Entanglement and Uncertainty in the Special-Relativistic Regime

Abstract

This paper seeks an interplay between entanglement and uncertainty in the special-relativistic regime. A pure quantum state describing the system of identical target particles in a special relativity scenario is adopted to be correlated if, for a pair of position and momentum observables, the quantum covariance functions do not vanish. We explore how the relativistic uncertainty principle (RUP) is modified in the presence of quantum entanglement within the bipartite and multipartite systems. In addition, the separability criteria established upon the total variance of a pair of Einstein-Podolski-Rosen type operators are derived. These criteria making use of RUP specify necessary and sufficient conditions for factorizing the two-mode and multi-mode continuous-variable states.

Appendices

Appendix A

Using the same properties applied to (4), the third expression on the right-hand side of (9) can be reduced to

$$\begin{aligned}{} & {} \frac{1}{2}\left\langle \sqrt{1+\alpha \left( \hat{p}_1^2+\hat{p}_2^2 +\alpha \hat{p}_1^2\hat{p}_2^2\right) }\right\rangle \nonumber \\\ge & {} \frac{1}{2}\sum _{n=0}^\infty f(n)\alpha ^n\left\langle \left( \hat{p}_1^2+\hat{p}_2^2+\alpha \hat{p}_1^2\hat{p}_2^2\right) ^n\right\rangle \nonumber \\\ge & {} \frac{1}{2}\sum _{n=0}^\infty f(n)\alpha ^n\left\langle \left( \hat{p}_1^2+\hat{p}_2^2\right) ^n+\alpha n\hat{p}_1^2\hat{p}_2^2\left( \hat{p}_1^2+\hat{p}_2^2\right) ^{n-1}\right\rangle \nonumber \\\ge & {} \frac{1}{2}\sum _{n=0}^\infty f(n)\alpha ^n\left( \left\langle \hat{p}_1^2+\hat{p}_2^2\right\rangle ^n+\alpha n\left\langle \hat{p}_1^2\hat{p}_2^2\right\rangle \left\langle \hat{p}_1^2+\hat{p}_2^2\right\rangle ^{n-1}\right) \nonumber \\\ge & {} \frac{1}{2}\sqrt{1\!+\!\alpha \left( (\Delta \hat{p}_1)^2\!+\!(\Delta \hat{p}_2)^2\right) }\!+\!\frac{\alpha (\Delta \hat{p}_1)^2(\Delta \hat{p}_2)^2}{2\left( (\Delta \hat{p}_1)^2\!+\!(\Delta \hat{p}_2)^2\right) }\sum _{n=0}^\infty nf(n)\alpha ^n\left( (\Delta \hat{p}_1)^2\!+\!(\Delta \hat{p}_2)^2\right) ^n \nonumber \\\ge & {} \frac{1}{2}\sqrt{1+\alpha \left( (\Delta \hat{p}_1)^2+(\Delta \hat{p}_2)^2\right) }\left( 1+\alpha \frac{(\Delta \hat{p}_1)^2(\Delta \hat{p}_2)^2}{(\Delta \hat{p}_1)^2+(\Delta \hat{p}_2)^2}\right) \nonumber \\\ge & {} \frac{1}{2}\left( 1+\alpha \left( \frac{(\Delta \hat{p}_1)^2}{2}+\frac{(\Delta \hat{p}_2)^2}{2}+\frac{(\Delta \hat{p}_1)^2(\Delta \hat{p}_2)^2}{(\Delta \hat{p}_1)^2+(\Delta \hat{p}_2)^2}\right) \right) \end{aligned}$$

(A1)

Appendix B

The last expression on the right-hand side of (20) can be found as

$$\begin{aligned}{} & {} \sum _{k\ne l=1}^N\left\langle \sqrt{1+\alpha \left( \hat{p}_k^2+\hat{p}_l^2 +\alpha \hat{p}_k^2\hat{p}_l^2\right) }\right\rangle \nonumber \\\ge & {} \sum _{k\ne l=1}^N\sum _{n=0}^\infty f(n)\alpha ^n\left\langle \left( \hat{p}_k^2+\hat{p}_l^2 +\alpha \hat{p}_k^2\hat{p}_l^2\right) ^n\right\rangle \nonumber \\\ge & {} \sum _{k\ne l=1}^N\sum _{n=0}^\infty f(n)\alpha ^n\left\langle \left( \hat{p}_k^2+\hat{p}_l^2\right) ^n+\alpha n\hat{p}_k^2\hat{p}_l^2\left( \hat{p}_k^2+\hat{p}_l^2\right) ^{n-1}\right\rangle \nonumber \\\ge & {} \sum _{k\ne l=1}^N\sum _{n=0}^\infty f(n)\alpha ^n\left( \left\langle \hat{p}_k^2+\hat{p}_l^2\right\rangle ^n+\alpha n\left\langle \hat{p}_k^2\hat{p}_l^2\right\rangle \left\langle \hat{p}_k^2+\hat{p}_l^2\right\rangle ^{n-1}\right) \nonumber \\\ge & {} \sum _{k\ne l=1}^N\left( \sqrt{1+\alpha \left( (\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2\right) }+\alpha \frac{(\Delta \hat{p}_k)^2(\Delta \hat{p}_l)^2}{(\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2}\sum _{n=0}^\infty nf(n)\alpha ^n\left( (\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2\right) ^n\right) \nonumber \\\ge & {} \sum _{k\ne l=1}^N\sqrt{1+\alpha \left( (\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2\right) }\left( 1+\alpha \frac{(\Delta \hat{p}_k)^2(\Delta \hat{p}_l)^2}{(\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2}\right) \nonumber \\\ge & {} \sum _{k\ne l=1}^N\left( 1+\alpha \left( \frac{(\Delta \hat{p}_k)^2}{2}+\frac{(\Delta \hat{p}_l)^2}{2}+\frac{(\Delta \hat{p}_k)^2(\Delta \hat{p}_l)^2}{(\Delta \hat{p}_k)^2+(\Delta \hat{p}_l)^2}\right) \right) \end{aligned}$$

(B1)