AcademyAtomic Quantum Structure
Academy
Quantum Entanglement
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
Entanglement occurs when a combined quantum state cannot be described as independent states for its parts. Measurements on one part are correlated with measurements on the other, even when the parts are far apart.
Entanglement gives stronger correlations than ordinary ignorance about pre-existing values, but it does not allow faster-than-light signalling.
Notation
\(|0\rangle,|1\rangle\)
two-state basis vectors
1
\(|\uparrow\rangle,|\downarrow\rangle\)
spin basis states
1
\(|\Psi\rangle\)
combined quantum state
1
\(P\)
measurement probability
1
\(A,B\)
two subsystems
1
Method
Derivation 1: Compare product and entangled states
A product state factors into a state for subsystem A times a state for subsystem B. An entangled state does not.
Product state
\[|\Psi\rangle=|\psi_A\rangle|\psi_B\rangle\]
Bell-like entangled state
\[|\Psi\rangle=\frac{1}{\sqrt2}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)\]
Derivation 2: Extract probabilities
The squared magnitude of each amplitude gives the probability of that joint outcome.
Equal amplitudes
\[\left|\frac{1}{\sqrt2}\right|^2=\frac12\]
Allowed outcomes
\[P(01)=\frac12,\quad P(10)=\frac12\]
Derivation 3: Interpret correlations
Measuring one subsystem updates the state used to predict the other subsystem. It does not send a controllable message.
Marginal randomness
\[P(A=0)=P(A=1)=\frac12\]
Perfect anticorrelation
\[A=0\Rightarrow B=1\]
Rules
Product state
\[|\Psi\rangle=|\psi_A\rangle|\psi_B\rangle\]
Bell-like state
\[|\Psi\rangle=\frac{1}{\sqrt2}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)\]
Born rule
\[P=|\mathrm{amplitude}|^2\]
Examples
Question
For
\[|\Psi\rangle=(|01\rangle+|10\rangle)/\sqrt2\]
what are the possible joint outcomes?Answer
The possible outcomes are
\[01\]
and \[10\]
each with probability \[1/2\]
Checks
- Entanglement is a property of the combined state.
- Each local result can be random even when joint results are strongly correlated.
- Entanglement cannot be used to send controllable faster-than-light messages.
- Bell tests distinguish entanglement from simple local hidden-variable models.