Quantum entanglement via nilpotent polynomials

We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed.

Figure 1

(a) Canonic form of the entangled state for three qubits. By a local transformation the amplitudes of the lowest excited states and the phases of the second highest exited states are set to zero. The amplitude of the lowest state is taken as the common factor determining the normalization and the global phase. (b) In the case of qudits, the group of local transformations is richer, and the canonic form can be chosen such that it corresponds to maximum population of the symmetric states $\mid k,k,k⟩$ where $k=0,\dots ,d-1$. (c) The structure of the tanglemeter for a 4-qubit system in the case of $\mathrm{SL}(2,\mathbb{C})$ local transformations. The scaling factors $q_i$ can be chosen in many different ways, in particular such that some of the nonzero coefficients equal one.

Figure 2

The polynomial invariant $\mid I^{\left(2\right)}\mid$ plotted versus the nonunitarity measure $\exp \mid \Sigma \ln {\mid \psi \mid }^2\mid$ for a set of $\sim {10}^2$ randomly chosen pure states in a $n=4$ qubit assembly.

Figure 3

Diagram illustrating the dynamics of the coefficients ${\beta }_i$ of the nilpotential $f$ for a four-qubit system subject to $\mathrm{SL}(2,\mathbb{C})$ transformations. The coefficients corresponding to quadric, cubic, and bilinear terms are shown by black, gray, and white circles, respectively. (a) Flux directions. The $\mathrm{SL}(2,\mathbb{C})$ transformations are applied in such a way, that the polynomials remain in the canonic form with respect to su(2) transformations, with all linear terms ${\beta }_i{\sigma }_i^{+}=0$. The dashed arrows show the linear contributions of different states to the time derivatives of the neighboring states, and the dotted arrows jointed by asterisk depict two types of bilinear contributions. The other, similar terms may be constructed by symmetry. (b) Canonic form for the generic state and for singular classes corresponding to one (c), two (d), three (e), and four (f) vanishing eigenvalues of the determinant Eq. (69).

Figure 4

(a) A possible strategy for identifying the tanglemeter $f_c$ for three qutrits. The choice of the parameters $P_i^{\kappa }$ of local unitary transformations is done in such a way that the transformations sequentially eliminate the terms in $F_c$ that are coupled to the states $\mid k,k,k⟩$ by just one local operator. The SU(3) transformations maximize the population of the vacuum $\mid \mathrm{O}⟩=\mid 0,0,0⟩$. Next, the SU(2) transformations not acting on the states $\mid 0_i⟩$ shown by dotted circles, and only the states shown by dashed and solid circle are mixed, maximize the population of the state ∣1,1,1⟩. The population of the state ∣2,2,2⟩ cannot be further maximized by local unitary operations. However, SL(3) and SL(2) dynamic transformations can eliminate some of the coefficients $\beta$ directly connected to such states. (b) The same diagrams for the case of elements with different dimensions. The arrows show the tanglemeter coefficients that “absorb” the coefficients vanishing in the course of the controlled dynamics.

Figure 5

Bipartite entanglement among the qubits 1 and 2. (a) Direct, (b) indirect, (c) configurations without entanglement, and (d) configurations that can result in cancellations of entanglement.

Figure 6

(a) Tripartite entanglement in two different configurations. (b) The three classes of entanglement for three qubits. $B$ stands for biseparable, $W$ for the singular class containing the Werner state and $G$ for the the general orbit.

Figure 7

(Color online) Function $\mid \zeta \left(z\right)\mid$ as in Eq. (D2). In the limit $\mid z\mid \to \infty$, $\mid \zeta \left(z\right)\mid \to 4$.