I n QE: Quantum Computation, Quantum Information, and Irreducible n -Qubit Entanglement

1. InQE:Quantum Computation, Quantum Information, andIrreducible n-Qubit Entanglement Daniel A. Pitonyak Lebanon Valley College

2. Quantum Computation & Quantum Information • Quantum particles are analogous to traditional computer bits • Quantum bit space differs from classical bit space

3. Classicalvs.Quantum 1-bit space1-qubit space {0, 1}{c0e0 + c1e1} 3-bit space3-qubit space {000, 001, . . . , 110, 111}{c000e000 +  + c111e111 } Note: The c’s are complex numbers and the e’s are basis vectors

4. Quantum computations have the potential to occur exponentially faster than traditional computations

5. Fundamental Concepts • An n-qubit system is a system of n qubits • An n-qubit density matrix is a positive semi-definite Hermitian matrix with trace = 1 and is represented by ρ

6. The Kronecker product • 2  2 Example

7. If ρ =  † for some n 1 matrix , then ρ is considered pure • Otherwise, ρ is considered mixed • A density matrix ρ is pure if and only if tr(ρ2) = 1

8. Example of a 2-qubit pure density matrix

9. If ρ can be written as the Kronecker product of a k-qubit density matrix and an (n – k)-qubit density matrix, then ρ is a product state • Otherwise, ρ is a non-product state and is said to be entangled

10. Example of a 2-qubit product state

11. Example of a 2-qubit entangled state

12. Two states have the same type of entanglement if we can transform one state into another state by only operating on the former state’s individual qubits • Such states are said to be LU equivalent

13. Given a 1-qubit state c0e0 + c1e1 = , a 2  2 unitary matrix operates by ordinary matrix multiplication

14. Given an n-qubit state, a Kronecker product of 2  2 unitary matrices operates on the state as a whole • Each individual 2  2 unitary matrix in the Kronecker product acts on a certain qubit

15. Key Questions: • To what degree is a specific state entangled? • How do we determine which states are the most entangled?

16. Irreducible n-Qubit Entanglement (InQE) • We can “trace over” a subsystem of qubits and consider the state composed only of those qubits not in that subsystem • Called a partial trace

17. 2-qubit example of the partial trace

18. The matrix ρ(2) = tr2(ρ) = τ = is called a reduced density matrix • In general, the matrix ρ (k) denotes the (n – 1)-qubit reduced density matrix found by tracing over the kth qubit of ρ

19. If we are given all of an n-qubit density matrix’s (n – 1)-qubitreduced density matrices, can we “reconstruct” the original n-qubitdensity matrix?

20. If another n-qubit state has all the same reduced density matrices as the n-qubit state just considered, then the answer is NO • We say such an n-qubit state has InQE

21. An n-qubit state, with associated density matrix ρ, has InQE if there exists another n-qubit state, with associateddensity matrix τ ≠ ρ, such that τ(k) = ρ(k) for all k

22. An n-qubit state, with associated density matrix ρ, has LU InQE if there exists another n-qubit state, with associated density matrix τ ≠ ρ, such that τ is LU-equivalent to ρ and τ(k) = ρ(k) for all k

23. Which states have InQE? • All 2-qubit states, except those that are completely unentangled, have InQE • Most mixed states have InQE • Most pure states do not have InQE

24. What higher numbered qubit pure states have InQE ? • A 3-qubit pure state τ has InQE if and only if τ is LU-equivalent to the pure state ρ = †, where  = for some real numbers, .

25. A Result on n-Cat &InQE • n-cat is the following n 1 matrix:

26. FACT. Let τ =  † be an n-qubit pure state density matrix. Let  be the density matrix for n-cat, where n ≥ 3. Then τ(k) = (k) for all k if and only if  = for some real numbers , θ. (Note:  is an n 1 matrix)

27. PROOF.The proof of this fact follows directly from the complete solution to a matrix equation that represents the n ∙ 2n – 1 equations in 2n variables that simultaneously must be true in order for a density matrix to have all the same reduced density matrices as n-cat.

28. Main Research Goal • BIG QUESTION: For n 3, which pure states have InQE? • BIG CONJECTURE: For n 3, an n-qubit pure state τ has InQE if and only if τ is LU-equivalent to the pure state ρ =  †, where  = , for some real numbers , θ.

29. Research Approach • Let Y be the Kronecker product of n 2  2skew Hermitian matrices with trace = 0 • We say YKρ ,where ρ is an n-qubit density matrix, if [Y, ρ] = Yρ – ρY = 0 • The structure of Kρ is closely connected with the idea of InQE

30. 2-qubit example of Kρ

31. 2-Qubits dim(Kρ) Non-Product Basis for Kρ 0 x x 1 ψ = (1, 1, 1, 0) {(-iσ3 - 2iσ1, iσ3 + 2iσ1)} 2 x x 3 ψ = (1, 0, 0, 1) {(-iσ1, iσ1), (iσ2, iσ2), (-iσ3, iσ3)} dim(Kρ) Product Basis for Kρ 0 x x 1 x x 2 ψ = (1, 0, 0, 0) {(iσ3, 0), (-iσ3, iσ3)} 3 x x

32. Current Research Direction • Meaningful relationships have been established between Kand LUInQE • We believe the following to be true:  is a pure state that has LU InQE  is LU equivalent to generalized n-cat (Note: generalized n-cat = for some real numbers , θ)

33. Conclusion • If our conjecture is true then we would know generalized n-cat and its LU-equivalents are the only states that have LU InQE • Strong indication thatInQEand LUInQE are one in the same • Question would still remain as to whether or not other states have InQE (This research has been supported by NSF Grant #PHY-0555506)