Quantum entanglement definition

Question

1. How can we define Quantum entanglement (in QFT)?

2. What are the known mathematical settings and special physical (or logical) conditions of QE applied to Quantum computing?

Answer 1

Quantum entanglement is the property of two objects $A,B$ – more precisely two subsystems – or a relationship between these two objects whose quantities or observables aren't independent of each other. It means that there exist some quantities $a_j$ and $b_k$ describing $A,B$, respectively, such that the probability distribution for these observations doesn't factorize, as expected for "independent propositions": $$ P(a_j=\lambda_c, b_k=\mu_d) \neq P(a_j=\lambda_c) \times P(b_k=\mu_d) $$ In other words, there exists at least one measurement that may be done on $A$ and one measurement done on $B$ such that the results of the two measurements are predicted to be correlated.

In quantum mechanics, such state (situation of the two objects) almost always results from the interaction of the systems $A,B$ in the past – when they were in contact or close enough to influence each other – and the mathematical description of the pure (maximally known) state of $A,B$ in quantum mechanics is in terms of superpositions: $$ |\psi\rangle = \sum_{m=1}^N c_m |\alpha_m\rangle \otimes |\beta_m\rangle $$ Whenever at least $N\geq 2$ terms on the right hand side are needed to express the state $|\psi\rangle$, we say that this state $|\psi\rangle$ is entangled. As I said, it's almost always the case when the two objects interacted in the past but weren't observed separately so far.

Quantum entanglement is nothing else than the correlation of the two objects $A,B$ in the "quantum regime" i.e. when the description in terms of state vectors is needed because the quantum coherence (information about the relative phases of the probability amplitudes) is preserved.

So quantum entanglement may be perhaps said to be a particular feature of the "organization of information", although the definition of the entanglement is in no way given by the words "organization of information". While "organization of information" is at least slightly correct, the phrase "random data exchange" isn't appropriate for the quantum entanglement in any way.

The entanglement is a correlation that resulted from some interactions in the past and doesn't imply any exchange in the present. The correlations between the two measurements are consequences of the entanglement which is a consequence of the contact in the past; the correlations are not a consequence of any information exchange at the present.

Answer 2

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently — instead, a quantum state must be described for the system as a whole.

Measurements of physical properties such as position, momentum, spin, polarization, etc., performed on entangled particles are found to be appropriately correlated. For example, if a pair of particles are generated in such a way that their total spin is known to be zero, and one particle is found to have clockwise spin on a certain axis, then the spin of the other particle, measured on the same axis, will be found to be counterclockwise, as to be expected due to their entanglement. However, this behavior gives rise to paradoxical effects: any measurement of a property of a particle can be seen as acting on that particle (e.g., by collapsing a number of superposed states) and will change the original quantum property by some unknown amount; and in the case of entangled particles, such a measurement will be on the entangled system as a whole. It thus appears that one particle of an entangled pair "knows" what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by arbitrarily large distances.