THE START Necessary Sufficient Condition for 4 D

THE START Necessary & Sufficient Condition for 4 -D Entanglement Pace University rfrank@pace. edu 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 1

1. Table of Contents [1/1] 1. TOC & Header [1 -2] 2 2. Background Notation [3 -5] 3 3. A typical proof ofentanglement by contradiction [6] 1 4. Proof of the NASC [7 -8] 2 5. Examples: [9 -11] 3 Computational Basis States are not entangled [9] The Bell States for reference [10] The Bell States are Each Entangled [11] 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 2

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. 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 3

2. 2 Background Notation The other 3 possible interaction states are: These form the basis of the 4 -D states of 2 -qubit interactions. 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 4

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). 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 5

3. Typical Proof of Entanglement 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 6

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 10/19/2021 contrapositiveof an implication Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 7

4. 2 Proof of the NASC This is the Necessary & Sufficient Entanglement Condition. 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 8

5. 1 Examples We apply the NASEC to the 4 -D basis vectors. This shows that the 2 -D computational basis is not entangled. 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 9

5. 2 Examples Bell states for reference: 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 10

5. 3 Examples Bell states are each entangled: 10/19/2021 Necessary & Sufficient Condition for 4 -D Entanglement © Ronald I Frank 2019 11