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

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

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

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

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

The Kronecker product 2  2 Example

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

Example of a 2-qubit pure density matrix

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

Example of a 2-qubit product state

Example of a 2-qubit entangled state

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

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

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

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

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

2-qubit example of the partial trace 1/2 -i/2 i/2 1/2 -i/2 i/2

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

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?

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

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

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

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

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 , .

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

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)

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.

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.

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

2-qubit example of Kρ

are the standard Pauli matrices The matrices σ0 = , σ1 = , σ2 = , and σ3 = are the standard Pauli matrices

{(-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)}

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)

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-0555506)