1. Introduction to Entanglement Allan Solomon, Paris VI

2. Mathematics Prelude Paradigm: Quantum mechanics & Maths Paradigm: Quantum mechanics & Maths Quantum Field Theory For example, Feynman Diagrams involve Analysis Riemann Zeta fns and extensions Algebra Hopf Algebra Braid Groups Combinatorics Graph Theory CountingTopology Knot Theory

3. Mathematics Prelude Entanglement Entanglement (Fr.) Intrication (Eng.) Intricate = Complexity involves Analysis, Algebra, Topology,Combinatorics (Fr.) Intrication (Eng.) Intricate = Complexity involves Analysis, Algebra, Topology,Combinatorics Borromean Rings – example of an entangled system

4. Physics Prelude: EPR paradox Source emits spin singlet Measurement by Alice on |  > determines Bob’s measurement. The electrons are entangled.

5. “Entanglement is the characteristic trait of quantum mechanics” Erwin Schroedinger, 1935. Erwin Schroedinger, 1935.

6. “ Interference (in Classical and Quantum Mechanics) is just the fact that the sum of squares is not the square of the sum. ” Richard Feynman Richard Feynman

7. 1 Vectors, Vector Spaces A basic operation for vectors is addition. For mathematicians therefore, vector addition presents no surprises. For physicists, vector addition is such a remarkable property that in quantum mechanics the phenomena it gives rise to it go by many names, superposition rule, interference, entanglement,… A basic operation for vectors is addition. For mathematicians therefore, vector addition presents no surprises. For physicists, vector addition is such a remarkable property that in quantum mechanics the phenomena it gives rise to it go by many names, superposition rule, interference, entanglement,…

8. Vector notation |0> <1| Maths Notation Physics Notation Dirac

9. 2 Bipartite Spaces Maths Notation Physics Notation If V 1 has basis {e i } and V 2 has basis {f j } then {e i  f j, i=1..m,j=1..n} is a basis for V 1  V 2 Every vector in V 1  V 2 is a sum of products; but not every vector is a product. If it is a product, then it is said to be non-entangled.

10. Bipartite Spaces: Entanglement Example: Example: Not entangled Entangled “entangled” means not factorizable

11. 3 States, Pure and Mixed (a) Pure States Vectors correspond to Pure states: example We may equally represent a Pure State  by the Operator (Projector)  which projects onto that state: NOTE: trace P=1 (Normalization) and P is Hermitian with (semi-)positive eigenvalues (0 and 1 ).

12. States (b) Mixed States We define a (mixed) state  as a positive matrix of trace 1 Note: A mixed state is a mixture of pure states. which is a (convex) sum of pure states.

13. Mixed state is not a unique sum of pure states Example: Example:

14. 4 Entropy of a State (Von Neumann Entropy) State  has eigenvalues i usually log 2 Example(a): Pure state Example(b): Mixed state Every Pure State has Entropy Zero. Maximum entropy 1 for maximally random state. (E is entropy here!) Claude Shannon John Von Neumann

15. 5 Measures of Entanglement Intuitively we expect (1) (Pure) state (1/  2)(|0,0> + |0,1>) No0 (2) Bell state (1/  2)(|0,0> + |1,1>) Yes1 (3)  |0,0> +  (1- ) |1,1>) Yes 0 .?.  1 0 .?.  1 Entangled? E Measure It turns out that the (VN) Entropy gives a measure of entanglement for pure states; but not directly, as all pure states have entropy zero. We must first take the Partial Trace over one subsystem of the bipartite system.

16. 6 Partial Trace If V = V A  V B then tr B (Q A  Q B )=Q A tr(Q B ) Extend to sums by linearity. Pure States  Q A =|u1> <v2| then tr B (Q A  Q B )= |u1> then tr B (Q A  Q B )= |u1> Example (non-entangled state): Entanglement (Entropy of partially traced state) is 0.

17. Example: Bell state |b>=(1/   b =(1/  Tr B  b  =(1/    John Stewart Bell

18. Example (Entangled state): Example (Entangled state): Entropy of partially traced state is non-zero (=.55)

19. Entangled state (intermediate) Pure state cos(  )|0,0>+sin(  )|1,1> =  So this measure of entanglement gives an intuitively correct variation from 0 (non-entangled) to maximum of 1 (Bell state) for PURE States.

20. Recap: Definitions A pure state may be represented by a vector or a positive matrix (Projection matrix) with eigenvalues 1, 0, … 0, 0, 0 A pure state may be represented by a vector or a positive matrix (Projection matrix) with eigenvalues 1, 0, … 0, 0, 0 A mixed state is a (convex) sum of pure states and may be represented by a positive matrix of trace 1. A mixed state is a (convex) sum of pure states and may be represented by a positive matrix of trace 1. A pure state is separable (non-entangled) if it can be written as a product of vectors (factorizable). A pure state is separable (non-entangled) if it can be written as a product of vectors (factorizable). A mixed state is separable if it can be written as a (convex) sum of separable (factorizable) pure states. A mixed state is separable if it can be written as a (convex) sum of separable (factorizable) pure states. The expression of a mixed state as a convex sum of pure states is not unique. The expression of a mixed state as a convex sum of pure states is not unique.

21. Measure of Entanglement The entanglement   of a pure bipartite state  V A  V B is given by the Entropy of the Partial Trace of  The entanglement   of a pure bipartite state  V A  V B is given by the Entropy of the Partial Trace of  The entanglement   of a mixed bipartite state  V A  V B is given by   (  )=min{             The entanglement   of a mixed bipartite state  V A  V B is given by   (  )=min{            

22. This definition of entanglement measure for Mixed States is very difficult to apply, requiring infinite tests. This definition of entanglement measure for Mixed States is very difficult to apply, requiring infinite tests. Example (revisited) Example (revisited) E(E( (this IS the min and therefore the entanglement )

23. 7 Concurrence (Physics viewpoint) Spin Flip Operation Spin Flip Operation qubit 2-qubit Concurrence C = Factorizable state Bell state C=0 C=1 So Concurrence gives a measure of entanglement for PURE states

24. Concurrence (Maths viewpoint) Partial trace of Pure state  gives 2X2 matrix so Entanglement determined by eigenvalue equation    ±  Concurrence C 2 =4 C varies from 0 to 1 so Concurrence gives a measure of entanglement for PURE states Equivalently (for pure states) C 2 =tr Recall

25. Wootters’ Concurrence Wootters[1,2] has shown that the form for C Wootters[1,2] has shown that the form for C where the ’s are the square roots of the eigenvalues of in descending order Gives the entanglement for mixed states; i.e. it gives the correct minimum over Pure States. (Note the formula coincides with the previous for PURE states.) [1] Hill, S and Wootters, WK, PhysRevLett 78,26,5022(1997) [2] Wootters, WK, PhysRevLett 80,10,2245(1998)

26. Feynman Nobel Lecture The Development of the Space-Time View of Quantum Electrodynamics The Development of the Space-Time View of Quantum Electrodynamics “We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on.” “We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on.”

27. Tripartite entanglement “Naïve Solution” “Naïve Solution” Extend Concurrence to 3-subspace Pure states by summing over (3) Partial Traces Extend Concurrence to 3-subspace Pure states by summing over (3) Partial Traces Example Example Three Partial Traces are equal, each with Concurrence 2/3, leading to a 3-concurrence of (1/3) (2/3+2/3+2/3)=2/3

28. Tripartite states However ………. However ………. The entangled state (  /  |000>+|111>) has 3 equal partial traces which is separable (concurrence =0). Borromean Rings analogy – every cut leaves a separable system

29. A funny “Resource” Physicists do experiments on the principle that they can be replicated in other laboratories – invariance under transformations. Physicists do experiments on the principle that they can be replicated in other laboratories – invariance under transformations. In Quantum mechanics, we expect our measurable quantities to be invariant under Unitary Transformations (or anti- unitary – Wigner) In Quantum mechanics, we expect our measurable quantities to be invariant under Unitary Transformations (or anti- unitary – Wigner) This is NOT the case for Entanglement! This is NOT the case for Entanglement! Eugene Wigner

30. Open Problems What is the significance of Entanglement for Quantum Computing? What is the significance of Entanglement for Quantum Computing? Find a measure of Entanglement for 3 (or more) qubits (tripartite spaces,..). Find a measure of Entanglement for 3 (or more) qubits (tripartite spaces,..). Interaction with the Environment (Dissipation of Entanglement) Interaction with the Environment (Dissipation of Entanglement)