Abstract
For any quantum state representing a physical system of identical particles, the density operator must satisfy the symmetrization principle (SP) and conform to super-selection rules (SSR) that prohibit coherences between differing total particle numbers. Here we consider bi-partitite states for massive bosons, where both the system and sub-systems are modes (or sets of modes) and particle numbers for quantum states are determined from the mode occupancies. Defining non-entangled or separable states as those prepared via local operations (on the sub-systems) and classical communication processes, the sub-system density operators are also required to satisfy the SP and conform to the SSR, in contrast to some other approaches. Whilst in the presence of this additional constraint the previously obtained sufficiency criteria for entanglement, such as the sum of the and variances for the Schwinger spin components being less than half the mean boson number, and the strong correlation test of being greater than are still valid, new tests are obtained in our work. We show that the presence of spin squeezing in at least one of the spin components , and is a sufficient criterion for the presence of entanglement and a simple correlation test can be constructed of merely being greater than zero. We show that for the case of relative phase eigenstates, the new spin squeezing test for entanglement is satisfied (for the principle spin operators), whilst the test involving the sum of the and variances is not. However, another spin squeezing entanglement test for Bose–Einstein condensates involving the variance in being less than the sum of the squared mean values for and divided by the boson number was based on a concept of entanglement inconsistent with the SP, and here we present a revised treatment which again leads to spin squeezing as an entanglement test.
Highlights
Since the work of Einstein et al [1] on local realism, the famous cat paradox of Schrodinger [2] and the derivation of inequalites by Bell [3] and others [4], entanglement has been recognized as being one of the essential features that distinguishes quantum physics from classical physics
Bose–Einstein condensates involving the variance in Sz being less than the sum of the squared mean values for Sx and Sy divided by the boson number was based on a concept of entanglement inconsistent with the symmetrization principle (SP), and here we present a revised treatment which again leads to spin squeezing as an entanglement test
We have shown that spin squeezing in any spin component is a test of entanglement—entanglement being defined here in terms of separable states satisfying the local particle number super-selection rules (SSR)
Summary
Since the work of Einstein et al [1] on local realism, the famous cat paradox of Schrodinger [2] and the derivation of inequalites by Bell [3] and others [4], entanglement has been recognized as being one of the essential features that distinguishes quantum physics from classical physics. For systems of identical bosons we argue that to define the separable states (and the entangled states) one should take into account the presence of the particle-number super-selection rule (SSR) at the global level—as required for any quantum state and crucially at the level of the local sub-systems in the case of separable states. We work from the viewpoint of the second observer Phase reference systems such as BEC with large boson numbers—described by the second observer as being in statistical mixtures of Glauber coherent states with large fixed amplitudes and all phases having equal weight, and which satisfies the SSR for the BEC mode—are often involved in protocols such as Ramsay interferometry, dense coding, Bell inequalities. B, which can be achieved in ultra-cold gas experiments
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