantum mechanics. According to Bell's theorem, without entanglement, the statistics of measurements on a pair of systems would have to satisfy certain inequalities.

Even if the spin states are more complicated than they appear, these inequalities still hold true, providing a way to confirm the presence of entanglement independent of the specific theory being used. This method offers a robust way to verify the production of entangled pairs without relying on the underlying assumptions of the spin theory.

In conclusion, while traditional measurement methods often disturb the system being measured, techniques like quantum tomography and Bell's theorem offer non-invasive ways to measure quantum entanglement. These methods provide a deep insight into the nature of entangled systems and help in validating the principles of quantum mechanics.