Narrow Results

Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of $n$-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the $n$-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability.

General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and $n$-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that $\rho$ (MDS) is not separable is that it has an eigenvalue larger than $1∕d$ for a qubit and a qudit, and larger than $1∕2^{n-1}$ for $n$-qubit system. The PTU transformation does not change the eigenvalues of the $n$-qubit MDS density matrices for odd $n$. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even $n$.

The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the $W$ state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8GHZ density matrices with probabilities $p_i(i=1,2,\dots ,8)$, is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.

We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0\le r_A,r_B\le 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A)<p(r_A,1-r_A)$, except for rather narrow “crossover” intervals $[{\tilde{r}}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in $4\times K$-dimensions, for $K=3,4,5$, with $K=4$ corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) $K=4$, the $X$-states $K=5$, and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two $X$-states cases, we derive analytical results; for $K=3,4$, we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower ${\tilde{r}}_A$) as $K$ increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A48 (2015) 035306.] from a univariate ($r_A$) framework — in which they found separability probabilities to hold constant with $r_A$ — to a bivariate ($r_A,r_B$) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form $p(r_A,\frac{1}{2})$. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.