Sequential generation of polynomial invariants and N-body non-local correlations

Abstract

We report an inductive process that allows for sequential construction of local unitary invariant polynomials of state coefficients for multipartite quantum states. The starting point can be a physically meaningful invariant of a smaller part of the system. The process is applied to construct a chain of invariants that quantify non-local N −way correlations in an N −qubit pure state. It also yields the invariants to quantify the sum of N −way and (N − 1)-way correlations. Analytic expressions for four-way and three-way correlation quantifiers for four qubit states, as well as, fiveway and four-way correlation quantifiers for five qubit pure states are given.

Upper bound on three-tangles of reduced states of four-qubit pure states

“…Using the notation from ref. [20], we define D 00 (A3) i 3 = a 00i3 a 11i3 − a 10i3 a 01i3 , (i 3 = 0, 1) as determinant of a two-way negativity font and D 00i3 = a 00i3 a 11i3+1 − a 10i3 a 01i3+1 , (i 3 = 0, 1) is determinant of a three-way negativity font. Three tangle of pure state |Ψ 3 as defined in ref.…”

“…We shall also use the three and four qubit invariants constructed in section V of ref. [20]. Three-qubit invariants of interest for : m = 0 to 4 are invariant with respect to local unitaries on qubits A 1 , A 2 , and A 3 .…”

“…Three-qubit invariants of interest for : m = 0 to 4 are invariant with respect to local unitaries on qubits A 1 , A 2 , and A 3 . Four-qubit invariant that quantifies the sum of three-way and four-way correlations [20] of triple A 1 A 2 A 3 , reads as…”

“…Being independent of the choice of focus qubit, I 4,8 can be written in alternate forms. Four-tangle based on degree eight invariant is defined [16,20] as : m = 0 to 4 for qubits A 1 A 3 A 4 can be constructed from two-qubit invariants of properly selected pair of qubits [20]. In the following sections, the subscript 4 in (I 3,4 )…”

“…[20] that one can construct local unitary invariants characterizing a pure state of N two-level quantum systems (qubits) in terms of N-1 qubit invariants. A natural question is, can we write entanglement of (N-1)-qubit marginals of an N-qubit pure state in terms of invariants of the N-qubit pure state?…”