Bound entangled singlet-like states for quantum metrology

Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered very weakly entangled. Since no pure entanglement can be distilled from them, they are also called bound entangled. In this paper, we present two classes of $(2d\times 2d)$-dimensional PPT entangled states for any $d\ge 2$ which outperform all separable states in metrology significantly. We present strong evidence that our states provide the maximal metrological gain achievable by PPT states for a given system size. When the dimension $d$ goes to infinity, the metrological gain of these states becomes maximal and equals the metrological gain of a pair of maximally entangled qubits. Thus, we argue that our states could be called “PPT singlets.”

Figure 1

Metrological performance of the bound entangled states presented in this paper. (solid) The dependence of the quantum Fisher information ${\mathcal{F}}_Q[{\varrho }_{\mathrm{F}n},H]$ on the dimension $d,$ given also analytically in Eq. (2). $\left(2d\right)\times \left(2d\right)$ PPT quantum states are considered here, and the Hamiltonian is defined in Eq. (3). Note that the dependence is the same for the two families of states denoted by ${\varrho }_{\mathrm{F}n}$ for $n=1,2.$ (dashed) Maximum for the quantum Fisher information for bipartite quantum states. (dotted) Maximum for the quantum Fisher information for separable quantum states. Since the states have quantum Fisher information larger than this value for all $d,$ they are evidently entangled.

Figure 2

Positive partial transpose (PPT) states and quantum Fisher information in a section of the state space. (gray area) Points corresponding to PPT states. (white area) Points corresponding to non-PPT states. (solid curves) Contour plot of the quantum Fisher information ${\mathcal{F}}_Q[\varrho ,\mathcal{H}].$ In the figure, $f_n<f_{n+1}$ holds. (point E) Point corresponding to an extremal state of the PPT set maximizing the quantum Fisher information within the set. Note that it is also imaginable that the quantum Fisher information is maximized by states that are on the boundary of the set but not extremal or by states that are within the set. Still, at least one of the states with a maximal quantum Fisher information must be extremal.

Figure 3

(solid) The quantum Fisher information for the ${\varrho }_{\mathrm{F}n}$ state of size $2d\times 2d$ mixed with white noise given in Eq. (11) as a function of $p$ for (bottom to top) $d=2,8,\infty .$ (dashed) Noise limits for metrological usefulness for the same dimensions. If $p$ is larger than this bound, then the state is more useful for metrology than separable states. It can be seen that, for large $d,$ the state remains useful even for relatively small $p.$

Figure 4

Metrological gain for diagonal local Hamiltonians $H_1$ and $H_2$ with $+1$ and $-1$ elements, as a function of the number of $+1$ in $H_1$ and $H_2$. On the axes, we have excess of 1 compared with $D/2,$ i.e., the dimension over two, in $H_k.$ Thus, 0 corresponds to equal number of $-1$ and $+1$. The gain is maximal if the number of $+1$ is half of the dimension. That is, half of the elements are $+1,$ half of them are $-1.$ (a) $4\times 4,$ (b) $6\times 6,$ and (c) $8\times 8$ systems.

Figure 5

Maximal metrological gain for a Hamiltonian of the type given in Eq. (80) if the number of $+1$ and $-1$ is equal in the diagonals of the local Hamiltonians, and we are changing $c_2$ while we keep $c_1=1$ for (bottom to top) $4\times 4,6\times 6,$ and $8\times 8$ systems. (crosses) The maximum metrological gain obtained numerically. (solid) The metrological gain for the bound entangled states based on private states ${\varrho }_{\mathrm{F}1},$ given in Eq. (25). The $u_{ij}$ are chosen based on the quantum Fourier transform. As it can be seen, the bound entangled state ${\varrho }_{\mathrm{F}1}$ has the largest gain among PPT quantum states even if $c_2\ne 1.$