Convexity properties of superpositions of degenerate bipartite eigenstates

TL;DR

This paper investigates the convexity properties of entanglement measures in superpositions of degenerate bipartite eigenstates, revealing predictable convexity behavior based on shared entropy, with exact analysis of specific two-particle systems.

Contribution

It demonstrates the convexity properties of von Neumann entropy in superpositions of degenerate eigenstates and provides a method to predict these properties using shared entropy measures.

Findings

Von Neumann entropy exhibits definite convexity or concavity as a function of superposition parameter.

Shared entropy at superposition extremes predicts the convexity behavior.

Exact analysis of two-particle systems confirms theoretical predictions.

Abstract

The entanglement content of superpositions of pairs of degenerate eigenstates of a bipartite system are considered in the case that both are also eigenstates of the $z$ component of the total angular momentum. It is shown that the von Neumann entropy of the state that is obtained tracing out one of the parts of the system has a definite convexity (concavity) as a function of the superposition parameter and that its convexity (concavity) can be predicted using a quantity of information that measures the entropy shared by the states at the extremes of the superposition. Several examples of two particle system, whose eigenfunctions and density matrices can be obtained exactly, are analyzed thoroughly.

Tables

Table 1. Table 1: Entropies, convexity and the Q c subscript 𝑄 𝑐 Q_{c} value for the two cases of non-interacting oscillators, λ = 0 𝜆 0 \lambda=0 , shown in Figure  1 ( | m + p | = 1 𝑚 𝑝 1 |m+p|=1 and | m + p | = 2 𝑚 𝑝 2 |m+p|=2 ). Note that in the first two columns a negative quantum number, − n 𝑛 -n , is denoted as n ¯ ¯ 𝑛 \underline{n} .

$nmlp$	$n^{\prime }{m}^{\prime }{l}^{\prime }{p}^{\prime }$	convexity	$Q_c$	$S_{vn}$	$S_{NS}$	$S_R$
$\left|m+p\right|=1$
0031	0031	convex	+	3.907	0	3.907
0031	3100	convex	+	3.907	0	3.907
1120	0031	concave	–	3.704	1.959	1.745
$\left|m+p\right|=2$
0022	0022	convex	+	3.94	0	3.94
0022	1111	concave	–	3.94	2.85	1.09
0022	1111	concave	–	3.94	2.85	1.09
1111	1111	convex	+	2.94	0	2.94

Table 2. Table 2: Entropies, convexity and the Q c subscript 𝑄 𝑐 Q_{c} value for the two cases of interacting oscillators, λ = 0.7 𝜆 0.7 \lambda=0.7 , shown in Figure  2 ( | m + p | = 1 𝑚 𝑝 1 |m+p|=1 and | m + p | = 2 𝑚 𝑝 2 |m+p|=2 ).

$nmlp$	$n^{\prime }{m}^{\prime }{l}^{\prime }{p}^{\prime }$	convexity	$Q_c$	$S_{vn}$	$S_{NS}$	$S_R$
$\left|m+p\right|=1$
1100	1100	convex	+	2.717	1.034	1.683
2100	2100	convex	+	3.487	1.121	2.366
$\left|m+p\right|=2$
0200	0200	concave	–	1.776	1.123	0.653
1200	1200	concave	–	3.006	1.740	1.266

Table 3. Table 3: The values of the convexity, the criterion prediction and entropies of the curves shown in Figure  3 .

$LM$	$L{M}^{\prime }$	convexity	$Q_c$	$S_{NS}$	$S_R$
21	21	convex	+	0.917	1.584
22	22	concave	–	1.422	0.578

Table 4. Table 4: The values of the convexity, the criterion prediction and the entropies of the curves shown in Figure  4 .

$lm$	$l^{\prime }{m}^{\prime }$	convexity	$Q_c$	$S_{NS}$	$S_R$
11	11	convex	+	0	0.796
21	21	convex	+	0	0.894
21	22	convex	+	0.194	0.700
11	22	convex	+	0.187	0.608
33	33	convex	+	0	1.323

Table 5. Table 5: The values of the convexity, the criterion prediction and the entropies of the curves shown in Figure  5 .

$L$	$\left|M\right|$	convexity	$Q_c$	$S_{NS}$	$S_R$
1	1	convex	+	0.196	1.558
2	2	convex	+	0.550	0.997
3	3	concave	–	1.031	0.299
4	4	concave	–	1.067	0
5	5	concave	–	0.693	0
6	6	concave	0	0	0

Equations

$${\rho }_A=\text{mbox}{Tr}_B(|\psi ⟩⟨\psi |),{\rho }_B=\text{mbox}{Tr}_A(|\psi ⟩⟨\psi |).$$

$${\rho }_A{\phi }_i^A={\lambda }_i^A{\phi }_i^A,$$

$${\rho }_B{\phi }_i^B={\lambda }_i^B{\phi }_i^B,$$

$$S(\rho \Vert \sigma )=-\text{mbox}Tr\left(\rho \log \sigma \right)-S(\rho )=\mathrm{Tr}\rho (\log \rho -\log \sigma ),$$

$$S(\rho )=-\text{mbox}Tr\rho \log \rho .$$

$$\rho =\sum _i{p}_i|v_i⟩⟨v_i|,\sigma =\sum _i{q}_i|w_i⟩⟨w_i|,$$

$$S(\rho \Vert \sigma )=\sum _i{p}_i\left(\log p_i-\sum _j(\log q_j)P_{ij}\right),$$

$$|{\psi }_{\alpha }⟩=\sqrt{\alpha }|{\psi }_0⟩+\sqrt{1-\alpha }|{\psi }_1⟩;0\le \alpha \le 1.$$

$$S_{ns}({\rho }_{A,0})=-\sum _{{\lambda }_i^{A,0}>0}\Theta \left[{\lambda }_i^{A,0}-{⟨{\rho }_{A,1}⟩}_i\right]{\log }_2({\lambda }_i^{A,0}),$$

$${\rho }_{A,\alpha }=\text{mbox}{Tr}_B(|{\psi }_{\alpha }⟩⟨{\psi }_{\alpha }|),$$

$$S_{ns}({\rho }_{A,0})$$

$$S_{NS}({\rho }_{A,0})$$

$$\tilde{S}=\min \left\{\Theta \left[\lambda -{⟨{\rho }_{A,1}⟩}_1\right]+\Theta \left[\lambda -{⟨{\rho }_{A,1}⟩}_2\right]\right\}{\log }_2(1/\lambda ),$$

$$\tilde{S}=\left\{\begin{array}[]{lcl}(2\lambda-(\lambda_{1}+\lambda_{2}))\log_{2}(1/\lambda)&&\mbox{if}\quad\lambda>\displaystyle{\frac{\lambda_{1}+\lambda_{2}}{2}}\\ 0&&\mbox{otherwise}\end{array}\right..$$

$$S_R({\rho }_{A,0})=S({\rho }_{A,0})-S_{NS}({\rho }_{A,0}).$$

$$S({\rho }_{A,\alpha })\le \alpha S({\rho }_{A,0})+(1-\alpha )S({\rho }_{A,1})\text{mbox}if{S}_{NS}({\rho }_{A,0})<S_R({\rho }_{A,0})$$

$$S({\rho }_{A,\alpha })\ge \alpha S({\rho }_{A,0})+(1-\alpha )S({\rho }_{A,1})\text{mbox}if{S}_{NS}({\rho }_{A,0})>S_R({\rho }_{A,0}).$$

$$Q_c=sgn(S_R({\rho }_{A,0})-S_{NS}({\rho }_{A,0})),$$

$$S_R=-\sum _i\min \left({\lambda }_i^{A,0},{\lambda }_i^{A,1}\right){\log }_2({\lambda }_i^{A,0}),$$

$$H(\overrightarrow{x_1},{\vec{x}}_2)=-\frac{1}{2}{▽}_1^2-\frac{1}{2}{▽}_2^2+\frac{1}{2}|{\vec{x}}_1{|}^2+\frac{1}{2}|{\vec{x}}_2{|}^2+\lambda |{\vec{x}}_1-{\vec{x}}_2{|}^2$$

$$\vec{R}={\vec{x}}_1+{\vec{x}}_2,\vec{r}={\vec{x}}_1-{\vec{x}}_2,$$

$$H(\vec{R},\vec{r})$$

$$H_{\omega }|n,m⟩$$

$$L_z^{(1)}|n,m⟩$$

$$H|n,m,l,p⟩$$

$$L_z|n,m,l,p⟩=(m+p)|n,m,l,p⟩,$$

$${\rho }_A({\vec{x}}_1,{\vec{x}}_1^{\prime })=\int d^2{x}_2{\psi }_{n,m,l,p}^{\ast }({\vec{x}}_1,{\vec{x}}_2){\psi }_{n,m,l,p}({\vec{x}}_1^{\prime },{\vec{x}}_2).$$

$$H=-\frac{1}{2}{\nabla }_1^2-\frac{1}{2}{\nabla }_2^2+V(r_1,r_2,r_{12}).$$

$$\psi ({\text{r}}_1,{\text{r}}_2)=\sum _{l_1=d_0}^L{\Phi }_{l_1{l}_2}(r_1,r_2,r_{12}){\Omega }_{l_1,l_2}^{L,M}({\hat{r}}_1,{\hat{r}}_2),$$

$$d_{0}=L+1-d,\qquad\mbox{and}\quad d=\left\{\begin{array}[]{lcl}L+1&\mbox{if}&\pi=(-1)^{L}\\ L&\mbox{if}&\pi=(-1)^{L+1}\end{array}\right..$$

$${\mathbf{L}}^2{\Omega }_{l_1,l_2}^{L,M}({\hat{r}}_1,{\hat{r}}_2)=L(L+1){\Omega }_{l_1,l_2}^{L,M}({\hat{r}}_1,{\hat{r}}_2),$$

Full text

I Introduction

As the old recipe goes, the only thing that someone needs to prepare a pure state with well defined quantum numbers is a complete set of compatible observables of the quantum system whose states are to be prepared. Then, applying a given sequence of one-dimensional projectors to an arbitrary pure state, it is possible to label the resulting state with the quantum numbers associated to each projector. The sequence contains only one projector for each observable which is taken from the corresponding spectral decomposition. Only if the observable set is complete the state at the end of the sequence will be an element of the basis that expands the Hilbert space of the system and, consequently, can not be written as a superposition of states generated following the recipe but with different projectors. Of course, one-dimensional projectors are elusive objects to be actually constructed, in particular when the observable has a continuum spectrum.

The drive to process quantum information, to control the states where it is stored and to palliate the unwanted effects of decoherence mechanisms has resulted in a host of methods to produce pure or almost pure states in a reliable and repetitive way in different quantum systems. A short list of examples includes, effective pure states in Magnetic Nuclear Resonance Gershenfeld1997 ; Cory1997 ; Vandersypen2001 , state preparation of coupled electron states in quantum dots Petta2005 and synthesis of arbitrary states in superconducting qubits Hofheinz2009 . Moreover, the availability of quantum states made of specific superpositions of pure states or measurement-prepared states could improve the means to perform a given quantum information task Knill2001 ; Abrams1999 ; Obrien2009 ; lofranco .

Some years ago, Vaziri, Weihs and Zeilinger Vaziri2002 showed that for a particular Quantum Information task, Quantum Cryptography, the availability of photon states made of superpositions of orbital angular momentum eigenstates resulted in an expanded alphabet to be used to code information, with possibilities beyond the simpler alphabet formed by the two polarization states. In Reference Vaziri2002 superpositions between Gaussian and Laguerre-Gaussian states were considered, but over the years numerous other examples of states that are composed of different orbital angular momentum states have been analyzed including photon pairs with entangled orbital angular momentum Jack2009 , composite Laguerre-Gaussian beams with tunable intensity and phase distribution Parisi2014 and elliptic Gaussian optical vortices Kotlyar2017 . Besides, it has been shown experimentally that photonic states with large orbital angular momentum, often called qudits, can be cycled among them using standard optical elements Schlederer2016 .

In other physical systems, to our knowledge, there are neither the requirement of an specific superposition of orbital angular momentum states to perform a task, nor a protocol to achieve it, as it is for the neat example for photons presented above. Anyway, this scenario is changing because of the appearance of experiments and theoretical proposals in which the properties of a photonic state are transferred to a condensate system Dall2014 .

Nevertheless, there are at least two reasons to consider superpositions of degenerate states. One comes from the properties of many-body eigenstates, and another one arises from considerations about the Hilbert space expanded by all the superpositions that can be formed using a set of pure states. When considering many-body models the eigenstates of the Hamiltonian are, most commonly, degenerate and depending on the problem that is under study the entanglement content could be calculated for a mixture or for a superposition of them. This subtle point was early acknowledged by Osborne and Nielsen Osborne2002 . They were studying the behaviour of the entanglement in the quantum phase transition that appears in the transverse Ising model and discussed the differences in entanglement content between the low temperature limit of the Gibbs state, that is a equally weighted mixture of the two degenerate ground states that the system admits, and the entanglement of one of those ground states.

Since the work of Osborne and Nielsen Osborne2002 , the entanglement content of the degenerate eigenstates of many-body problems has been addressed to asses the relationship between long-range interactions and the entanglement of spin pairs separated at different lengths Gaudiano2008 , the relationship between entanglement and symmetry in permutation-symmetric states Markham2011 , the difference between symmetrical superpositions of ground states and symmetry breaking ones using mutual information Hamma2016 , the entanglement properties of the whole set of eigenstates of different Hamiltonians Vidmar2017a ; Vidmar2017b . For instance, in Reference Gaudiano2008 the entanglement was calculated for equally weighted mixtures of degenerate eigenstates, while Markham in Reference Markham2011 considers superpositions of states where its coefficients are given by complex numbers with modulus equal to one, so the normalization constant of the superposition is equal to the number of states that enters in it.

On the other hand, when all the superpositions of a set of $N$ eigenstates are considered, since they are contained in a compact space, a given continuous functional of the states should reach a set of extrema. So, any given entanglement measure will reach a number of extrema. But, for which superpositions are those extrema achieved? Moreover, the entanglement measure of superpositions of states is convex or concave as a function of the superposition parameter? Besides, can these convexity properties be predicted from the knowledge of the states that are being superposed?

Some of us started to contemplate these questions after working with the Calogero model Garagiola2016 . The Calogero model has many well documented properties and applications varias-Calogero , but the one that caught our attention and was the reason to start this work is the following: the two-particle two dimensional case has a degenerate ground state if the total wave function considered is anti-symmetric under particle permutation. Most commonly, the eigenfunctions are chosen as eigenfunctions of the angular momentum operator component perpendicular to the two dimensional plane where the system inhabits, let us call them $\psi_{\pm}$. Other usual choice for the basis eigenfunctions are the combinations $(\psi_{+}\pm\psi_{-})/\sqrt{2}$. Denoting by $S(\psi)$ the von Neumann entropy of the reduced density operator obtained tracing out one of the particles from the two-particle density operator, $\rho=|\psi\rangle\langle\psi|$, we obtained Garagiola2016 that $S(\xi)\leq S(\psi_{+})=S(\psi_{-})$ for all $\xi=\sqrt{\alpha}\psi_{+}+\sqrt{1-\alpha}\psi_{-}$, where $0\leq\alpha\leq 1$. This finding led us to analyze more general examples.

Suppose that ${L_{z}}$ commutes with $H$, where $H$ is the Hamiltonian with degenerate eigenstates labeled as $\left|\psi_{i}\right\rangle$, and ${L_{z}}\left|\psi_{i}\right\rangle=m_{i}\left|\psi_{i}\right\rangle$, where $m_{i}$ are some eigenvalues. Consider a superposition $\chi=\sum_{i}\sqrt{\alpha_{i}}\psi_{i}$, such that $S(\chi)$ is convex or concave as a function of the parameters $\alpha_{i}$, which satisfy that $\sum_{i}\alpha_{i}=1$ and $0\leq\alpha_{i}\leq 1$. In this work we study only superpositions of degenerate states with eigenvalues of the form $\pm m\hbar$. We give a criterion that predicts the convexity or concavity of the entropy of the superposition. To avoid the uncertainty and difficulties that arise when numerical solutions are required, we consider a two-particle exactly solvable model, two-interacting harmonic oscillators; a quasi-exactly solvable one, the spherium spherium ; dehesa_spherium ; a one-particle exactly solvable model, the Laguerre-Gaussian one photon wave function kok ; and the sum of two angular momentum operators. This set of examples has been chosen because the one-party reduced density matrix can be exactly calculated and its eigenvalues can be obtained to arbitrary precision. For bipartite two-particle models one particle must be traced out from the whole two-particle density matrix, while for the one-particle model what is traced out is one of the two relevant spatial degrees of freedom. This procedure has been used to assess the separability of a given wave function as a product of two functions that depend on separate variables Garagiola2018 and has been also considered to assess the entanglement between degrees of freedom of one particle systems, for instance polarization and spatial degrees of freedom in one-photon states one-particle-entanglement or Rydberg-like harmonic states dehesa_rydberg .

The paper is organized as follows, in Section II a criterion to predict the convexity properties of the entropy of superposition of eigenstates is presented. In Sections III, IV, V and VI we present the calculation of the eigenstates, entropies and the criterion for two interacting oscillators, two electrons confined in the surface of a sphere (the spherium), one photon Laguerre-Gaussian states and the addition of two angular momentum operators, respectively. We defer a number of mathematical details to the Appendices. Finally in Section VII we discuss our results and perspectives of the research.

II The criterion

Consider a bipartite composite system with Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are the Hilbert spaces of the two subsystems $A$ and $B$, whose dimensions are equal, $dim(\mathcal{H}_{A})=dim(\mathcal{H}_{B})$.

For a given pure state in $\mathcal{H}$, $\left|\psi\right\rangle$, the reduced density matrices $\rho_{A}$ and $\rho_{B}$ are given by

[TABLE]

Both reduced density matrices are isospectral and have associated an eigenvalue problem, i.e.

[TABLE]

and

[TABLE]

where $\lambda_{i}^{A}$ ($\lambda_{i}^{B}$) are the eigenvalues and $\varphi_{i}^{A}$ ($\varphi_{i}^{B}$) the eigenvectors of $\rho_{A}$ ($\rho_{B}$). It is convenient to introduce other two Hilbert spaces, $V_{A}=span(\{\varphi_{j}^{A}\})$ where the only eigenvectors $\varphi_{j}^{A}$ that enter in the $span$ are those such that its corresponding eigenvalues satisfy $\lambda_{j}^{A}>0$, and equivalently for $V_{B}$.

The considerations over the Hilbert spaces defined above become clear when computing the quantum relative entropy, a quantity commonly used to compare two quantum states, $\rho$ and $\sigma$. The quantum relative entropy of $\rho$ with respect to $\sigma$ is given by relative-entropy

[TABLE]

where $S(\rho)$ is the von Neumann entropy

[TABLE]

If $\mbox{supp}(\rho)\bigcap\mbox{ker}(\sigma)\neq 0$ then $S(\rho\|\sigma)=\infty$, where $\mbox{supp}(\rho)$ y $\mbox{ker}(\sigma)$ stand for the support and kernel of the $\rho$ and $\sigma$ states, respectively. The divergence and other properties of the relative entropy can be analyzed more directly using the spectral decompositions

[TABLE]

in terms of which the quantum relative entropy can be calculated as

[TABLE]

where $P_{ij}=|\langle v_{i}|w_{j}\rangle|^{2}$.

Some remarks are in order. If the states $\rho$ and $\sigma$ have degenerate eigenvalues, each one of the decompositions in Equation 6 are not unique, since the corresponding eigenvectors $|v_{i}\rangle$ or $|w_{i}\rangle$ can be chosen in different ways. This possibility leads to the undesirable result that the relative entropy could depend on a particular election of the eigenvectors that corresponds to a degenerate eigenvalue. Since we intend to introduce an information-like quantity that allow us to compare the reduced density matrices of different superpositions of Hamiltonian eigenstates, the quantity to be defined must be calculable even when the two drawbacks mentioned above are present.

If $H$ is the Hamiltonian of the composite system, we will focus in some particular superpositions of degenerate bound states which are also eigenstates of the $z$ component of the total angular momentum $L_{z}$. In particular, if $\left|\psi_{0}\right\rangle$ and $\left|\psi_{1}\right\rangle$ are two such eigenstates, we will consider the superposition

[TABLE]

Let us define the not-shareable entropy of $\rho_{A,0}$ with respect to $\rho_{A,1}$ as

[TABLE]

where

[TABLE]

$\left\langle\rho_{A,1}\right\rangle_{i}=\mbox{Tr}(P^{A,0}_{i}\rho_{A,1})$, $P^{A,0}_{i}$ is a one-dimensional projector associated to $\lambda_{i}^{A,0}$, and $\Theta[x]=x\,\theta(x)$, where $\theta(x)$ is the Heaviside step function. It is clear that when the eigenvalues of $\rho_{A,0}$ are degenerate the projectors $P^{A,0}_{i}$ are not uniquely defined. So, decomposing the sum in Equation 9 in terms of the degenerate and non-degenerate eigenvalues we get that

[TABLE]

where the first sum runs over the different eigenvalues and the second one over the degeneracy of the corresponding eigenvalue. With the previous definitions, we now define the entropy in which we will found our study. The not-shared entropy of $\rho_{A,0}$ with respect to $\rho_{A,1}$ is given by

[TABLE]

where the minimum must be obtained in each degenerate subspace associated to a degenerate eigenvalue of $\rho_{A,0}$. The minimum can be obtained analytically for low degeneracy proceeding as follows. To simplify the notation let us call $\lambda$ the degenerate eigenvalue of interest and $\lambda_{1}$ and $\lambda_{2}$ the eigenvalues of $\rho_{A,1}$, it is clear that we are assuming a twofold degeneracy. In the corresponding subspace, $\left\langle\rho_{A,1}\right\rangle_{1}+\left\langle\rho_{A,1}\right\rangle_{2}=\lambda_{1}+\lambda_{2}$, so

[TABLE]

can be calculated explicitly as

[TABLE]

Once the not-shared entropy is defined we introduce, in a similar fashion, the remaining entropy of state $\rho_{A,0}$, which is given by

[TABLE]

We are now in conditions to state the criterion about the convexity properties of the entropy of a superposition of two degenerate states, like the one defined in Equation 8. If both states, $\psi_{0}$ and $\psi_{1}$, are eigenfunctions of the $z$ component of the total angular momentum operator, $L_{z}$, with eigenvalues $\pm m\hbar$, then

[TABLE]

and

[TABLE]

The criterion predicts exactly the convexity properties expected for the entropy, and this can be noted by the quantity,

[TABLE]

which has the value $+1$ when the entropy $S(\rho_{A,\alpha})$ is convex and $-1$ when it is concave.

Then, the criterion establishes a relationship between the convexity of the entropy of a superposition state with an entropic quantity that depends only on the extremal states of such superposition. In the following sections we will test the criterion in several systems and show how it correctly predicts how state superposition can lead to more or less information entropy.

It is worth to point that the criterion can be stated also in terms of the state $\rho_{A,1}$ with respect to the state $\rho_{A,0}$. Even more, its seems desirable to state a criterion in which both states play more symmetrical roles. For the moment we prefer to use the criterion as stated from Equation 9 trough Equation 18. The criterion compares a given state, say $\rho_{A,0}$ with another, say $\rho_{A,1}$, so the not-shared entropy contains information of $\rho_{A,0}$ that can not be obtained from state $\rho_{A,1}$ measuring the last in the basis of eigenprojectors of the former. If $V_{A}\bigcap V_{B}=\phi$ then $S_{NS}(\rho_{A,0})=S(\rho_{A,0})$. In this sense, the not-shared entropy measures how different are the states when both are measured using the projectors associated to one of them.

The remaining entropy quantifies how much information can be obtained about $\rho_{A,0}$ when $\rho_{A,1}$ is measured using the projectors associated to $\rho_{A,0}$. This can be seen rather directly when both states, $\rho_{A,0}$ and $\rho_{A,1}$, have non-degenerate eigenvalues and the same set of eigenvectors. In that case

[TABLE]

and the sum runs over the eigenvalues $\lambda_{i}^{A,0}$ whose eigenvectors belong to $V_{A}\bigcap V_{B}$.

The not-shared entropy, as proposed in Equation 12, depends on the spectral decomposition of both reduced density operators $\rho_{A,0}$ and $\rho_{A,1}$. This dependency does not imply a restriction from the point of view of the calculations in concrete systems but, as is the case for entanglement measures, a definition in terms of a minimization over a Hilbert space can offer a more general point of view or an operational procedure to determine the not-shared entropy. We will return to this subject once the example considering states in finite dimensional Hilbert spaces, Section VI, has been analyzed thoroughly.

In the next Sections we apply the criterion to several bipartite systems. The different Hamiltonians allow us to carry the calculations to obtain the reduced density matrices analytically.

III Two interacting two-dimensional oscillators

A well known exactly solvable problem consists of two particles interacting harmonically confined in an harmonic trap. We use units such that $\hbar=1$, $m=1$, $\omega=1$, where $m$ is the mass and $\omega$ is the frequency of both oscillators. The Hamiltonian is given by

[TABLE]

where $\vec{x}_{i}=(x_{i},y_{i})$, $\bigtriangledown_{i}^{2}$ is the two-dimensional Laplacian, and $i=1,2$ is the index numbering the particles.

The parameter $\lambda$ allows to switch from a non-interacting system, $\lambda=0$, to an interacting one, $\lambda>0$. The symmetries and quantum numbers of the wave functions can be chosen at convenience. Since the superpositions in which we are interested are composed of eigenfunctions whose energy eigenvalues are degenerate but with different angular momentum eigenvalue, it is useful to derive expressions for the wave functions in coordinates where the system can be recast as a non-interacting one. So, introducing the centered and relative coordinates

[TABLE]

respectively, the Hamiltonian in Equation 20 can be written as

[TABLE]

where $H_{\omega}$ is a two-dimensional harmonic oscillator Hamiltonian, with frequency $\omega$ and mass equal $1/2$. The oscillator that depends on the relative coordinates has $\omega_{r}=\sqrt{4\lambda+1}$, while the other, that depends on the centered coordinates, has $\omega_{R}=1$. The exact eigenfunctions and eigenvalues of the one-particle Hamiltonians in Equation III are well known in several coordinate systems Cohen . The two-particle eigenfunctions can now be written as a product of a pair of one-particle ones.

The one-particle cylindrical eigenvectors, $\left|m,n\right\rangle$, satisfy

[TABLE]

where $\omega$ is either $\omega_{R}$ or $\omega_{r}$, and $L^{(1)}_{z}$ is the $z$-component of the one-particle angular momentum.

Collecting the results given above, the two-particle eigenvector, $\left|n,m,l,p\right\rangle=\left|n,m\right\rangle\left|l,p\right\rangle$, satisfy

[TABLE]

and

[TABLE]

where $L_{z}=L_{z_{R}}+L_{z_{r}}$.

Equations 24 and 25 show that, when $\sqrt{4\lambda+1}$ is a natural number, there are many different ways to combine the quantum numbers $n,m,l$ and $p$ to obtain degenerate eigenfunctions with different values of angular momentum quantum number.

The traces in Equation 1 over $A$ and $B$ have the meaning of particle $1$ and $2$. Hence the reduced density matrix of a particular angular momentum state is

[TABLE]

The states $\left|\psi_{0}\right\rangle$ and $\left|\psi_{1}\right\rangle$ are now identified with two different angular momentum states $\left|n,m,l,p\right\rangle$ that have the same energy and opposite eigenvalues for $L_{z}$. The reduced density matrix for the mixed state $\rho_{A,\alpha}$ is defined analogously to $\rho_{A}(\vec{x}_{1},\vec{x}_{1}^{\prime})$ in Equation 26. We explain how to obtain an exact expression for $\rho_{A}(\vec{x}_{1},\vec{x}_{1}^{\prime})$ and how to compute its eigenvalues in Appendix B.

III.1 Non-interacting harmonic oscillators, $\lambda=0$

The bipartite states are labeled with the four quantum numbers $n,m,l,p$. In this section we will consider superpositions of states that satisfy $|\,m+p\,|=1$ or $|\,m+p\,|=2$, there is no a priori restriction to the values of the quantum numbers $n$ and $l$ but the one imposed by Equation 24. However, since we use a finite basis from which we obtain the eigenvalues of $\rho_{A,\alpha}$, we consider the states for $n,l\leq 3$.

Figure 1 shows the behaviour of the von Neumann entropy as a function of the superposition parameter $\alpha$. Both possible curvatures, or convexities, can be clearly appreciable according with the states that enter in the superposition. In Table 1 the values of von Neumann, remaining and not-shared entropies, and $Q_{c}$ are listed for all the cases shown in Figure 1. The criterion predicts correctly what convexity is to be expected.

III.2 Interacting harmonic oscillators, $\lambda\neq 0$

Despite that a two-particle system, both confined by an harmonic potential and interacting harmonically, is exactly solvable, the evaluation of the necessary matrix representation of a given wave function (see Appendix B) becomes quite taxing. Because of this, it is simpler, and computationally faster, to obtain very accurate approximate variational wave functions using basis set functions that are products of one-particle functions. Once these variational eigenfunctions are calculated they can be used to evaluate matrix representations and approximate eigenvalues of the reduced density matrix corresponding to a given variational eigenfunction. The procedure to calculate each one of these quantities has been described elsewhere, see for instance Garagiola2016 ; Ho and References therein. The accuracy of the whole procedure can be assessed in different ways, mainly comparing the variational eigenvalues with the corresponding exact values computed using Equation 24. For all the cases that are discussed in this Section the variational eigenvalues corresponding to the variational eigenfunctions used to construct superpositions differ from the exact ones in less than $1\times 10^{-4}$.

One of the drawbacks of the variational method comes from the fact that assigning quantum numbers to the variational eigenfunctions is usually complicated. In the present case, this assignment is simplified by choosing a particular value of the interaction parameter, $\lambda=0.7$, that separates adequately the frequencies $\omega_{R}=1$ and $\omega_{r}\simeq 1.949$, and using basis sets with well defined values of $L_{z}$. All in all, for the lowest eigenvalues it is possible to unequivocally label the variational eigenfunction using the set of quantum numbers in Equation 24. The results for the interacting system can then be presented using the same conventions that those used for the non-interacting one. The von Neumann entropy as a function of the superposition parameter is shown in Figure 2 for several superpositions and the corresponding values for the remaining entropy, the not-shared one and the criterion are collected in Table 2.

The current example shows all the features that are characteristic of what can be observed in superpositions of degenerate states. The changes in the convexity are correctly predicted by the criterion presented in Section II. The example is valuable because it is exactly soluble and its numerical (variational) implementation is simple. But, to some extent, the peculiar behaviour of systems of harmonic oscillators, that can be cast as interacting or non-interacting, limits the scope and validity of the example. So in Sections IV and V we present further examples that can be analyzed analytically and present, in the case of the Spherium model, strong correlations between the particles.

IV The Spherium model

The number of exactly solvable two interacting particle models in different spatial dimensions is remarkably low, which explains why so many studies of entanglement entropies are about systems of harmonic oscillators or variants of the Calogero model. As has been said in the Introduction, the study of the Calogero model was what triggered the formulation of the criterion that is the object of the present work. Fortunately, there is a growing number of quasi-exactly solvable models quasi-exactly-varios that can be used to study properties of strongly interacting two-particle models Pont2018 .

The spherium model Spherium-varios , i.e. two electrons interacting via the Coulomb potential and confined to the surface of a $D$ dimensional sphere, was proposed to study the properties of electronic correlations in a confining geometry. It has been studied using different approaches and as a benchmark to test numerical approximations. As was shown in Reference spherium , this model is quasi-exactly solvable, with analytical solutions for particular values of the radius $R$ of the sphere and the dimension $D$. These solutions can be found writing the two-electron wave function as a bipolar expansion bipolar , which allows to calculate eigenfunctions with small total orbital angular momentum numbers, $L=0,1$ or $2$, but it is quite cumbersome to implement for larger values of $L$.

To test the criterion we will construct two-electron wave functions with $L=0,1$ and $2$, following the work of Pestka Petska , which can be applied in a systematic way. In this Section we present the main details of the derivation of the wave functions, the reduced matrix elements and its eigenvalues.

IV.1 Description of the General Solution for $D=3$

Consider a two-particle Hamiltonian of the form

[TABLE]

Note that the potential depends on the radial coordinates of both particles, $r_{1}$ and $r_{2}$, and the distance between them $r_{12}$. The bipolar decomposition assumes that the solutions that are simultaneous eigenfunctions of the Hamiltonian, the total orbital angular momentum and the $z$ component of the orbital angular momentum, with eigenvalues $E$, $L(L+1)$ and $M$, respectively, can be written as

[TABLE]

with the constraint $l_{2}=d-l_{1}$. The inferior limit of the sum must be determined using the parity of the solution $\pi=(-1)^{l_{1}+l_{2}}$, according to

[TABLE]

The functions $\Omega_{l_{1},l_{2}}^{L,M}(\hat{r}_{1},\hat{r}_{2})$ are the eigenfunctions of $\mathbf{L}^{2}$ and $L_{z}$

[TABLE]

and

[TABLE]

Note that the wave function in Equation 28 has $L+1$ or $L$ radial functions (depending on the parity) $\Phi_{l_{1}\,l_{2}}(r_{1},r_{2},r_{12})$. Some algebra can be simplified noting that

[TABLE]

where $\nabla^{2}_{i}$ is any of the Laplacian operators that enter in Equation 27, and $f(r_{1},r_{2},r_{12})$ is any function that depends only on the variables $r_{1},r_{2}$ and $r_{12}$. The curly brackets indicate that the operators $\hat{X}_{i}^{\tilde{l}_{i},l_{i}}$ are applied only over the radial terms.

The operators $\hat{X}_{i}^{\tilde{l}_{i},l_{i}}$ have the property that

[TABLE]

and can be written as

[TABLE]

when $0\leq l_{i}\leq L$ and

[TABLE]

for $1\leq l_{i}\leq L$. In both cases the value of $j$ is such that $j\neq i$.

When the particles are confined to the surface of a sphere, $r_{1}=r_{2}=R$, the last two expressions can be further simplified, for $0\leq l_{i}\leq L$

[TABLE]

where a constant term is omitted, and for $1\leq l_{i}\leq L$

[TABLE]

All in all, the method proposed by Petska reduce the calculation of the two-electron wave function to another problem which consists in a set of $L$ (or $L+1$) coupled equations for the quantities $\Phi_{l_{1}\,l_{2}}$.

IV.2 Wave functions with $L=2$

The wave function with angular momentum number $L=2$ and parity $\pi=(-1)^{L+1}$, depends on just two radial functions, since there are two combinations of possible values $l_{1}=1$, $l_{2}=2$ and $l_{1}=2$, $l_{2}=1$. Using this, we get

[TABLE]

and

[TABLE]

These pair of coupled equations have a solution of the form $\Phi_{12}=\Phi_{21}=\Phi$. In this case Equations 38 and 39 reduce to

[TABLE]

Using the ansatz

[TABLE]

we found that this function is a solution with $\alpha=4$ and $R^{2}=6$.

Collecting again the partial results, we get the whole set of wave functions with $L=2$ and $M=0,\pm 1,\pm 2$

[TABLE]

where $\mathcal{Y}_{1,2}^{2,M}(\Omega_{1},\Omega_{2})$ are the usual total angular momentum eigenfunctions defined in terms of the spherical harmonics and the Clebsch-Gordan coefficients.

The one-electron reduced density matrix can be straightforwardly obtained using Perkins’ formula perkins

[TABLE]

This expression is valid for $k=-1,0,1,..$, and the coefficients $L_{1}^{k}$ and $L_{2}^{k,l}$ are given by

[TABLE]

and

[TABLE]

The coefficients $C_{klt}$ are given by

[TABLE]

From Equations 41 and 43 it is clear that the wave functions can be completely written in terms of spherical harmonics of both solid angles, $Y_{lm}(\Omega_{1})$ and $Y_{lm}(\Omega_{2}$).

So, if $\Psi(1,2)$ is any given superposition of two-electron wave functions, the corresponding reduced density matrix is

[TABLE]

To calculate the necessary eigenvalues, the matrix representation of the density operator in Equation 47 is calculated in a basis. It is natural to use the spherical harmonics

[TABLE]

Fortunately, to evaluate explicitly and exactly both Equations 47 and 48 only integrals involving two and three spherical harmonics are required,

[TABLE]

together with the formula

[TABLE]

Figure 3 shows the results for the $L=2$ case. The superposition state $\left|\psi_{\alpha}\right>$ of Equation 8 is in this case

[TABLE]

As can be seen in Figure 3, the two possible elections for $|M|=1,2$ render a convex and a concave entropy curve for the superposition state $\left|\psi_{\alpha}\right\rangle$, respectively. The criterion, shown in Table 3, correctly predicts the convexity in both cases.

V Laguerre-Gaussian one-photon states

Two-photon states can be constructed, from a theoretically point of view, just applying two creation operators to the vacuum state. From an experimental point of view, the most used method is the spontaneous parametric down conversion (SPDC) kok . It is well known that the SPDC mechanism provides a couple of photons in an entangled state that can be used to perform different quantum information tasks. Nevertheless, the two-photon state depends on the mode function of the pump and the phase matching conditions. So, to analyze a simpler case we focus in one-photon states and use the concept of single particle entanglement one-particle-entanglement where one spatial degree of freedom of the photon wave-function is traced out. Since we are interested in states that are eigenstates of $L_{z}$ the Laguerre-Gaussian states are an obvious choice to test the criterion. They are given by kok ; one-particle-entanglement

[TABLE]

where $r,\phi,z$ are the usual cylindrical coordinates, $m$ is the quantum number of $z$ component of the orbital angular momentum, $L_{l}^{|m|}$ are the modified Laguerre polynomials and the waist function $s(z)$ is a classical quantity that quantifies the width of the beam along the $z$ direction

[TABLE]

In the following we drop the superscript in $u_{lm}^{LG}$ to simplify the notation and make the change of variable $x=r/s(z)$. Let us recall that the Laguerre-Gaussian states are transverse modes that describe the free propagation of a photon with energy $\hbar\omega_{k}$, and are solutions of the Helmholtz equation in the paraxial approximation.

At this stage the procedure and the quantities to be calculated are well known so, in this Section, we include which states are studied and the formal expression for the reduced density matrix.

For a superposition given by

[TABLE]

where both $l,m$ and $l^{\prime},m^{\prime}$ are quantum numbers compatible with Equation 52, we calculate the reduced density matrix

[TABLE]

where $z$ is considered as a parameter and $x,y$ are the usual Cartesian coordinates perpendicular to $z$. We consider reduced density matrices at constant values of $z$ since the LG states in Equation 52 are not square-integrable functions. To avoid this kind of assumption it is possible to implement the calculation of reduced density matrices and the corresponding entropies using LG modes in a cavity cavity1 ; cavity2 , which are square-integrable and very similar to those in Equation 52, so the dependency on the $z$ variable can be traced out completely.

To test the criterion it is necessary to obtain the matrix elements

[TABLE]

where the one-coordinate basis functions used are the Hermite functions

[TABLE]

Figure 4 shows the results obtained for these states. As the results included in Table 4 show, the criterion correctly predicts the convexity of the superpositions considered.

The one-photon eigenstates can not provide a superposition with a concave von Neumann entropy because the one-photon wave function depends on the two transversal coordinates, $x$ and $y$, in exactly the same way. It is clear from Equation 52 that the state with quantum number $m$ depends on the same set of Laguerre polynomials than the state with quantum number $-m$. So, when tracing one or the other coordinate the corresponding reduced density matrices that enter in the calculation of the criterion both ”occupy” the same portion of the one-coordinate Hilbert space. As a consequence their remaining entropy always overcome their not-shared entropy.

VI Two-particle total angular momentum eigenstates

So far, the examples analyzed in the previous Sections provide a strong evidence of the validity of the criterion stated in this work about the convexity of superpositions of degenerate states. Regrettably, all of them have reduced density matrices $\rho_{A,0}$ with non-degenerate eigenvalues.

An exact example showing reduced density matrices with degenerate eigenvalues can be constructed from the addition of two angular momentum operators. As usual we consider

[TABLE]

where $\mathbf{L}^{2}=(\mathbf{L}_{1}+\mathbf{L}_{2})^{2}$, $L_{z}=L_{z}^{(1)}+L_{z}^{(2)}$. For each particle the square of the angular momentum operator, $\mathbf{L}_{i}^{2}$, and its $z$ component, $L_{z}^{(i)}$, have common eigenfunctions which satisfy

[TABLE]

for $i=1,2$.

To fix ideas, let us consider the Hamiltonian for two interacting spins given by

[TABLE]

and choose two spins with the same angular quantum number, $l_{1}=l_{2}=\ell$. Consistently with the superpositions analyzed in previous Sections, we consider states given by

[TABLE]

where

[TABLE]

and $C(\ell,m_{1};\ell,m_{2};L,M)$ are the Clebsch-Gordan coefficients.

The states $\mathcal{Y}^{L,L}_{\ell,\ell}$ and $\mathcal{Y}^{L,-L}_{\ell,\ell}$ are degenerate since

[TABLE]

and for fixed $L$ they are the states with minimum energy, besides $0\leq L\leq 2\ell$.

We include in Appendix A the necessary algebraic details to evaluate explicitly and exactly the entries of the different reduced density matrices and their eigenvalues.

Figure 5(a) shows the explicit evaluation of $S(\rho_{A,\alpha})$ for the case $\ell=3$, $|M|=L$ and $L=1,2,3,4,5$ and $6$. It can be appreciated that the cases $L=3,4,5$ and $6$ have a different convexity than the $L=1$ and $2$ cases. On the other hand, the solid circular dots in Figure 5(b) correspond to the values of the not-shared and remaining entropies calculated using Equations 12 and 15. The lines are included as a guide to the eye. It is clear that using these entropies the criterion detects the correct convexity of all the cases, as it is shown in Table 5.

It is interesting to analyze in some detail the case $\ell=3$ and $L=2$. The reduced density matrices can be calculated explicitly and, in the standard one-particle angular momentum basis $\left\{|m=3\rangle,|m=2\rangle,\ldots,|m=-3\rangle\right\}$, they are diagonal matrices

[TABLE]

and

[TABLE]

It is clear that the matrix $\rho_{A,0}$ has two pairs of degenerate eigenvalues, the first and fifth one are the first pair ($\lambda=5/42$), and the second and fourth are the second one ($\lambda=5/21$). Accordingly with Equation 14, to contribute to the not-shared entropy we must compare two times the eigenvalue of $\rho_{A,0}$ with the sum of eigenvalues of $\rho_{A,1}$ that lie in the same sub-space. But since $2\times\frac{5}{42}=\frac{5}{6}\times\frac{2}{7}<\frac{2}{7}$,

[TABLE]

On the other hand, the term associated to the eigenvalue $5/21$ contributes with

[TABLE]

Collecting these results, the not-shared entropy is equal to

[TABLE]

It is instructive, and simple to do, to check what happens if the minimization implied in the definition of the not-shared entropy is not performed, i.e. what values are obtained for different choices of the one-dimensional projectors associated to the degenerate eigenvalues. Choosing the canonical one-particle angular momentum basis to generate the one-dimensional projectors gives the not-shareable entropy

[TABLE]

where the eigenvalues, for the case $L=2$, are those in Equations 64 and 65. The values obtained using Equation 69 are shown in Figure 5(b) as square solid blue dots. Not surprisingly, the values are larger than those of $S_{NS}$, but more interestingly, it is clear that for $L=2$ the not-shareable entropy is larger than the remaining one, which could lead to an incorrect assessment of the convexity. Sometimes, choosing a particular basis to obtain the projectors $P_{i}^{A,0}$ could give very good results for particular values of $\ell$ and $|M|=L$, for instance, in Figure 5 the values obtained using a particular basis are shown using triangular dots. This election provides value of the not-shareable entropy larger than those of $S_{NS}$ and predicts correctly the curvature.

We have tested a very large number of cases, up to $\ell=12$, which can be done quite fast and efficiently given the simplicity of the bipartite states $\mathcal{Y}_{\ell,\ell}^{L,M}$ and for all these cases the criterion predicts correctly the curvature of the superposition of states.

VII Discussion and conclusions

From a theoretical point of view, the amount of analytical work involved in the examples presented, the two two-dimensional harmonic oscillators and the spherium, indicates how difficult it is to construct exact cases to test the convexity criterion. All the quantities involved, in particular the matrix elements of the reduced density matrices, involve a large number of nested sums, so its evaluation time grows as $M^{nes+1}$, where $nes$ is the number of nested sums and $M$ is the largest one-particle basis set size that it is necessary to use in order to guarantee the normalization of the reduced density matrix.

The criterion could be tested using pure states of many-body models (spin chains) which, in some cases, have exact solutions. The reduced matrices can be obtained using the adequate spin correlation functions for small subsystems. Since so far we have studied only bipartite systems made of two susbsystems whose Hilbert spaces have the same dimension it is not clear if some amendments are in order for the criterion to work in the spin chain setting. Work along this line is in progress.

Our results imply that, most likely, there should be a theorem about the convexity of the von Neumann entropy of superpositions of pure states but, so far, we have not been able to formulate the precise hypotheses that make it work, i.e. we know that the superpositions of pure degenerate eigenstates satisfy the requirements to have a defined convexity but we do not have an algorithm that allows us to generate a number of eigenvalues, eigenfunctions (or projectors) to construct $\rho_{A,0}$ and $\rho_{A,1}$ and $\rho_{A,\alpha}$ and guarantee that $S(\rho_{A,\alpha})$ will be convex (or concave). In this sense, the conditions that the superposition is made of two degenerate states with quantum number $\pm m$, where $m$ is the quantum number associate to the $z$-component of the total angular momentum, seem to be sufficient for states defined over hyper-spheres or that have the same asymptotic behavior that the harmonic oscillator eigenfunctions.

In the same sense that in the paragraph above, it is not necessary that $S(\rho_{A,0})=S(\rho_{A,1})$ to ensure that the von Neumann entropy, $S(\rho_{A,\alpha})$ has a well defined convexity (concavity). Nevertheless, it is worth to point out that if $\rho_{A,0}=\mbox{Tr}_{B}(|\psi_{L}\rangle\langle\psi_{L}|)$ or $\rho_{A,0}=\mbox{Tr}_{B}(|\psi_{-L}\rangle\langle\psi_{-L}|)$ the criterion predicts exactly the same convexity (concavity). In other words, the criterion holds even when $\rho_{A,0}$ and $\rho_{A,1}$ are not isospectral.

In contradistinction to what happens to the entropy of one-photon states, for the eigenstates of the total angular momentum it is possible to find concave and convex functions. For example, for the states $|2\ell,2\ell,\ell,\ell\rangle$ and $|2\ell,-2\ell,\ell\ell\rangle$, once a particle is traced out, they result in orthogonal states that do not share entropy and, consequently, the von Neumann entropy of the superposition is concave. Besides, for some value $k$ the states $\mbox{Tr}_{B}(|2\ell-k,2\ell-k,\ell,\ell\rangle\langle 2\ell-k,2\ell-k,\ell,\ell|)$ and $\mbox{Tr}_{B}(|2\ell-k,-(2\ell-k),\ell,\ell\rangle\langle 2\ell-k,-(2\ell-k),\ell,\ell|)$ are not longer orthogonal and share some entropy. At some larger value $k_{c}>k$ the remaining entropy overcomes the not-shared and the von Neumann entropy $S(\rho_{A,\alpha})$ becomes convex.

In Reference Garagiola2016 it was envisaged that certain superpositions of degenerate bipartite states could have definite convexity (concavity) and that the extremal states would also be eigenstates of other observable of the system Garagiola2016 , it was required that the system under study had at least two conserved quantities. In this work we have restricted ourselves to the case where those quantities are the energy and the $z$ component of the total angular momentum, which is very reasonable for systems with a preferred direction. For particle systems it is difficult to construct other conserved quantities beyond the Hamiltonian, the total angular momentum or some of its components, unless that some superintegrable system is considered. There are some examples of two- and three-body superintegrable problems in dimensions two and three, where the conserved quantities are polynomials of the momentum operator Cartesian components. Currently, we are studying the convexity properties of superpositions of degenerate states in this kind of problems.

As a final comment, we want to return to the a subject that we raised at the end of Section II, where we stated that it could be desirable to formulate the not-shared entropy, Equation 12, without resorting to the spectral decompositions of the reduced density operators. Here, we discuss some numerical tests that we implemented on the examples considered in Section VI, i.e. orbital angular momentum states with quantum number $L$. Consider a set of one-dimensional projectors $\{P_{\alpha}\}_{\alpha=1}^{2L+1}$ that are mutually orthogonal and such that $\sum_{\alpha}P_{\alpha}=I$. Besides, consider the quantity

[TABLE]

where $\langle\rho_{A,i}\rangle_{\alpha}=\mbox{Tr}(\rho_{A,i}P_{\alpha})$. For very small values of $L$ it is numerically feasible to show that

[TABLE]

where the minimum was obtained generating randomly families of orthogonal projectors and evaluating $\tilde{S}$. For larger values of $L$ the number of random families of projectors necessary to pick up approximately the value of the minimum grows so fast that a more educated sampling becomes mandatory. Choosing random sets of projectors close enough to the eigen-projectors of $\rho_{A,0}$ Equation 70 was verified for moderate values of $L$ as the ones studied in Section VI. Note that if for a set of projectors $S-2\tilde{S}<0$ then this is sufficient to affirm that the superposition will be concave. Further work along these lines is under progress.

Acknowledgements

We acknowledge CONICET (PIP-11220150100327CO) for partial financial support. N.G. and O.O. also acknowledges SECYT-UNC for partial financial support.

Appendix A Angular momentum reduced density matrices example

The simpler example that allow to construct bipartite states to test if the von Neumann entropy of a given superposition $S(\rho_{A,\alpha})$ is a convex function can be constructed from the states

[TABLE]

where $C(l_{1},m_{1};l_{2},m_{2};L,M)$ are the Clebsch-Gordan coefficients and $Y_{l_{1},m_{1}}$ are the usual spherical harmonics. For superpositions of the form

[TABLE]

the reduced density operator is

[TABLE]

The reduced density matrix above is function of both solid angles $\Omega_{1}$ and $\Omega_{1}^{\prime}$, so it is logical to look for its expression in the spherical harmonics basis, where its elements are given by

[TABLE]

After some tedious, but straightforward algebra, it can be shown that

[TABLE]

where $|i|,|j|\leq l_{1}$.

Appendix B Two harmonic oscillators related expressions

In this Appendix we present explicit expressions of the two harmonic oscillator states that allow the computation of the density matrix eigenvalues. The angular momentum two-particle eigenstates in Equations 24 and 25, can be written as

[TABLE]

where the sub-indexes indicate for which oscillator the vector sate is an eigenstate, following the convention that $R$ designates the centered coordinates oscillator and $r$ the relative coordinates one.

Each one of the one-particle angular momentum eigenvectors in Equation 76 can be written as linear combination of Cartesian oscillator states, $\left|n_{x},n_{y}\right\rangle^{c}=\left|n_{x}\right\rangle^{c}\left|n_{y}\right\rangle^{c}$ Cohen , as follows

[TABLE]

where $\tilde{r},\tilde{\phi}$ stands for $R,\phi_{R}$ or $r,\phi_{r}$, and the same convention holds for the Cartesian coordinates, $sgn(m)=1$ for $m\geq 0$ and $sgn(m)=-1$ for $m<0$. The one-particle Cartesian eigenfunctions corresponding to the vector state $\left|n_{\tilde{x}},n_{\tilde{y}}\right\rangle^{c}=|n_{\tilde{x}}\rangle^{c}|n_{\tilde{y}}\rangle^{c}$ are given by

[TABLE]

where $H_{n}$ is the Hermite polynomial of $n-$th degree, with $n=0,1,2.\ldots$ The energy of these two-dimensional harmonic oscillator states is $E_{n_{\tilde{x}},n_{\tilde{y}}}=\omega_{\tilde{r}}(n_{\tilde{x}}+n_{\tilde{y}}+1)$.

Collecting the results above, we get that

[TABLE]

the notation indicates which oscillator and coordinates must be used to obtain the corresponding eigenfunctions

[TABLE]

where $f_{n}^{\omega}(x)$ are the eigenfunctions of a one dimensional harmonic oscillator of frequency $\omega$ and quantum number $n$ and the last equation defines $\Upsilon_{a,b,c,d}$.

The one-particle reduced density matrix for an eigenfunction $\psi_{n,l,m,p}$ can be obtained exactly from

[TABLE]

where $\psi_{n,m,l,p}(\vec{x}_{1},\vec{x}_{2})$ is the two-particle wave function $\left|n,m,l,p\right\rangle$ written in terms of the original particle coordinates, Equation 80. Actually, to implement the calculation of the eigenvalues, $\lambda_{i}^{A}$, required to obtain the different entropies that enter in the convexity criterion, Equations 12, 15, 16 and 17, it is useful to calculate the matrix elements of $\psi_{n,m,l,p}$ in a one-particle one-coordinate basis functions $\phi_{i}(z)$, where $z$ stands for $x$ or $y$, i.e. we first calculate

[TABLE]

and then solve the eigenvalue problem

[TABLE]

where $\mathbf{K}$ is the matrix whose entries are given by Equation 82. Since $\psi_{n,m,l,p}$ is a symmetric kernel (under particle exchange), the eigenvalues of the reduced density matrix satisfy that symmetric-kernel

[TABLE]

The algebra involved in the calculation of the elements $K^{n,m,l,p}_{i_{1},j_{1},i_{2},j_{2}}$ is rather cumbersome, but direct, so we write them explicitly. We compute the expressions for the wave function of two non-interacting harmonic oscillators, $\lambda=0$, and also the elements of its reduced density matrix, Equation 82. The corresponding quantities for $\lambda\neq 0$ can be obtained following a completely equivalent procedure.

Then, using the expansion of the Hermite polynomials Hermite , we get that

[TABLE]

Now, for each $\Upsilon$ function we construct a kernel ${K_{\Upsilon}}$ by means of Equation 82, and using the one-particle basis functions $\phi_{k}=f^{1}_{k}(x)$, we get the matrix-representation of each kernel as

[TABLE]

All the integrals that appear in the last expression are obtained in terms of the Gamma function Gamma . Finally, the matrix representation of $\mathbf{K}$ in Equation 83 can be written as

[TABLE]