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Local Unitary Invariants for N-qubit Pure State

We obtain local unitary invariant polynomials for N qubit quantum state from first principles. A basic unit of entanglement, referred to as negativity font, is defined as a two by two matrix of probability amplitudes that determines the negative eigen value of a four by four submatrix of partially transposed state operator. Transformation properties of determinants of negativity fonts under local unitary (LU) transformations are exploited to obtain multi qubit invariants written in terms of such determinants.

In 1935 Schrödinger [1] coined the term 'entanglement' to describe quantum correlations that make it possible to alter the properties of a distant system instantaneously by acting on a local system. A spin singlet is an example of entangled state of two spin half particles. A qubit is any two level quantum system with basis states represented by |i , i = 0 and 1. The spin singlet is an entangled state of two qubits. For a pure state of bipartite quantum system consisting of two distinguishable parts A and B, each of arbitrary dimension, negativity [2] of partially transposed state operator [3] is known to be an entanglement monotone [4]. How properties of one part of a multipartite quantum system are altered by local operations on other parts at distinct remote locations is a complex question. In this letter, we present a novel approach to construct meaningful LU invariants for multi-qubit systems from first principles that is by examining the effect of local unitaries on different parts of the composite system. Our method, illustrated for four qubit case in ref. [5] introduces basic units of entanglement, referred to as negativity fonts. A negativity font is defined as a two by two matrix of probability amplitudes that determines the negative eigen values of a specific four by four submatrix of a partially transposed state operator. It was shown earlier [6] that a partial transpose can be written as a sum of K−way (2 ≤ K ≤ N ) partial transposes. A K−way partial transpose contains information about K body corelations of multipartite system. Contibutions of partial transposes to global negativity, referred to as partial K−way negativities are not unitary invariants, but when calculated for canonical states for three qubits [7,8] and four qubit [9] coincide with entanglement monotones. This article complements our earlier work by outlining a direct method to obtain multiqubit invariants relevant to the construction of entanglement monotones without reaching the canonical state to calculate partial K−way negativities. Multi qubit unitary invariants are obtained by examining the transformation properties of negativity fonts present in global partial transpose [3] and K−way (2 ≤ K ≤ N ) partially transposed matrices [9] constructed from N −qubit state operator. The mathematical form of resulting multiqubit invariants for a given state reveals the entanglement microstructure of the state.
Multi qubit invariants, written in terms of determinants of negativity fonts, are essentially relations between intrinsic negative eigenvalues of selected 4 × 4 submatrices of K−way partially transposed matrices. In the case of four qubits, the standard approach from invariant theory has lead to the construction of a complete set of SL-invariants [10,11] and algorithm for constructing N-qubit invariants is given in ref. [12]. Results for five qubits have also been reported [11]. The N qubit invariants for even number of qubits have been reported earlier in [15] and for even and odd number of qubis in [16]. Focus is on geometric aspects of such invariants in Refs. [17][18][19]. Independent of these approaches, a method based on expectation values of antilinear operators with emphasis on permutation invariance of the global entanglement measure [13,14], has been suggested. The number of polynomial invariants is known to increase very fast with number of qubit. However, in general, a small number of invariants is needed to qualify and quantify the entanglement. The advantage of our approach is that it is easily applied to obtain relevant invariants for any state at hand not necessarily the general state or canonical state. Our results bring out the structural difference between LU invariants for N −odd and N −even qubits through the nature of K−way negativity fonts present in respective invariants. For multipartite case, one needs in-equivalent entanglement measures [15,20,21]. To show that the method can be used to construct entanglement monotones that detect the entanglement of specific parts of the composite system, four qubit invariants to detect entanglement of a pair of qubits due to four-way correlations are obtained.
The entanglement of qubits A 1 and A 2 in pure state ρ A1A2 = Ψ A1A2 Ψ A1A2 where is measured by the negativity of four by four matrix ρ A1A2 T1 G obtained by partially transposing the state of qubit A 1 in ρ A1A2 . We refer to two by two matrix ν 00 = a 00 a 01 a 10 a 11 as a negativity font of ρ A1A2 T1 G . The squared negativity of ρ A1A2 T1 G , is given by N A1 G 2 = 4 det ν 00 2 . If det ν 00 = 0, the state is separable. A general N-qubit pure state reads as where |i 1 i 2 ...i N are the basis vectors spanning 2 N dimensional Hilbert space, and A p is the location of qubit p.
If ρ is a pure state, then the negative eigenvalue of 4×4 sub-matrix of ρ Tp G in the space spanned by distinct basis vectors is defined as a K−way negativity font. The subscript K is used to group together the negativity fonts arising due to K−way coherences of the composite system that is the correlations responsible for GHZ state like entanglement of a K−partite system. For a given value of K, the negativity of K−way partial transpose ρ Tp K with respect to subsystem p, as defined in Ref. [9], arises solely from K−way negativity fonts. Determinants of negativity fonts are, in a sense, intrinsic negative eigenvalues of a global or a K−way partial transpose of state operator. Global partial transpose of an N −qubit state is a combination of K−way partially transposed operators (2 ≤ K ≤ N ) [9] and can be expanded as where ρ 1 is the trace norm of ρ, arises due to all possible negativity fonts present in ρ Tp G . Since K qubits may be chosen in ways the form of a K−way font must specify the set of K qubits it refers to. To distinguish between different K−way negativity fonts we shall replace subscript K in Eq. (4) by a list of qubit states for which δ im,jm = 1. In other words a K−way font and its determinant is represented by Here i m + 1 = 0 for i m = 1 and i m + 1 = 1 for i m = 0. In this notation no subscript is needed for an N −way negativity font that is ν is an invariant of local unitary U Ap acting on qubit A p . After applying unitary transformation Using primed symbols for determinants of negativity fonts calculated from coefficients b i1i2...iN , we can write four transformation equations relating N −way and (N − 1) −way negativity fonts. Eliminating variable x, invariants of U Ap U Aq are found to be Relevant multiqubit invariants for a given value of N can be written down from these general results. Invariants of U Ap U Ar for K−way fonts (2 ≤ K ≤ N ) with qubits p and r in the superscript and N − K subscripts are analogous to those for N −way fonts.

II. N-EVEN N-WAY INVARIANT
Invariant of U A1 U A2 U A3 is obtained by taking a combination of N −way invariants of U A1 U A2 such that Eq. (13) is satisfied for the third qubit, for example and the N −way invariant for N qubits reads as Noting that D 00i3...iN = −D 01i3+1...iN +1 , we have giving I N −odd = 0, while for N-even The invariant for N −even has permutation symmetry, as such may be used to define N −tangle as Degree four invariants for N qubits are obtained by starting with N − 2 qubit N way invariants and using Eq. (14) to obtain an N qubit invariant. Four qubit 4−way invariant with negativity fonts lying solely in 4−way partial transpose is written from Eq. (20) as I 4 = D 0000 + D 0011 − D 0010 − D 0001 . We identify I 4 with invariant H of degree two as given in ref. [10]. A four qubit state with four qubit entanglement arising due to quantum correlations of the type present in a four qubit GHZ state, is distinguished from other entangled states by a non zero I 4 . This entanglement is lost without leaving any residue, on the loss of a single qubit. The entanglement monotone based on I 4 is called four-tangle in analogy with three tangle [20]. Four tangle τ 4 vanishes on W −like state of four qubits, however, fails to vanish on product of two qubit entangled states.
We now apply the method to construct entanglement monotones that detect the entanglement of specific parts of the composite system, an entangled qubit pair in this case. To obtain degree four invariants that detect products of two qubit states, consider the combination of 4−way fonts J = D 0000 − D 0100 + D 0010 − D 0110 , which is an invariant of U A1 U A2 . Using (Eq. (14)), applied to four-way and three way fonts the four qubit invariant is found to be Similarly, invariant obtained by starting with four-way U A1 U A3 invariant form is and starting with U A1 U A4 invariant we get with corresponding entanglement monotones defined as β A1Ai = 4 3 J A1Ai , i = 2 − 4. By construction J A1Ai detects entanglement between qubits A 1 A i , provided the pair A 1 A i is entangled to its complement in four qubit state. For qubit A 1 the invariants J A1A2 , J A1A3 , and J A1A4 satisfy the relation (I 4 ) 2 = 1 3 J A1A2 + J A1A3 + J A1A4 . An interesting four qubit state reported in Ref. [22] is which is known to have maximal entanglement of the pair A 1 A 2 with pair of qubits A 2 A 4 . The state can be rewritten as an entangled state of A 1 A 4 and A 2 A 3 Bell pairs however, is not reducible to a pair of Bell states. We verify that for this state I 4 = 0, J A1A2 = J A1A3 = J A2A4 = J A3A4 = − 1 4 , and J A1A4 = J A2A3 = 1 2 . Therefore the state is characterised by τ 4 = 0, β A1A2 = β A1A3 = β A2A4 = β A3A4 = 1 3 , while β A1A4 = β A2A3 = 2 3 , indicating that the entanglement of state |χ is distinct from that of GHZ state of four qubits having τ 4 = 1, β A1A2 = β A1A3 = β A1A4 = 1 3 as well as β A2A3 = β A2A4 = β A3A4 = 1 3 . The degree four invariants for four qubits denoted as L, M , and N in Ref. [11] are combinations of J A1A2 , J A1A3 , J A1A4 and (I 4 ) 2 . Additional invariants are easily constructed to detect all possible types of Four qubit entanglement. One can verify that different types of four qubit entanglement detected by antilinear operators of ref. [13] are quantified by entanglement monotones constructed from four qubit invariants.

III. N-ODD N-WAY INVARIANT
Since I N −odd = 0, there is no degree two invariant of N −way fonts for a general state of N −odd qubits. But we can single out a qubit, write N − 1 qubit invariants and then use Eq. (14) to obtain N −qubit invariant. If we single out N th qubit and look at negativity fonts of ρ TA 1 , then two (N − 1) qubit N −way invariants are and . For N = 5, the five-way invariants of local unitaries on qubits A 1 , A 2 , A 3 , and A 4 and corresponding four-way invariants combine to give which is a five qubit invariant of degree four with fonts in five way and four-way partial transpose with respect to qubit A 1 . In general, one can construct I for each choice of p and q value. Degree four invariants to detect entanglement of two entangled qubits with their compliment in a five qubit state are combinations of two qubit invariants of five way, four-way, three way and two way fonts and can be obtained in a way analogous to that for five tangle.
To conclude, local unitary polynomial invariants for N qubit quantum state have been obtained from basic units of entanglement, referred to as negativity fonts. The method exploits the transformation properties of determinants of K−way negativity fonts under local unitary transformations. The entanglement monotone based on square of degree two invariant for N even (Eq. (21)) and degree four invariant of Eq. (28) for N −odd is referred to as N −tangle in analogy with three-tangle [20]. The method aims at obtaining LU invariants, that are relevant to classifying multiqubit entangled states. To illustrate the construction of entanglement monotones that detect entanglement of specific parts of the composite system, degree four invariants to detect entanglement of entangled pairs in a four qubit state are reported. Our method can be used to generate the relevant invariants obtained by using different approaches in references [10-12, 15, 16] and also to generate additional invariants necessary to detect specific entanglement modes. Entanglement monotones constructed from invariants can identify the class to which a given state belongs. Local unitary transformations redistribute the negativity fonts amongst K−way partial transposes and may also reduce the number of negativity fonts in a given partial transpose. To determine unitary transformations that relate two unitary equivalent states is an important question in quantum information. The key to determine the unitary transformations relating two states belonging to the same class lies in the numerical value of invariants, number and type of negativity fonts and transformation equations that the determinants of negativity fonts for each state satisfy. The transformation equations for negativity fonts can be used directly to identify the unitaries that relate two unitary equivalent states. The method can be easily extended to qutrits and higher-dimensional systems.
Financial support from CNPq, Brazil and Fundação Araucária, Brazil is acknowledged.