Convergence Rates for the Quantum Central Limit Theorem

Abstract

Abstract Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n -splitter, which is fed with n copies of a centred state $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> with finite second moments, converges to the Gaussian state with the same first and second moments as $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate $$\mathcal {O}\left( n^{-1/2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mfenced> </mml:mrow> </mml:math> in the Hilbert–Schmidt norm whenever the third moments of $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m -mode state enters a cascade of n beam splitters of equal transmissivities $$\lambda ^{1/n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$\chi _\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>ρ</mml:mi> </mml:msub> </mml:math> is uniformly bounded by some $$\eta _\rho &lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>ρ</mml:mi> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> outside of any neighbourhood of the origin; also, $$\eta _\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>ρ</mml:mi> </mml:msub> </mml:math> can be made to depend only on the energy of the state $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> .

Topics & Concepts