Particle-wave duality: a dichotomy between symmetry and asymmetry

Abstract

Symmetry plays a central role in many areas of modern physics. Here, we show that it also underpins the dual particle and wave nature of quantum systems. We begin by noting that a classical point particle breaks translational symmetry, whereas a wave with uniform amplitude does not. This provides a basis for associating particle nature with asymmetry and wave nature with symmetry. We derive expressions for the maximum amount of classical information we can have about the symmetry and asymmetry of a quantum system with respect to an arbitrary group. We find that the sum of the information about the symmetry (wave nature) and the asymmetry (particle nature) is bounded by log(D), where D is the dimension of the Hilbert space. The combination of multiple systems is shown to exhibit greater symmetry and thus a more wavelike character. In particular, a class of entangled systems is shown to be capable of exhibiting wave-like symmetry as a whole while exhibiting particle-like asymmetry internally. We also show that superdense coding can be viewed as being essentially an interference phenomenon involving wave-like symmetry with respect to the group of Pauli operators.

Measuring quantum coherence in multislit interference

“…The duality relation (1) was also experimentally verified [6]. Wave-particle duality in two-slit experiments has also been connected to things like entropic uncertainty relations [7], and the dichotomy between symmetry and asymmetry [8].…”

“…From (8) it is clear that in order to tell which slit the quanton went through, one has to be able to tell which path-detector state, out of the set {|d j } materialized. If one is only interested in the path-dector and is not concerned with what happens to the quanton after passing through the slits, one can write the density operator corresponding to the state (8), and trace out the states of the quanton. The resulting reduced density operator for the path detector is…”

“…Since coherence is a property of the quanton alone, we will first trace out the path-detector states, to obtain a reduced density matrix of the quanton. Writing the corresponding density operator for the state (8), and taking a trace over an orthonormal set of path-detector states, it is straightforward to obtain the reduced density operator of the quanton . Schematic diagram of a n-slit interference experiment.…”

“…Let us calculate the coherence C for the state (8). Since coherence is a property of the quanton alone, we will first trace out the path-detector states, to obtain a reduced density matrix of the quanton.…”

Observable Measure of Quantum Coherence in Finite Dimensional Systems

“…The latter is the ability of a state to act as a reference frame under a superselection rule, being widely investigated in recent years [13,14,[31][32][33][34][35][36][37][38][39][40][41][42][43][44]. One can observe that asymmetry is the quantum coherence lost by applying a phase shift w.r.t.…”