Indistinguishability of Identical Bosons from a Quantum Information Theory Perspective

Using tools from quantum information theory, we present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. Our results do neither rely on additional assumptions on the input state of the interferometer, such as, for instance, a fixed mode occupation, nor on any assumption on the degrees of freedom that potentially make the particles distinguishable. We identify the expectation value of the projector onto the $N$-particle symmetric subspace as an operationally meaningful measure of indistinguishability, and derive tight lower bounds on it that can be efficiently measured in experiments. Moreover, we present a consistent definition of perfect distinguishability and characterize the corresponding set of states. In particular, we show that these states are diagonal in the computational basis up to a permutationally invariant unitary. Moreover, we find that convex combinations of states that describe partially distinguishable and perfectly indistinguishable particles can lead to perfect distinguishability, which itself is not preserved under convex combinations.

Figure 1

The experimental setup we consider: $N$ photons are injected into a linear optical interferometer followed by particle number detection on the output modes. Description in first quantization (orange), and description in second quantization (blue), as explained in the main text.

Figure 2

Tight upper and lower bounds on $p_N$, for different $N$, as a function of $p_2$, cf. Theorem 2. The dashed area are all states for $N=10$ that are compatible with the respective value for $p_2$. A large value of $p_2$ implies high indistinguishability. For a given particle number $N$, $p_2$ provides a nontrivial lower bound on $p_N$ if $p_2\ge (N-2)/(N-1)$. That is for a larger number of particles, higher values of $p_2$ are required to obtain a nontrivial lower bound on $p_N$.

Figure 3

(right) The set of $N$-particle external density matrices according to their weight on the symmetric subspace. (left) The subset fulfilling Eq. (3), contains the nonconvex set of perfectly distinguishable states (blue). Mixing a perfectly distinguishable state $\varrho \propto {\sum }_{\pi }\pi |1\dots N⟩⟨1\dots N|{\pi }^{†}$ and $\tilde{\varrho }=U^{\otimes N}\varrho (U^{†}{)}^{\otimes N}$ for a suitable $U$ results in a state $\tau$ not satisfying Theorem 3. It is unclear if there exist states $\omega$ (yellow region), outside the convex hull of perfectly distinguishable states, that satisfy Eq. (3). Moreover, it is possible to mix perfectly (${\varrho }_1$) and partially indistinguishable (${\varrho }_2$) states, where $|{\mathrm{\Psi }}^{\pm }⟩=(|12⟩\pm |21⟩)/\sqrt{2}$, to obtain perfectly distinguishable states. For instance, $|{\mathrm{\Psi }}^{+}⟩⟨{\mathrm{\Psi }}^{+}|+|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|=|12⟩⟨12|+|21⟩⟨21|$ is perfectly distinguishable.