Indistinguishability of subsystems is a requirement for quantum interference, which is in turn the workhorse of extra‐classical demonstrations such as those proposed in quantum information protocols. However, indistinguishability does not arise in the standard axioms of quantum mechanics. Once the standard textbook approach is assumed, the pure states of the system in question are conventionally restricted to a Hilbert subspace with permutation symmetry. From the tomographic perspective, this is unacceptable, as such an assumption of symmetry can be too restrictive. Here we report two approaches to the problem of multiple bosons that admit both distinguishable and indistinguishable states of the same system. In doing so we define two notions of distinguishability. Both approaches are shown to give the same results in the sense that they predict the same restriction on density matrices as would be reconstructed tomographically. The number of unknown elements is polynomial in the number of particles, in stark contrast to the distinguishable case which scales exponentially. We use this picture to investigate possible measurement schemes which could detect and/or characterise indistinguishability.
Indistinguishability of subsystems is a requirement for quantum interference, which is in turn the workhorse of extra‐classical demonstrations such as those proposed in quantum information protocols. However, indistinguishability does not arise in the standard axioms of quantum mechanics. Once the standard textbook approach is assumed, the pure states of the system in question are conventionally restricted to a Hilbert subspace with permutation symmetry. From the tomographic perspective, this is unacceptable, as such an assumption of symmetry can be too restrictive. Here we report two approaches to the problem of multiple bosons that admit both distinguishable and indistinguishable states of the same system. In doing so we define two notions of distinguishability. Both approaches are shown to give the same results in the sense that they predict the same restriction on density matrices as would be reconstructed tomographically. The number of unknown elements is polynomial in the number of particles, in stark contrast to the distinguishable case which scales exponentially. We use this picture to investigate possible measurement schemes which could detect and/or characterise indistinguishability.
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.
While the concept of entanglement for distinguishable particles is well established, defining entanglement and nonlocality in systems of indistinguishable particles, which require the use of the (anti)symmetrization postulate, remains challenging, and multiple approaches have been proposed to address this issue. In this work we study the problem of detecting genuine tripartite entanglement among systems of indistinguishable bosons. A genuine entangled state is one that cannot be separable under any bipartition, where separability in the indistinguishable regime is defined by the existence of single-particle properties within each subsystem, without the possibility of knowing which property belongs to which subsystem. We use an algorithm that allows us to search for these single-particle properties and consequently rank states according to their degree of separability. In particular, we introduce a state of indistinguishable bosons with properties analogous to those of the standard Greenberger-Horne-Zeilinger state.
In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.
The principle of tomographic locality states that the operational state of a multipartite system can be fully characterized by the statistics obtained from measurements that are local to the individual subsystems. This property holds in quantum theory and features prominently in axiomatic reconstructions of the theory, where it serves to rule out a wide class of alternatives. For instance, quantum theory with Hilbert spaces defined over the real field (rather than the complex field) is an example of a theory that is ruled out in this fashion. Given its foundational importance, it is worthwhile to subject this principle to a direct experimental test. Specifically, we consider an experiment on the polarization degrees of freedom of a pair of photonic modes in a prepare-and-measure scenario and analyze the resulting data within the framework of generalized probabilistic theories. The signature of a failure of tomographic locality is that there are pairs of states on the bipartite system that can only be distinguished by the statistics they yield for non-separable measurements. In the full quantum setting, we find no evidence of a violation of tomographic locality. As a test of our analysis method, we also verify that if we restrict attention to those states and measurements that lie within the fragment described by quantum theory over the real field, then a clear signature of the failure of tomographic locality is observed.
Partial distinguishability affects multiphoton interference, which is central to large-scale photonic experiments such as Boson Sampling and to applications of multiphoton states across quantum technologies. Characterizing the internal degrees of freedom responsible for this behavior is challenging: in many scenarios they cannot be accessed directly and even their dimension or physical nature may be unknown \emph{a priori}. Here we introduce Gram-matrix tomography for independently prepared photons with pure or nearly pure internal states. For pure states, the Gram matrix contains all information required to predict the statistics of experiments that are insensitive to the internal degrees of freedom. We reconstruct this matrix, up to physically irrelevant phase choices, from a quadratic number of measurable Bargmann invariants (quantities encoding overlap magnitudes and collective phases). We access these quantities using two complementary approaches: targeted Fourier interferometers and two- and three-body photon-number correlations measured in a single random interferometer. We derive explicit error bounds that control the total-variation distance between predicted and actual outcome distributions for any subsequent experiment that is insensitive to the internal states. These guarantees account for finite sampling and deviations from purity. Crucially, the total sample complexity of the protocol is independent of the internal Hilbert-space dimension and polynomial under mild assumptions on overlaps of internal states. Our protocol uses standard interferometric measurements and turns them into a robust and operationally grounded characterization of partially distinguishable photons.
What theoretical structures characterise our notions of what it means for a whole physical system to be the sum of its parts? We explore various aspects of this question, with emphasise on quantum theory, from the following perspective: A physical theory constitutes the definition of a set of propositions with logical relationships between them such that probabilities can be used to judge the merits of the theory, given empirical evidence. The logical relationships between the propositions characterise our semantic notions such as individuation. We distinguish between the ontological and epistemological notions of individuation. We introduce the notion of ontological individuation within the context of ontological models. A subsystem is defined as a proposition that provides necessary and sufficient information for making useful predictions. This definition to some extent generalises the notions of separability and local causality to define systems that may not be local. It highlights the unified motivations for many assumptions in the literature, such as measurement independence, and points to hidden assumptions of causality and mutual exclusivity that occur when defining systems. An epistemic no-signalling criterion gives rise to a tensor product structure for epistemic systems in quantum theory. We utilise a symmetry emergent from this structure to derive a bound on the observability of a large class of operations, dependent on only the purity of the quantum state. In particular, we derive an upper bound on the trace distance between an untransformed state and the state transformed by any trace preserving, unital operation. Our notions of epistemic systems also relate to distinguishability. It is shown that for measurements of any two (_nite outcome) particle quantum numbers on multi-fermion states, the total uncertainty, minimised over all states, is zero. This is because despite indistinguishability characterised as a constraint on the observable algebra, the appropriate correspondence between states of distinguishable and indistinguishable particles is one where the Hilbert space of indistinguishable particles relates to particles with an extra quantum number such that effective distinguishability can emerge. This Hilbert space is larger than the one for distinguishable particles and hence can (does) contain states for which the uncertainty relations for distinguishable particles are no longer valid. We conclude by arguing that in general, epistemic systems are not necessarily distinguishable; They may be only effectively distinguishable. This would imply a theory of indistinguishable modes. Several motivations are presented and the form of the theory suggested. It is hypothesised that such a theory may provide a relativistic quantum theory, alternative to quantum field theory, that satisfies cluster decomposition.