1. Quantum Key Distribution
December 2009
THE START
Necessary & Sufficient Condition for 4-D Entanglement
Pace University
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

2. Quantum Key Distribution
1. Table of Contents [1/1]
December 2009
TOC & Header [1-2] 2
Background Notation [3-5] 3
A typical proof of entanglement
by contradiction [6] 1
Proof of the NASC [7-8] 2
Examples: [9-11] 3
Computational Basis States are not entangled [9]
The Bell States for reference [10]
The Bell States are Each Entangled [11]
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

3. Quantum Key Distribution
December 2009
2.1 Background Notation
The 0 and 1 ket in Dirac Notation [and in matrix form] stand for two independent states of a quantum system. They are orthonormal. A general state of the system is a linear superposition of the basis states [called the computational basis]. This is a qubit.
The tensor product of 2 qubits forms a complex interaction state.
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

4. Quantum Key Distribution
December 2009
2.2 Background Notation
The other 3 possible interaction states are:
These form the basis of the 4-D states of 2-qubit interactions.
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

5. Quantum Key Distribution
December 2009
2.3 Background Notation
More generally:
“Entanglement” means the 4-D state can not be derived as a tensor product of 2, 2-D states (2 qubits).
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

6. 3. Typical Proof of Entanglement
Quantum Key Distribution
December 2009
3. Typical Proof of Entanglement
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
(C) Ronald I. Frank (V.18.)

7. 4.1 Proof of the NASC Notice in 2.3 (pg. 5) that AD=BC = abgd.
This provides the basis for the NASEC.
If we have a tensor product of 2, 2-D qubits,
AD=BC.
Reminder about the contrapositive of an implication ,
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019

8. Necessary & Sufficient Entanglement Condition.
4.2 Proof of the NASC
This is the
Necessary & Sufficient Entanglement Condition.
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019

9. 5.1 Examples We apply the NASEC to the 4-D basis vectors.
This shows that the 2-D computational basis is not entangled.
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019

10. 5.2 Examples Bell states for reference:
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Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019

11. 5.3 Examples Bell states are each entangled:
6/5/2019
Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019
This shows that the 4 Bell states are each entangled.