Tensorization of the strong data processing inequality for quantum chi-square divergences

It is well-known that any quantum channel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi></mml:mrow></mml:math> satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>κ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:math> divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> does not increase under the action of any quantum channel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi></mml:mrow></mml:math>. For a fixed channel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi></mml:mrow></mml:math> and a state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math>, the divergence between output states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> might be strictly smaller than the divergence between input states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math>, which is characterized by the strong data processing inequality (SDPI). Among various input states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math>, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>κ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:math> divergence for arbitrary quantum channels and also for a family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>κ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:math> divergences (with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>) for arbitrary quantum-classical channels.