Entanglement gain in measurements with unknown results

We characterize nonselective global projective measurements capable of increasing quantum entanglement between two particles. In particular, by choosing negativity to quantify entanglement, we show that entanglement of any pure nonmaximally entangled state can be improved in this way (but not of any mixed state) and we provide detailed analysis for two qubits. It is then shown that Markovian open system dynamics can only approximate such measurements, but this approximation converges exponentially fast as illustrated using the Araki-Żurek model. We conclude with numerical evidence that macroscopic bodies in a random pure state do not gain negativity in a random nonselective global measurement.

Figure 1

Entanglement gain in nonselective global measurement as a black box problem. A state $\rho$ is input to a black box measuring device which conducts von Neumann measurement along projectors $\left\{{\mathrm{\Pi }}_j\right\}$. Measurement results are unknown and therefore the state emerging from the box is ${\rho }^{\prime }={\sum }_j{\mathrm{\Pi }}_j\rho {\mathrm{\Pi }}_j$. We study conditions under which entanglement in the state ${\rho }^{\prime }$ is higher than entanglement of the input state $\rho$.

Figure 2

Graphical depiction of an LOCC protocol to implement nonselective measurement in basis (14). Here ${\mathrm{\Pi }}_{01}$ denotes $\left|0\right.〉\left.〈0\right|+\left|1\right.〉\left.〈1\right|$, and ${\mathrm{\Pi }}_i=\left|i\right.〉\left.〈i\right|$. The LOCC procedure for the nonselective Bell measurement is described around Eq. (5).

Figure 3

Random sampling supports our claim that for the initial state (26) the measurement giving rise to highest entanglement gain is given by (28). Each point represents negativity gain for a random input state measured along random basis. Both states and measurements are sampled uniformly at random according to the Haar measures. One million trials are recorded. The upper boundary is given by (33) derived for the candidate optimal measurement, i.e., given by (28).

Figure 4

Distribution of negativity between systems of different dimensions. Dashed lines enclosing blue color show the distributions for the postmeasurement state whereas solid lines enclosing orange color show the distribution for random pure input states.