Bound entanglement in symmetric random induced states

TL;DR

This work shows that symmetric random induced states (RIS) offer a practical route to generate PPT bound entangled states in multipartite systems for $N>3$, by tracing out ancillas from random pure states. It introduces two RIS-construction methods, MI and MII, and demonstrates that the probability of obtaining PPT BE can approach unity as system size grows, with distinct entanglement patterns across bipartitions. A Hilbert–Schmidt distance analysis reveals that MI yields a broader diversity of PPT BE states, while MII produces ensembles concentrated near the maximally mixed state and with wider parameter tolerance, making it more robust experimentally. Overall, symmetric RIS provide a versatile toolkit for generating large families of PPT BE states without heavy numerical optimization, with implications for entanglement theory, metrology, and quantum communication.

Abstract

Bound entanglement, a weak -- yet resourceful -- form of quantum entanglement, remains notoriously hard to detect and construct. We address this in this paper by leveraging symmetric random induced states, where positive partial transpose (PPT) bound entanglement arises naturally under partial tracing when proper parameters are selected. We investigate the probability of finding PPT bound entanglement in symmetric random induced states constructed via two methods: partial tracing of symmetric multiqubit pure states on the one hand (MI) and tracing out a qudit ancilla on the other hand (MII). For $N > 3$ qubits, we demonstrate that bound entanglement naturally emerges under optimal parameters, with a probability of occurrence very close to 1. We show that the two methods produce different varieties of PPT bound entangled states, and identify the contexts in which each method offers distinct advantages. These methods provide a versatile toolkit for the generation of large families of random PPT bound entangled states without complex numerical optimization.

Figures (9)

• Figure 1: Set of bipartite states : Separable states (SEP), PPT bound entangled states (PPT BE), and NPT entangled states (NPT). The union of the PPT BE and SEP states yield all PPT states. It is still an open question whether there exists NPT bound entangled states (NPT BE) IQOQI. For $2 \times 2$ and $2 \times 3$ systems, the PPT BE set is empty PERES1996HORODECKI1996.
• Figure 2: Generation of RIS. Method I (a): a RIS $\rho_1$ is created from an initial pure state $\ket{\psi_1}$ randomly generated on the symmetric subspace $\mathcal{H}_S^{N + N_a + 1}$ of $N+N_a$ qubits after tracing out $N_a$ qubits. Method II (b): a RIS $\rho_2$ is created from an initial pure state $\ket{\psi_2}$ randomly generated on the tensor product space $\mathcal{H}_S^{N+1} \otimes \mathcal{H}_a^{d_a}$ of the symmetric subspace of $N$ qubits and a $d_a$-dimensional qudit after tracing out the ancilla qudit.
• Figure 3: Bound entanglement in Method I. Probabilities that the generated random induced states have NPT entanglement (red), PPT bound entanglement (yellow), or are separable (green) using Method I with respect to the number of qubits $N_a$ in the ancilla system, for $N = 4,\dots,9$. $N_a$ starts at 1 and increases with a step of 1. The maximal PPT BE probability quickly grows to 1 as $N$ increases, showing the efficiency of the method. The brown curves are a refinement of the PPT BE yellow curve. They show the probabilities to get PPT bound entanglement across the different bipartitions: PPT BE$_1$, PPT BE$_{1,2}$, and PPT BE$_{1,2,3}$ as the ancilla parameter increases. The PPT BE$_1$ curve and the global PPT BE curve (yellow) are almost identical for $N=4$ and $5$. The bottom frame exhibit the 3 probabilities of getting PPT BE$_X$, PPT BE$_\mathrm{all}$, and UNK states. The lines between the dotted points are to guide the eye. The background colors indicate the regions where the probability to get NPT entanglement (red), PPT bound entanglement (yellow), or separable states (green) is the highest compared to the other two.
• Figure 4: Bound entanglement in Method II. Probabilities that the generated random induced states have NPT entanglement (red), PPT bound entanglement (yellow), or are separable (green) using Method II with respect to the dimension of the ancilla qudit $d_a$ for $N = 4,\dots,7$. The value of the ancilla starts at $d_a = 2$ for each $N$. For $N = 4,\dots,7$, the following values of $N_a$ are multiples of $10, 25, 100$, and $200$, respectively. The brown curves are a refinement of the PPT BE yellow curve. They show the probabilities to get PPT bound entanglement across the different bipartitions: PPT BE$_1$ and PPT BE$_{1,2}$ as the ancilla parameter increases. The PPT BE$_1$ curve and the global PPT BE curve (yellow) are almost identical for $N=4$ and $5$. The bottom frame exhibit the 3 probabilities of getting PPT BE$_X$, PPT BE$_\mathrm{all}$, and UNK states. The lines between the dotted points are to guide the eye. The background colors indicate the regions where the probability to get NPT entanglement (red), PPT bound entanglement (yellow), or separable states (green) is the highest compared to the other two.
• Figure 5: Entanglement Phase Diagram. Diagram highlighting where NPT (red), PPT BE (orange) and SEP (green) RIS are the most likely obtained for Method I, as a function of $N$ and $N_a$. The boundaries correspond to the intersections between the probability curves of figure \ref{['fig:MethodI']}. The lines are linear and quadratic fits of the intersection points.