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Physical Review A

Theorem on nonclassical states

Ching Tsung Lee

Published Oct 1, 1995• 110 citations
Publisher

Abstract

Let the density matrix of an arbitrary state be expressed in terms of number states as \ensuremath{\rho}^=${\ensuremath{\sum}}_{\mathit{n}=0}^{\mathrm{\ensuremath{\infty}}}$${\ensuremath{\sum}}_{\mathit{m}=0}^{\mathrm{\ensuremath{\infty}}}$\ensuremath{\rho}(n,m)\ensuremath{\Vert}n〉〈m mU. We prove the following theorem: If \ensuremath{\rho}(0,0)=0, then the state is as nonclassical as possible according to the measure introduced by Lee [Phys. Rev. A 44, R2775 (1991)].

DOI
10.1103/physreva.52.3374

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In a recent paper, C. T. Lee [ Phys. Rev. A 52, 3374 (1995) ] showed that if \ensuremath{\rho}^=${\ensuremath{\sum}}_{\mathit{n},\mathit{m}}$\ensuremath{\rho}(n,m)|n〉〈m| is the density operator describing an arbitrary state, with \ensuremath{\rho}(0,0)=〈0|\ensuremath{\rho}^|0〉=0, the state is as nonclassical as possible. Here we exhibit an example in apparent contradiction with the above theorem. \textcopyright{} 1996 The American Physical Society.

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Let the density matrix of an arbitrary state be expressed in terms of number states as \ensuremath{\rho}^={\ensuremath{\sum}}_{\mathit{n}=0}^{\mathrm{\ensuremath{\infty}}}$${\ensuremath{\sum}}_{\mathit{m}=0}^{\mathrm{\ensuremath{\infty}}}\ensuremath{\rho}(n,m)\ensuremath{\Vert}n〉〈m mU. We prove the following theorem: If \ensuremath{\rho}(0,0)=0, then the state is as nonclassical as possible according to the measure introduced by Lee [Phys. Rev. A 44, R2775 (1991)].
A general theorem is proved on the density matrix of quantum statistics. Let the density matrix be ρ{M}=exp[−β ∑ l=1MHl],where the operators Hl's are not always commutable. ρ{M} can be expanded in series of the form ρ{M}=(ΣM≧ ∑ i=1nmi, n≧0Σ{m1}M···{mn}M)ρ*{m1}···ρ*{mn}ρ̄{M−m1−···−mn},where {m1}M, ··· {mn}M are subsets of operators chosen from the set {M} and operators belonging to different subsets are commutable. The summation is over all the possible choices of the sets {m1}, ··· {mn}. ρ̄{k} is defined by ρ̄{k}=[Π{k}exp(−βHi)]s,where suffix s means the symmetrization by changing the order of the products. And ρ*{m} is defined by ρ*{m}=(Σ{k}m)(−)m−kρ{k}ρ̄{m−k},which is proved to be O(βm+1). Some possible applications are briefly discussed.
We introduce a quantum-optical notion of nonclassicality that we call the process output nonclassicality for multimode quantum channels. The motivation comes from an information-theoretic point of view and the emphasis is on the output states of a channel. We deem a channel to be ``classical'' if its outputs are always classical irrespective of the input, i.e., if the channel is nonclassicality breaking, and nonclassical otherwise. Our condition is stronger than the one considered by Rahimi-Keshari et al., [Phys. Rev. Lett. 110, 160401 (2013)] and we compare the two approaches. Using our framework we then quantify the nonclassicality of a quantum process by introducing a noise-robustness type of measure that we call the nonclassicality depth of a channel. It characterizes a certain threshold noise beyond which a given channel outputs only classical states. We achieve this by generalizing a prescription by Lee [Phys. Rev. A 44, R2775 (1991)] to multimode states and then by extension to multimode channels.
In a recent paper Gamayun et al. [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] studied the dynamics of a mobile impurity weakly coupled to a one-dimensional Tonks-Girardeau gas of strongly interacting bosons. Employing the Boltzmann equation approach, they, in particular, arrived at the following conclusions: (i) a light impurity, being accelerated by a constant force F, does not exhibit Bloch oscillations, which were predicted and studied by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamanev, Ann. Phys. (N.Y.) 327, 639 (2012)]; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity v(D)∝√[F]. In this Comment we argue that result (i) is an artifact of the classical Boltzmann approximation. The latter misses the formation of the quasibound state between the impurity and a hole. Its dispersion relation E(b)(P,ρ) is a smooth periodic function of momentum P with the period 2k(F)=2πℏρ, where ρ is a density of the host gas. Being accelerated by a small force, such a bound-state exhibits Bloch oscillations superimposed with the drift velocity v(D)=μF. The mobility μ may be expressed exactly [M. Schecter et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of E(b)(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.