An Energetic Constraint for Qubit-Qubit Entanglement

Abstract

We analyze qubit-qubit entanglement from an energetic perspective and reveal an energetic trade-off between quantum coherence and entanglement. We decompose each qubit internal energy into a coherent and an incoherent component. The qubits' coherent energies are maximal if the qubit-qubit state is pure and separable. They decrease as qubit-qubit entanglement builds up under locally-energy-preserving processes. This yields a "coherent energy deficit" that we show is equal to a well-known measure of entanglement, the square negativity. In general, a qubit-qubit state can always be represented as a mixture of pure states. Then, the coherent energy deficit splits into a quantum component, corresponding to the average square negativity of the pure states, and a classical one reflecting the mixedness of the joint state. Minimizing the quantum deficit over the possible pure state decompositions yields the square negativity of the mixture. Our findings bring out new figures of merit to optimize and secure entanglement generation and distribution under energetic constraints.

Figures (3)

• Figure 1: The energetic splitting into coherent and incoherent components for a qubit with mean energy $E$. An illustration of the coherent energy (blue-solid line) is less than the maximum (nominal) value (black-dashed line) due to decoherence. The difference between these two lines is the coherent energy deficit ${\cal D}$ (See main text).
• Figure 2: Top: The maximum squared negativity for fixed energies of subsystem B. The amount of entanglement present in the system is dictated by the subsystem with the lowest nominal coherent energy. Bottom: The maximum efficiency of entanglement generation as a function of the energies of subsystem $A$ and $B$. The most efficient processes are those that distribute energies equally (or oppositely) between the two subsystems (dashed-grey line). For larger energies, slight differences in the subsystem energies are less detrimental to the efficiency than when only a small amount of energy is available. Note, the interval where $E_B\in[1/2,1]$ is a mirror image of the current plot about the $E_B = 1/2$ line.
• Figure 3: The entanglement that Alice (solid lines) and Eve (dotted lines) can extract from the energy encoded mixed state. In the [a]symmetric case (black [red] line), Alice's entanglement is greater than what Eve can obtain. Inset: The encoding efficiency for the two mixed state encoding. The black [red] line corresponds to the [a]symmetric case.