Entanglement, purity, and energy: Two qubits versus two modes

We study the relationship between the entanglement, mixedness, and energy of two-qubit and two-mode Gaussian quantum states. We parametrize the set of allowed states of these two fundamentally different physical systems using measures of entanglement, mixedness, and energy that allow us to compare and contrast the two systems using a phase diagram. This phase diagram enables one to clearly identify not only the physically allowed states, but the set of states connected under an arbitrary quantum operation. We pay particular attention to the maximally entangled mixed states of each system. Following this we investigate how efficiently one may transfer entanglement from two-mode to two-qubit states.

Figure 1

(Color online) The boundaries of the entanglement-purity-energy diagram for two-qubit states when the logarithmic negativity is used as the measure of entanglement. The MEMS are the Werner states from Eq. (7) in this case and form a line rather than a plane as in the case when concurrence is used to measure the entanglement. All quantities are dimensionless.

Figure 2

The concurrence vs purity phase diagram of two qubits. The lower curve $C_W$ indicates the Werner states, while the upper curve corresponds to the MEMSs. All quantities are dimensionless.

Figure 3

(Color online) Numerically determined entanglement-purity-energy phase diagram for 100 000 randomly chosen two-qubit states where the entanglement is measured in terms of the concurrence. All quantities are dimensionless.

Figure 4

(Color online) The entanglement-purity-energy phase diagram of the (i) pure (blue curve), (ii) separable (green curve), and (iii) maximally entangled mixed states (red curves) of two qubits. All quantities are dimensionless.

Figure 5

Graph depicting how the energy $E$ of pure two-mode Gaussian states depends on the amount of entanglement $\mathcal{E}$ between the two modes. The physical region lies in the shaded region above the curve. All quantities are dimensionless.

Figure 6

Graph showing how the energy $E$ of the separable two-mode Gaussian states depends on the purity $P$ for energy values up to 2. The physical region lies in the shaded region above the curve. All quantities are dimensionless.

Figure 7

Graph showing how the entanglement $\mathcal{E}$ of two-mode Gaussian states depends on the purity $P$ for a fixed energy value of 2. The upper curve indicates the maximally entangled mixed states for fixed energy and the physical region is the shaded area below this curve. All quantities are dimensionless.

Figure 8

Graph showing the GMEMS and GLEMS of two-mode Gaussian states for fixed energy value of 2. The GMEMS are those states lying on the upper curve while the GLEMS are those states which lie in the darker shaded region between the two curves. All quantities are dimensionless.

Figure 9

(Color online) The entanglement-purity-energy phase diagram boundaries showing the (i) pure (blue curve satisfying $P=1$) (ii) separable (green curve satisfying $\mathcal{E}=0$), and (iii) maximally entangled mixed (red curves starting on the pure curve) Gaussian states of a two-mode continuous variable system for energies $E$ in the range $0⩽E⩽2$. All quantities are dimensionless.

Figure 10

The dependence of the maximum concurrence $C_{\mathit{out}}$ generated between two two-level atoms by evolution under the Jaynes-Cummings Hamiltonian (81) on the initial energy $E_{\mathit{in}}$ in the two-mode field. The field is initially in the entangled superposition state of Eq. (87) characterized by the coherent state amplitude $\alpha$. All quantities are dimensionless.

Figure 11

The dependence of the maximum concurrence $C_{\mathit{max}}$ generated between two two-level atoms by evolution under the Jaynes-Cummings Hamiltonian (81) on the initial entanglement in the two-mode field. The field is initially in the entangled superposition state of Eq. (87) characterized by the coherent state amplitude $\alpha$. All quantities are dimensionless.

Figure 12

(Color online) The dependence of the maximum concurrence $C_{\mathit{out}}$ (lower curve) generated in a pair of two-level atoms by evolution under the Jaynes-Cummings Hamiltonian (81) and the purity $P$ (upper curve) of the state of the two atoms at this time on the initial energy $E_{\mathit{in}}$ of the two-mode field when the field is initially in a pure two-mode Gaussian state. All quantities are dimensionless.

Figure 13

The dependence of the maximum concurrence $C_{\mathit{out}}$ generated in a two-qubit pair by evolution under the Jaynes-Cummings Hamiltonian (81) on the initial entanglement in the two-mode field $S_{\mathit{in}}$ when the field is initially in a pure two-mode Gaussian state. All quantities are dimensionless.