On the Complexity of Finding the Maximum Entropy Compatible Quantum State
Abstract
:1. Introduction
2. Background and Problem Statement
 Input: A family of circuits$\mathcal{Q}\left(\mathcal{C}\right)$that construct the family of marginal density operators$\mathcal{C}$.
 Accept: if$\mathcal{C}$is admissible.
 Reject: if$\mathcal{C}$is not admissible.
 Input: A family of circuits $\mathcal{Q}\left(\mathcal{C}\right)$ promised to construct an admissible $\mathcal{C}$, and a real value k.
 Accept: if there exists a $\rho \in Comp\left(\mathcal{C}\right)$ such that $S\left(\rho \right)\ge k$
 Reject: otherwise.
3. Hardness of Comparing Entropy of a Compatible Chain
 Input: Two quantum circuits ${Q}_{0}$ and ${Q}_{1}$ that generate tripartite density operators ${\rho}_{0}$ and ${\rho}_{1}$, respectively, over the same Hilbert space of the form ${\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}\otimes {\mathcal{H}}_{C}$, promised that:
 ▸
 ${\mathrm{Tr}}_{A}\left({\rho}_{0}\right)={\mathrm{Tr}}_{A}\left({\rho}_{1}\right)$;
 ▸
 ${\mathrm{Tr}}_{C}\left({\rho}_{0}\right)={\mathrm{Tr}}_{C}\left({\rho}_{1}\right)$;
 ▸
 $S\left({\rho}_{0}\right)S\left({\rho}_{1}\right)\ge 1/2$;
then,  Accept: if $S\left({\rho}_{0}\right)S\left({\rho}_{1}\right)\ge 1/2$;
 Reject: if $S\left({\rho}_{1}\right)S\left({\rho}_{0}\right)\ge 1/2$.
 Input: Two quantum circuits ${Q}_{0}$ and ${Q}_{1}$, acting on m qubits, that prepare the states ${\rho}_{0}$ or ${\rho}_{1}$ promised that
 ▸
 either $\left\right{\rho}_{0}{\rho}_{1}{\left\right}_{tr}\ge \beta $;
 ▸
 or $\left\right{\rho}_{0}{\rho}_{1}{\left\right}_{tr}\le \alpha $;
then,  Accept:$\left\right{\rho}_{0}{\rho}_{1}{\left\right}_{tr}\ge \beta $,
 Reject:$\left\right{\rho}_{0}{\rho}_{1}{\left\right}_{tr}\le \alpha $.
 ${\rho}^{\prime}={\xi}_{0}^{A}\otimes {\zeta}^{AC}\otimes {\xi}_{0}^{C}\otimes 0\rangle {\langle 0}_{B}$;
 ${\rho}^{\u2033}={\xi}_{1}^{AC}\otimes {\xi}_{1}^{CA}\otimes 0\rangle {\langle 0}_{B}$.
4. Quantum Markov Chains and Trees
 it admits a QMT in the compatibility space iff every sub3chain is compatible with a QMC—Theorem 3;
 the QMT coincides with the density operator that maximizes the von Neumann entropy, constrained by the provided set of twobody marginals—Corollary 1;
 defining a proper order in the graph—constructive ordering—we can construct the unique compatible QMT directly from the marginals. The Lagrange multipliers in the optimization problem are then obtained through Theorem 2.
4.1. Background on Quantum Markov Chains
 1.
 ${\rho}_{ABC}$ is a QMC over the chain $ABC$.
 2.
 ${I}_{\rho}(A:CB)=0$, where ${I}_{\rho}(A:CB):=S\left({\rho}_{AB}\right)+S\left({\rho}_{BC}\right)S\left({\rho}_{B}\right)S\left({\rho}_{ABC}\right)$.
 3.
 ${\mathcal{P}}_{B\to BC}\left(X\right):={\rho}_{BC}^{\frac{1}{2}}(\left({\rho}_{B}^{\frac{1}{2}}X{\rho}_{B}^{\frac{1}{2}}\right)\otimes {id}_{C}){\rho}_{BC}^{\frac{1}{2}},\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}a\phantom{\rule{4.pt}{0ex}}\mathit{CPTP}\phantom{\rule{4.pt}{0ex}}\mathit{map}\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{any}\phantom{\rule{4.pt}{0ex}}X\in \mathcal{B}\left({\mathcal{H}}_{B}\right)$ and preserves the partial trace ${\rho}_{AB}$.
 4.
 $log{\rho}_{ABC}(log{\rho}_{AB})\otimes {id}_{C}={id}_{A}\otimes (log{\rho}_{BC}){id}_{A}\otimes (log{\rho}_{B})\otimes {id}_{C}$.
4.2. Definition of Quantum Markov Trees
4.3. QMT as MaxEntropy Density Operator
 (i)
 $log\rho ={\sum}_{\mathcal{C}}log{\rho}_{{X}_{i}{X}_{j}}{\sum}_{i=1}^{n}(deg\left(\right)open="("\; close=")">{X}_{i}1)$;
 (ii)
 we have$$\forall k=2,\dots ,n:\phantom{\rule{1.em}{0ex}}{\rho}_{k}={Tr}_{\overline{{V}_{k}}}\left[\rho \right]\phantom{\rule{4pt}{0ex}}\mathrm{is\; s.t.}\phantom{\rule{4pt}{0ex}}{I}_{{\rho}_{k}}\left(\right)open="("\; close=")">{X}_{k}:\overline{{Y}_{k}}{Y}_{k}$$for some constructive ordering ${X}_{1}<\cdots <{X}_{n}$.
 $log{\rho}_{j}={\sum}_{{\mathcal{C}}_{j}}log{\rho}_{{X}_{i}{X}_{t}}{\sum}_{i=1}^{j}({\mathrm{deg}}_{{\mathcal{G}}_{j}}\left(\right)open="("\; close=")">{X}_{i}1)$;
 ${I}_{{\rho}_{j}}\left(\right)open="("\; close=")">{X}_{j}:\overline{{Y}_{j}}{Y}_{j}$.
 $log{\rho}_{k+1}={\sum}_{{\mathcal{C}}_{k+1}}log{\rho}_{{X}_{i}{X}_{t}}{\sum}_{i=1}^{k+1}({\mathrm{deg}}_{{\mathcal{G}}_{k}}\left(\right)open="("\; close=")">{X}_{i}1)$ and,
 ${I}_{{\rho}_{k+1}}\left(\right)open="("\; close=")">{X}_{k+1}:\overline{{Y}_{k+1}}{Y}_{k+1}$.
4.4. Compatibility with a QMT
4.5. QMT and the MECM Problem
 1.
 ${\mathcal{G}}_{C}$ is a spanning tree
 2.
 ${\rho}_{ijk}$ is a QMC constructed in polynomialtime (with respect with the number of nodes n) where ${\rho}_{i,j},{\rho}_{j,k}\in \mathcal{C}$ and $i<j<k$ for some given constructive order of ${\mathcal{G}}_{C}$.
4.6. QMT and Chow–Liu Algorithm
Algorithm 1 Chow–Liu Algorithm 

5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
MDPI  Multidisciplinary Digital Publishing Institute 
DOAJ  Directory of open access journals 
TLA  Three letter acronym 
LD  linear dichroism 
QSZK  Quantum Statistical Zero Knowledge 
CPTP  Completely Positive Trace Preserving 
QMA  Quantum Merlin Arthur 
QCMP  Quantum Compatible Marginal Problem 
MECMP  Maximum Entropy Compatible Marginal Problem 
BQP  BoundedError Quantum Polynomial Time 
(3c)QED  (3chain) Quantum Entropy Difference 
QSD  Quantum Statistical Difference 
QMC  Quantum Markov Chain 
QMT  Quantum Markov Tree 
Appendix A. Lemmas for Theorem 1
Appendix B. Proof of the Central Lemma 1
 (i)
 ${\varphi}^{\u2020}\left(\right)open="("\; close=")">\varphi {\left(\right)}^{{\rho}_{2}}it$$t\in \mathbb{R}$;
 (ii)
 ${\varphi}^{\u2020}\left(\right)open="("\; close=")">log\varphi \left(\right)open="("\; close=")">{\rho}_{1}$;
 (3 ⇒ 1) This implication comes for free from the definition of QMC. Moreover, the map ${\mathcal{P}}_{B\to BC}\left(\xb7\right)$ is clearly CPTP. The complete positivity indeed comes for free from the Hermitianicity of ${\rho}_{B}\otimes {\mathrm{id}}_{C}/{d}_{C}$ and ${\rho}_{BC}$, then of their squareroots.
 (1 ⇒ 3) Equation (A8) for t = 0 gives exactly the Petz map in (3), so the implication comes as corollary of Theorem A3.
 (1 ⇔ 2) This follows from the statement of Theorem A3.
 (2 ⇔ 4) It comes as corollary of Theorem A2, using the settings in Corollary A1.
Appendix C. Lemmas for Theorem 2 and 3
 (a)
 If the associate graph${\mathcal{G}}_{\mathcal{C}}$is a chain ABCD (i.e.,$\mathcal{C}=\{{\rho}_{AB},{\rho}_{BC},{\rho}_{CD}\}$) then
 (b)
 If the associate graph${\mathcal{G}}_{\mathcal{C}}$is a star centered in B (i.e.,$\mathcal{C}=\{{\rho}_{AB},{\rho}_{BC},{\rho}_{BD}\}$)
 (i)
 $\exists \phantom{\rule{0.166667em}{0ex}}{\tilde{\rho}}_{CBD}\in Comp({\rho}_{BC},{\rho}_{BD})$, ${\tilde{\rho}}_{BCD}\in \mathcal{B}\left(\right)open="("\; close=")">{\mathcal{H}}_{BCD}$ s.t. ${I}_{\rho}\left(\right)open="("\; close=")">C:DB$ and
 (ii)
 $\exists \phantom{\rule{0.166667em}{0ex}}{\tilde{\rho}}_{ABD}\in Comp({\rho}_{AB},{\rho}_{BD})$, ${\tilde{\rho}}_{ABD}\in \mathcal{B}\left(\right)open="("\; close=")">{\mathcal{H}}_{ABD}$ s.t. ${I}_{\rho}\left(\right)open="("\; close=")">A:DB$.
Appendix D. Number of 3Chains
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Di Giorgio, S.; Mateus, P. On the Complexity of Finding the Maximum Entropy Compatible Quantum State. Mathematics 2021, 9, 193. https://doi.org/10.3390/math9020193
Di Giorgio S, Mateus P. On the Complexity of Finding the Maximum Entropy Compatible Quantum State. Mathematics. 2021; 9(2):193. https://doi.org/10.3390/math9020193
Chicago/Turabian StyleDi Giorgio, Serena, and Paulo Mateus. 2021. "On the Complexity of Finding the Maximum Entropy Compatible Quantum State" Mathematics 9, no. 2: 193. https://doi.org/10.3390/math9020193