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InQE: Quantum Computation, Quantum Information, and Irreducible n-Qubit Entanglement
Daniel A. Pitonyak
Overview of the Importance of Quantum Computation and Quantum Information
For over a decade, attempts have been made to replace conventional computers with more advanced computers that have faster and more sophisticated computing capabilities. Quantum particles (such as electrons and photons) intrinsically contain information that is analogous to the information contained by traditional computer bits. For example, a traditional bit can either be 0 or 1. Likewise, an electron can either be spin-up or spin-down. As a result, the construction of a computer that uses quantum particles as its “bits” has immense potential. A quantum computer based on these quantum bits, or qubits, would revolutionize the world of technology. The main reason why such an impact would occur can be found in the principles of quantum mechanics themselves. These principles cause quantum bit space to differ from classical bit space. For example, 1-bit space consists of the set {0, 1} while 1-qubit space consists of the set {c0e0 + c1e1}, where c0 and c1
are complex numbers and e0 and e1 are basis vectors. Likewise, 3-bit space consists of the set {000, 001, . . ., 110, 111} while 3-qubit space consists of the set {c000e000 + × × × + c111e111 }
. The fact that an n-qubit lies in this higher dimensional space gives quantum computations 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 (non-negative eigenvalues) Hermitian matrix (the Hermitian transpose of the matrix equals the matrix itself) with trace (sum of the entries along the diagonal of the matrix) = 1 and is represented by ρ. The density matrix contains all the information about a given n-qubit and allows us to work with that n-qubit mathematically.
The Kronecker product of a (m ´ n) matrix and a (p ´ q) matrix is a (mp ´ nq) block matrix found by scaling the (p ´ q) matrix by each entry in the (m ´ n) matrix. The Kronecker product between two 2 ´ 2 matrices is given below:
If ρ = y† Äy for some n ´ 1 matrix y, then ρ is considered pure (y † is the Hermitian transpose of the matrix y); otherwise, ρ is considered mixed. A density matrix ρ is pure if and only if tr(ρ2) = 1. For example,
ρ = = Ä
is 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. For example, the density matrix
is a 2-qubit product state. On the other hand, the density matrix
is a 2-qubit entangled state. We say 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. This operation is done by 2 ´ 2 unitary matrices (orthonormal columns). Given a 1-qubit state c0e0 + c1e1 = , a 2 ´ 2 unitary matrix operates on that state by ordinary matrix multiplication. Given an n-qubit state, a Kronecker product of 2 ´ 2 unitary matrices operates on that state as a whole, with each individual 2 ´ 2 unitary matrix acting on a certain qubit.
In quantum computation and quantum information, the key questions are to what degree is a specific state entangled and how do we determine which states are the most entangled. For 2-qubits and 3-qubits, the answer is well understood. However, for higher numbered qubits, the answer is less certain.
Irreducible n-Qubit Entanglement (InQE)
The idea of irreducible n-qubit entanglement (InQE) plays an important role in answering the key questions of quantum entanglement for higher numbered qubit states. Given an n-qubit state with its associated density matrix, we can “trace over” a subsystem of qubits and consider the state composed only of those qubits not in that subsystem. This is called a partial trace. For example, the partial trace over the second qubit in the 2-qubit system given by
ρ = / 1/2 / 0 / -i/2 / 00 / 0 / 0 / 0
i/2 / 0 / 1/2 / 0
0 / 0 / 0 / 0
is the 1-qubit system given by
ρ(2) = tr2(ρ) = τ = .
(Note: The entries of τ (on the right) are calculated from the entries of ρ (on the left) marked with red.) The density matrix 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.
Essentially, when we trace over a subsystem of qubits, we are removing the “information” those qubits contribute to the n-qubit state and are simply analyzing the information that is contributed by the qubits that remain. The question becomes 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 cannot hope to reconstruct the density matrix for that aforementioned n-qubit. That is, too much critical information has been lost about the original n-qubit state by tracing over each qubit, and the original density matrix cannot be uniquely determined. We say such an n-qubit state has InQE. That is, 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 (where τ (k) and ρ (k) are the (n – 1)-qubit reduced state density matrices found by tracing over the kth qubit of τ and ρ, respectively). Another version of InQE is LU InQE, which narrows the possible candidates for τ. 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.
The natural question becomes which states have InQE. We know all 2-qubits, except those that are completely unentangled, have InQE. Also, most mixed states have InQE. Surprisingly, most pure states do not have InQE. For the purposes of quantum computation and quantum information, we would like to know what higher numbered qubit pure states have InQE. For 3-qubits, the question has already been answered. A 3-qubit pure state τ has InQE if and only if τ is LU-equivalent to the pure state ρ = y † Äy, where y = (f and q are real numbers; we have written the column vectory as the transpose of a row vector for convenience).
A Result On n-Cat and InQE
The following fact, for which I wrote the proof, not only indicates that the n ´ 1 matrix n-cat = has InQE but also tells us exactly which states have all the same reduced density matrices as n-cat.
FACT. Let τ = y† Äy be an n-qubit pure state density matrix. Let r be the density matrix for n-cat, where n ≥ 3. Then τ(k) = r(k) for all k if and only if y = for some f, θ Î, where y 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.
Let xj = |cj|2 where cj is the jth entry of y. Let Sn = . Consider the matrix equation
Mn = (1)[*]
where Mn is the following n∙2n – 1 × 2n matrix[**].
We know the complete solution xC to (1) is given by xC = xP + xN, where xP is a particular solution to (1) and xN is any arbitrary linear combination of the vectors that compose a basis for the nullspace of Mn. Since (1, 0, . . ., 0, 1) is clearly a particular solution to (1), xC = (1, 0, . . ., 0, 1) + xN. Enforcing the constraint that xj ≥ 0 for all j, we can show that xC = (1, 0, . . ., 0, 1). Thus, τ(k) = r(k) for all k if and only if y = for some f, θ Î.
BIG QUESTION/Main Research Goal: For n 3, which pure states have InQE?
ANSWER/BIG CONJECTURE
For n 3, an n-qubit pure state τ has InQE iff τ is LU-equivalent to the pure state ρ = y† Äy where y = (f and q are real numbers). Progress has been made towards verifying the correctness of this answer. The following outlines the approach that has been taken in the attempted proof.
Let Y be the Kronecker product of n 2 ´ 2 skew Hermitian matrices (Hermitian transpose equals the negative of the matrix) with trace = 0. We say Y Î Kρ , where ρ is an n-qubit density matrix, if [Y, ρ] = Yρ – ρY = 0. We call Kρ
the kernel of the map Y [Y, ρ]. The kernels of these maps provide a powerful tool for analyzing n-qubit states. The structure of Kρ is closely connected with the idea of InQE
I gathered many examples of kernels for several n-qubit density matrices. The possible dimension of the kernel is dependent on the number of qubits in the system and whether the given state is a product or non-product. For 2-qubit non-product states, only 1-dimensional and 3-dimensional kernels are possible. For 2-qubit product states, only 2-dimensional kernels are possible. The possible kernel dimensions for a given n-qubit state was a fact proven by Dr. David W. Lyons and Dr. Scott N. Walck of the Mathematical Physics Group at Lebanon Valley College. Here is a 2-qubit example of Kρ. The matrices
σ0 =, σ1 = , σ2 = , and σ3 =
are the standard Pauli matrices, which are fundamental in the study of quantum mechanics.
2-Qubitsdim(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
Results
Analysis of the kernels for various states has allowed meaningful relationships to be established between Kr and LU InQE. We believe the following to be true:
r is a pure state that has LU InQE Û r is LU-equivalent to generalized n-cat
We are tackling this equivalence indirectly via the kernel of r and already have several one-way implications.
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. There would also be strong indication that InQE and LU InQE are one in the same. However, the question would still remain as to whether or not other states have InQE.
This research has been supported by NSF Grant #PHY-0555506.
[*] Equation (1) gives a set of equations that involve only the entries along the main diagonal of τ. These equations must be true if τ(k) is to equal r(k) for all k.
[**] The matrix Mn can be divided into n sections, each of which contains rows whose entries are block matrices. The kth section is the result of tracing over the kth qubit of τ. Also, the dimensions of each individual block matrix (IB Dims) in each row are given.