<p>Probability distributions play a central role in quantum mechanics, and even more so in quantum optics with its rich diversity of theoretically conceivable and experimentally accessible quantum states of light. Quantifiers that compare two different states or density matrices in terms of ‘distances’ between the respective probability distributions include the Kullback–Leibler divergence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41745_2025_474_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$D_\text{KL}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mtext>KL</mtext> </msub> </math></EquationSource> </InlineEquation>, the Bhattacharyya distance <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41745_2025_474_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$D_\text{B}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mtext>B</mtext> </msub> </math></EquationSource> </InlineEquation>, and the <i>p</i>-Wasserstein distance <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41745_2025_474_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$ W_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>. We present a novel application of these notions to a variety of photon states, focusing particularly on the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41745_2025_474_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> Wasserstein distance <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41745_2025_474_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$ W_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> as it is a proper distance measure in the space of probability distributions.</p>

Comparing Probability Distributions: Application to Quantum States of Light

期刊论文
发表日期：

2025年1月1日

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Soumyabrata Paul,
V. Balakrishnan,
S. Ramanan,
S. Lakshmibala

摘要

Probability distributions play a central role in quantum mechanics, and even more so in quantum optics with its rich diversity of theoretically conceivable and experimentally accessible quantum states of light. Quantifiers that compare two different states or density matrices in terms of ‘distances’ between the respective probability distributions include the Kullback–Leibler divergence $D_\text{KL}$ $D_{KL}$ , the Bhattacharyya distance $D_\text{B}$ $D_{B}$ , and the p-Wasserstein distance $ W_{p}$ $W_{p}$ . We present a novel application of these notions to a variety of photon states, focusing particularly on the $p=1$ $p = 1$ Wasserstein distance $ W_{1}$ $W_{1}$ as it is a proper distance measure in the space of probability distributions.