Speaker
Description
Einstein in 1905, in his explanation of the photoelectric effect, postulated that light, the quintessential wave, had to possess particle-like properties. In the course of 1923-24, de Broglie, analyzing electron scattering from metal surfaces, postulated that electrons, the quintessential particles, must possess wave-like properties. In 1928, Bohr made the first attempt to reconcile the two viewpoints and introduced the concept of complementarity (or, in a more restricted sense, wave-particle duality), and thus the by now nearly 100 years history of complementarity has started.
We begin with a brief overview of the history of quantitative complementarity relations. A particle going through an interferometer can exhibit wave-like or particle-like properties. The first quantitative duality relation was obtained by Greenberger and Yasin [1], between the strictly single-partite properties: predictability $P = |\rho_{11} − \rho_{22}|$ and visibility $V = 2|\rho_{12}|$ and has the form
$$ \qquad \qquad P^2 + V^2 \le 1. \qquad \qquad (1) $$ In a seminal study of the two-path interferometer, Englert introduced detectors into the interferometer arms and defined the path distinguishability, D, as the discrimination probability of the path detector states [2]. He derived a relation between this type of path information and the visibility $V = 2|\rho_{12}|$ of the interference pattern, in the form $$ \qquad \qquad D^2 + V^2 \le 1. \qquad \qquad (2) $$ In a follow-up [3], Englert and Bergou showed that $D$ is a joint property of the system and the meter to be clearly distinguished from predictability, which is a strictly single partite property. They showed that (2) corresponds to the so-called which-way sorting (post-selection) of the measurement data. They also introduced the quantum erasure sorting, which led to the duality relation $P^2 + C^2 \le 1$, where the coherence $C$ is a joint property of the system and detectors. Most importantly, they conjectured that $D$ should be related to an entanglement measure. Taking up this conjecture, the complete bipartite (particle-meter) complementarity relation, connecting complementarity, i.e., visibility of the interference pattern, $V$, and path predictability, $P$, to entanglement, was found in [4], in the form of a *triality relation*, $$ \qquad \qquad P^2 + C^2 +V^2 \le 1. \qquad \qquad (3) $$ Here $C$ is the concurrence, emerging naturally as part of the completeness relation for a bipartite system. In [5], this *triality relation* was further generalized to multi-path ($n$-path) interferometers. These works completed the research on quantitative complementarity and brought the Bohr-Einstein debate to a very satisfying closure. In particular, Eq. (3), which is a triality relation, displays explicitly that entanglement is the genuinely quantum contribution with no classical counterpart, whereas visibility, quantifying wave-like behavior, and predictability, quantifying particle-like behavior, can be regarded as classical contribution. In all of the works discussed above, the $l_2$ measure of coherence was employed. Recently, however, a resource theory of quantum coherence was developed and two new coherence measures were introduced [6]. The $l_1$ measure is the trace distance, the entropic measure is the entropic distance of a given state to the nearest incoherent state. In the second part of the talk we present our recent results for multi-path interferometers, employing the new measures. Using these measures, we derived entropic and $l_1$ based duality relations for multi-path interferometers [7, 8]. The $l_1$ based duality relation for n-path interferometers is $$ \qquad \qquad \left( \frac{C+D-\frac{n-2}{n-1}}{\frac{\sqrt{n}}{n=1}} \right)^2 + \left(\frac{C-D}{\sqrt{\frac{n}{n-1}}}\right)^2 \le 1, \quad C,D>0, \qquad \qquad (4) $$ where $C$ is the $l_1$ measure of coherence, generalizing the visibility $V$. To close, we will present recent results generalizing duality relations to finite groups [9], recent entropic duality relations [10], and discuss recent developments, showing that relations like Eq. (1) can be derived from intrinsic properties of quantum states, without referring to measurements [11, 12].
References
[1] D. M. Greenberger and A. Yasin, ”Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A
128, 391 (1988).
[2] B.-G. Englert, ”Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154 (1996).
[3] B.-G. Englert and J. A. Bergou, ”Quantitative quantum erasure,” Opt. Commun. 179, 337 (2000).
[4] M. Jakob and J. A. Bergou, ”Quantitative complementarity relations in bipartite systems: Entanglement as a physical
reality,” Opt. Commun. 283, 827 (2010) [also as arxiv:0302075].
[5] M. Jakob and J. A. Bergou, ”Complementarity and entanglement in bipartite qudit systems,” Phys. Rev. A 76, 052107
(2007).
[6] T. Baumgratz, M. Cramer, Tand M. B. Plenio, ”Quantifying Coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[7] E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, ”Relations between Coherence and Path Information,” Phys. Rev.
Lett. 116, 160406 (2016).
[8] E. Bagan, J. Calsamiglia, J. A. Bergou, and M. Hillery, ”Duality Games and Operational Duality Relations,” Phys. Rev.
Lett. 120, 050402 (2016).
[9] E. Bagan, J. Calsamiglia, J. A. Bergou, and M. Hillery, ”A generalized wave-particle duality relation for finite groups,”
Journal of Physics A: Math. Theor. 51, 414015 (2018).
[10] E. Bagan, J. A. Bergou, and M. Hillery, ”Wave-particle duality relations based on entropic bounds for which-way information,” Phys. Rev. A 102, 022224 (2020).
[11] X. L¨u, ”Quantitative wave-particle duality as quantum state discrimination”, Phys. Rev. A 102, 02201 (2020).
[12] X. L¨u, ”Duality of path distinguishability and quantum coherence”, Phys. Lett. A 397, 127259 (2021).
