To mathematically prove quantum entanglement, we demonstrate that a given quantum state cannot be expressed as a tensor product of individual subsystem states. We'll use the Bell state \(|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)\) as an example, showing it violates the separability condition. We include two methods: (1) direct decomposition and (2) reduced density matrix analysis.
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Method 1: Proof by Contradiction (Direct Decomposition)
Assume \(|\Phi^+\rangle\) is separable, meaning it can be written as a tensor product:
\[
|\Phi^+\rangle = (a|0\rangle + b|1\rangle) \otimes (c|0\rangle + d|1\rangle),
\]
where \(a, b, c, d \in \mathbb{C}\) and normalization requires \(|a|^2 + |b|^2 = 1\), \(|c|^2 + |d|^2 = 1\).
Expanding the tensor product:
\[
(a|0\rangle + b|1\rangle) \otimes (c|0\rangle + d|1\rangle) = ac|00\rangle + ad|01\rangle + bc|10\rangle + bd|11\rangle.
\]
Equate this to \(|\Phi^+\rangle\):
\[
ac|00\rangle + ad|01\rangle + bc|10\rangle + bd|11\rangle = \frac{1}{\sqrt{2}}|00\rangle + 0|01\rangle + 0|10\rangle + \frac{1}{\sqrt{2}}|11\rangle.
\]
This yields the system:
1. \(ac = \frac{1}{\sqrt{2}}\),
2. \(ad = 0\),
3. \(bc = 0\),
4. \(bd = \frac{1}{\sqrt{2}}\).
Solving the system:
- From (2): \(ad = 0\) \(\implies\) \(a = 0\) or \(d = 0\).
- From (3): \(bc = 0\) \(\implies\) \(b = 0\) or \(c = 0\).
Case 1: \(a = 0\)
- From (1): \(0 \cdot c = 0 = \frac{1}{\sqrt{2}}\) → Contradiction.
Case 2:\(d = 0\)
- From (4): \(b \cdot 0 = 0 = \frac{1}{\sqrt{2}}\) → Contradiction.
Conclusion:The system has no solution. Thus, \(|\Phi^+\rangle\) cannot be written as a tensor product → entangled.
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Method 2: Reduced Density Matrix Analysis
For a separable state, the reduced density matrix of a subsystem is pure. If mixed, the state is entangled.
Step 1: Full density matrix \(\rho\)
\[
\rho = |\Phi^+\rangle \langle \Phi^+| = \frac{1}{2} \left( |00\rangle\langle 00| + |00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle\langle 11| \right).
\]
Step 2: Compute reduced density matrix for subsystem A (first qubit)
Trace out subsystem B:
\[
\rho_A = \text{Tr}_B(\rho) = \sum_{k=0,1} \langle k_B | \rho | k_B \rangle.
\]
- Term for \(k=0\):
\[
\langle 0_B | \rho | 0_B \rangle = \frac{1}{2} \langle 0_B| \left( |00\rangle\langle 00| + |00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle\langle 11| \right) |0_B\rangle.
\]
Using \(\langle 0_B|00\rangle = |0_A\rangle\), \(\langle 0_B|11\rangle = 0\):
\[
= \frac{1}{2} \left( |0_A\rangle\langle 0_A| + 0 + 0 + 0 \right) = \frac{1}{2} |0_A\rangle\langle 0_A|.
\]
- Term for \(k=1\):
\[
\langle 1_B | \rho | 1_B \rangle = \frac{1}{2} \langle 1_B| \left( \cdots \right) |1_B\rangle.
\]
Using \(\langle 1_B|00\rangle = 0\), \(\langle 1_B|11\rangle = |1_A\rangle\):
\[
= \frac{1}{2} \left( 0 + 0 + 0 + |1_A\rangle\langle 1_A| \right) = \frac{1}{2} |1_A\rangle\langle 1_A|.
\]
Step 3: Combine terms
\[
\rho_A = \frac{1}{2} |0_A\rangle\langle 0_A| + \frac{1}{2} |1_A\rangle\langle 1_A| = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.
\]
This is the maximally mixed state with eigenvalues \(\frac{1}{2}, \frac{1}{2}\).
Conclusion:\(\rho_A\) is mixed → entangled.
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Key Points
- Entanglement criterion: A state is entangled if it is not separable (i.e., cannot be written as \(\bigotimes_i |\psi_i\rangle\)).
- Bell state violation:The Bell state \(|\Phi^+\rangle\) violates separability, proving entanglement.
- Generalization: For any bipartite pure state, entanglement occurs iff the Schmidt rank > 1. Here, the Schmidt decomposition is \(|\Phi^+\rangle = \frac{1}{\sqrt{2}}|0_A0_B\rangle + \frac{1}{\sqrt{2}}|1_A1_B\rangle\) (Schmidt rank 2 → entangled).
This proof confirms quantum entanglement through algebraic contradiction and mixed-state subsystems.
Explanation
Here's a step-by-step explanation of quantum entanglement using the Bell state example, translated into everyday language without formulas:
The Core Idea of Entanglement
Imagine two coins that are magically linked. When you flip them:
- They always land showing the same side (both heads or both tails)
- But until you look, they exist in a ghostly "both possibilities at once" state
- The moment you see one coin, the other instantly "chooses" its state
This mysterious connection that defies normal physics is entanglement.
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Method 1 Explained: The Impossible Puzzle (Direct Decomposition)
The Setup:
1. We have our quantum "coins" (qubits) in state:
"50% chance both heads | 50% chance both tails"
2. We try to describe them as independent objects(like normal coins):
- Coin A = (x% heads + y% tails)
- Coin B = (p% heads + q% tails)
The Contradiction:
- For our entangled state:
Both heads must have 50% probability
Both tails must have 50% probability
❌ Mismatched results (A-heads+B-tails or A-tails+B-heads) must have 0% probability
- But if they're independent:
- Probability of both heads = (A-heads%) × (B-heads%)
- Probability of both tails = (A-tails%) × (B-tails%)
The Impossible Math.
- To get 50% for both heads:
(A-heads%) × (B-heads%) = 50%
- To get 0% for mismatches:
Either A never shows heads OR B never shows tails...
...but then both tails would be (A-tails%) × (B-tails%) = ?
The Conclusion:
➡️ No combination works!
➡️ The coins can't be independent - their fates are mathematically linked.
➡️ This proves entanglement isn't just hidden coordination - it's fundamental connection.
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🪙 Method 2 Explained: The Phantom Coin (Reduced Density Matrix)
The Experiment:
1. We entangle two coins and mail one to Paris, one to Tokyo.
2. In Paris, scientists examine only their local coin.
What Paris Sees:
- Their coin appears completely random:
- 50% chance heads 🪙
- 50% chance tails 🪙
- Like flipping a normal coin*
The Quantum Twist:
- If the coins were truly independent:
- Paris's randomness would be "real"
- Tokyo's coin would be unrelated
- But in entanglement:
- The moment Paris looks...
- Their coin "collapses" to heads/tails
- Tokyo's coin instantly collapses to match!
Why This Proves Entanglement:
- Paris sees maximum randomness (50/50)
- Yet this randomness disappears when comparing results with Tokyo
- The randomness was actually shared quantum information - not true independence
The Conclusion:
➡️ Individual coins show perfect randomness
➡️ But together they show perfect correlation
➡️ This proves they share a single quantum state across distance
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💡 Key Intuitive Takeaways
1. The whole > sum of parts:
Entangled particles are like a single "quantum object" split across space - you can't describe one without the other.
2. Spooky action at distance:
Changing one particle instantly affects its partner, no matter how far apart (verified by experiments).
3. Not hidden variables:
Our math proves this isn't just pre-agreed coordination (like identical twins) - it's deeper quantum connection.
4. Usefulness:
This "quantum link" enables:
- Ultra-secure communication (quantum cryptography)
- Computers solving impossible problems (quantum computing)
- Teleporting quantum information
> Entanglement isn't weird - it's quantum reality. We're the weird ones for expecting particles to behave like billiard balls."
> - Adapted from Niels Bohr
