Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems

We establish a holographic duality between <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi></a:math>-dimensional mixed-state symmetry-protected topological (mSPT) phases and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo stretchy="false">(</c:mo><c:mi>d</c:mi><c:mo>+</c:mo><c:mn>1</c:mn><c:mo stretchy="false">)</c:mo></c:math>-dimensional subsystem symmetry-protected topological (SSPT) states. Specifically, we show that the reduced density matrix of the boundary layer of a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">(</g:mo><g:mi>d</g:mi><g:mo>+</g:mo><g:mn>1</g:mn><g:mo stretchy="false">)</g:mo></g:math>-dimensional SSPT state with subsystem symmetry <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi mathvariant="script">S</k:mi></k:mrow></k:math> and global symmetry <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mrow><n:mi mathvariant="script">G</n:mi></n:mrow></n:math> corresponds to a <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>d</q:mi></q:math>-dimensional mSPT phase with strong <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mrow><s:mi mathvariant="script">S</s:mi></s:mrow></s:math> and weak <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mrow><v:mi mathvariant="script">G</v:mi></v:mrow></v:math> symmetries. Conversely, we demonstrate that the wave function of an SSPT state can be constructed by replicating the density matrix of the corresponding lower-dimensional mSPT phase. This mapping links the density matrix in lower dimensions to the entanglement properties of higher-dimensional wave functions, providing an approach for analyzing nonlinear quantities and quantum information metrics in mixed-state systems. Our duality offers a new perspective for studying intrinsic mSPT phases that are unique to open quantum systems, without pure-state analogues. We show that strange correlators and twisted Rényi-<y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>N</y:mi></y:math> correlators can diagnose these nontrivial phases and we explore their connection to strange correlators in pure-state SSPT phases. Furthermore, we discuss several implications of this holographic duality, including a method for preparing intrinsic mSPT phases through the duality.