Quantum Key Distribution

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Quantum Key Distribution December 2009 THE START Necessary & Sufficient Condition for 4-D Entanglement Pace University rfrank@pace.edu 6/5/2019 Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019 (C) Ronald I. Frank 2004 - 2009 (V.18.)

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 2004 - 2009 (V.18.)

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 2004 - 2009 (V.18.)

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 2004 - 2009 (V.18.)

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 2004 - 2009 (V.18.)

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 2004 - 2009 (V.18.)

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

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

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

5.2 Examples Bell states for reference: 6/5/2019 Necessary & Sufficient Condition for 4-D Entanglement © Ronald I Frank 2019

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.