We discuss the critical point $x_c$ separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator $\\rho=x|\\tilde{\\phi}><\\tilde{\\phi}| + \\frac{1-x}{D^N} I_{D^N}$ where $x \\in [0,1]$ , $D =2S+1$ , ${\\bf I}_{D^N}$ is the $D^N \\times D^N$ unity matrix and $|\\tilde {\\phi}>$ is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states $|\\tilde{\\phi}>$ invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point $x_c$ was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of $x_c$ is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.We discuss the critical point $x_c$ separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator $\\rho=x|\\tilde{\\phi}><\\tilde{\\phi}| + \\frac{1-x}{D^N} I_{D^N}$ where $x \\in [0,1]$, $D =2S+1$, ${\\bf I}_{D^N}$ is the $D^N \\times D^N$ unity matrix and $|\\tilde {\\phi}>$ is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states $|\\tilde{\\phi}>$ invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point $x_c$ was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of $x_c$ is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.

Oct 10, 200110 citations

Entanglement and Non-Separability in Quantum Mechanics

G. Costa

Springer eBooksJan 1, 20033 citations

The Pareto Frontiers of Magic and Entanglement: The Case of Two Qubits

Alexander Roman, Marco Knipfer, Jogi Suda Neto +3 more

Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, $M_2$ , and entanglement by the concurrence $Δ$ . We find that the Pareto frontier of maximal magic $M_2^{(max)}(Δ)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(Δ)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, $M_2$, and entanglement by the concurrence $Δ$. We find that the Pareto frontier of maximal magic $M_2^{(max)}(Δ)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(Δ)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.

quant-pharXiv:2603.24902Mar 27, 20260 citations

Separable-entangled frontier in a bipartite harmonic system

Constantino Tsallis, Domingo P. Prato, Celia Anteneodo

arXiv:quant-ph/0202077The European Physical Journal BOct 1, 200215 citations

Properties of entanglement in interacting spin systems

Rosario Fazio, Chiara Macchiavello

Annalen der PhysikJan 10, 20054 citations

We discuss the critical point

x_c

separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator

\\rho=x|\\tilde{\\phi}><\\tilde{\\phi}| + \\frac{1-x}{D^N} I_{D^N}

where

x \\in [0,1]

D =2S+1

{\\bf I}_{D^N}

is the

D^N \\times D^N

unity matrix and

|\\tilde {\\phi}>

is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states

|\\tilde{\\phi}>

invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point

x_c

was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of

x_c

is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.

M_2

, and entanglement by the concurrence

Δ

. We find that the Pareto frontier of maximal magic

M_2^{(max)}(Δ)

is composed of three separate segments, while the boundary of minimal magic

M_2^{(min)}(Δ)

is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.