Quantum Information and Quantum Computation
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Updated 7 October 2026

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Entropy photon-number inequality

Does the entropy photon-number inequality hold for every pair of independent finite-energy bosonic inputs? Let \(n\geq1\) and \(0\leq\eta\leq1\). The input is \(\rho_A\otimes\rho_B\), where each factor is an \(n\)-mode state with finite total mean photon number. Correlations among the modes within either factor are allowed. Mix corresponding input modes on beam splitters and retain the outputs

\begin{equation} c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad j=1,\ldots,n. \tag{1} \end{equation}

Let \(\rho_C\) be the joint state of the modes in Eq. (1). Use natural logarithms, \(S(\rho)=-\operatorname{Tr}\rho\ln\rho\), and \(g(x)=(x+1)\ln(x+1)-x\ln x\) for \(x\geq0\), with \(0\ln0=0\). The proposed inequality is

\begin{equation} g^{-1}\!\left(\frac{S(\rho_C)}n\right) \geq\eta g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)g^{-1}\!\left(\frac{S(\rho_B)}n\right). \tag{2} \end{equation}

Prove Eq. (2) in this full domain or provide a physical input pair that violates it.

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The PPT-squared conjecture

Must the composition of any two compatible PPT completely positive maps be entanglement breaking? Let \(\Phi:M_{d_1}(\mathbb C)\to M_{d_2}(\mathbb C)\) and \(\Psi:M_{d_2}(\mathbb C)\to M_{d_3}(\mathbb C)\) be completely positive. The Choi operator of \(\Phi\) is

\begin{equation} J(\Phi) :=\sum_{i,j=1}^{d_1}\lvert i\rangle\!\langle j\rvert \otimes\Phi(\lvert i\rangle\!\langle j\rvert), \tag{1} \end{equation}

and \(J(\Psi)\) is defined analogously. In the convention of Eq. (1), a map \(\Phi\) is PPT when

\begin{equation} (T\otimes\operatorname{id})\bigl(J(\Phi)\bigr)\succeq0, \tag{2} \end{equation}

and it is entanglement breaking when \(J(\Phi)\) is separable; the same definitions apply to \(\Psi\). Under condition Eq. (2) for both maps, is \(J(\Psi\circ\Phi)\) necessarily separable? Equivalently, does postselection on any joint measurement outcome on the middle systems of two PPT bipartite states always leave a separable state on the two outer systems?

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