1. Daniel A. Pitonyak Lebanon Valley College
InQE: Quantum Computation, Quantum Information, and Irreducible 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. Classical vs. Quantum 1-bit space 1-qubit space
{0, 1} {c0e0 + c1e1}
3-bit space 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
1/2
-i/2
i/2
1/2
-i/2
i/2

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
1/2
-i/2
i/2

19. If we are given all of an n-qubit density matrix’s (n – 1)-qubit reduced density matrices, can we “reconstruct” the original n-qubit density 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 associated density 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
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  2 skew 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. are the standard Pauli matrices
The matrices
σ0 = , σ1 = , σ2 = , and σ3 =
are the standard Pauli matrices

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

33. Current Research Direction
Meaningful relationships have been established between K and LU InQE
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)

34. 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 that InQE and LU InQE 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 )