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

11 12 11 12

11 1221 22 21 2211 12 11 12

21 22 21 22 11 12 11 1221 22

21 22 21 22

b b b ba a

b b b ba a b b

a a b b b b b ba a

b b b b

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

1/ 2 0 / 2 0 1/ 2

0 0 0 0 01[ 0 0]

/ 2 0 1/ 2 0 2 2 / 20 0 0 0 0

i

i

i i

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

1 0 0 0

0 0 0 0 1 0 1 0

0 0 0 0 0 0 0 0

0 0 0 0

Example of a 2-qubit entangled state

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

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

0

1

c

c

0 10

0 11

ac bcca b

fc dccf d

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 0 -i/2 0

0 0 0 0

i/2 0 1/2 0

0 0 0 0

1/2 -i/2

i/2 1/2

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 1/2

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 , .

T0 0 0 0 0 0i ie e

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

T1 0 0 1

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)

T0 0i ie e

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.

T0 0i ie e

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

2-Qubitsdim(Kρ) Non-Product Basis for Kρ

0 x x1 ψ = (1, 1, 1, 0) {(-iσ3 - 2iσ1, iσ3 + 2iσ1)}

2 x x3 ψ = (1, 0, 0, 1) {(-iσ1, iσ1), (iσ2, iσ2), (-iσ3, iσ3)}

dim(Kρ) Product Basis for Kρ

0 x x

1 x x2 ψ = (1, 0, 0, 0) {(iσ3, 0), (-iσ3, iσ3)}

3 x x

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)T

0 0i ie e

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)