Enlarging the notion of additivity of resource quantifiers

Many quantum mechanical tasks of either foundational or practical relevance, involving a particular state $\varrho$, require the preparation of a large number of copies of this state to be accomplished. In some situations, it turns out that employing $\varrho^{\otimes N}$ and global operations over the large Hilbert-Schmidt space it inhabits proves to be advantageous in comparison with the isolated manipulation of $N$ copies of $\varrho$. This is an instance of the nonadditivity of the figure of merit that embodies the ability to execute the task under consideration.

Additivity is a special property satisfied only by a handful of measures or monotones of quantum resources comm0 , as for instance, the logarithmic negativity vidal and squashed entanglement squashed , for non separability horodecki . For these special quantifiers ${\cal E}(\varrho^{\otimes N})=N{\cal E}(\varrho)$. Thus, additivity renders an otherwise hard problem trivial: the evaluation of the amount of resources contained in $N$ copies of a given state $\varrho$, ${\cal E}(\varrho^{\otimes N})$, once we know ${\cal E}(\varrho)$. We remark that a stronger notion of additivity is ${\cal E}(\varrho\otimes\sigma)={\cal E}(\varrho)+{\cal E}(\sigma)$, for all pairs of states $\varrho$ and $\sigma$. In this manuscript, whenever we refer to additivity, we mean the weaker condition.

The majority of quantum-resource quantifiers, however, is nonadditive. Examples regarding nonseparability are the entanglement of formation formation ; hastings and the geometric measure of entanglement geom , among others nonadd ; dafa ; dafa2 . An extreme example of such a behavior is the phenomenon of superactivation, for which, given a particular quantifier ${\cal E}$, we may find states $\varrho$, such that ${\cal E}(\varrho)=0$ and ${\cal E}(\varrho^{\otimes N})>0$, for some $N>1$, as is the case of the distillable entanglement superact ; watrous .

The technical difficulty to determine ${\cal E}(\varrho^{\otimes N})$ is an important motivation behind the search for asymptotic limits, which are useful whenever one can assume that the number of states is large enough to make the limit $N\rightarrow\infty$ an acceptable approximation. However, often, the asymptotic regime dominates only for a unpractically high number of copies oneshot ; natcomm ; ieee . Therefore, one cannot always evade the question of evaluating ${\cal E}$ for a large, but finite number of copies.

An alternative attempt to deal with the problem comes from the concept of scalable quantifiers, introduced in scal1 , which is summarized in the next section. The remanining sections of this work explore a generalization of the additivity notion, stemming from scalability, which can prove useful in the evaluation of some relevant figures of merit in the large $N$ limit.

Let us initially formalize the notion of scalability scal1 . Consider an arbitrary function ${\cal E}$ with argument $\varrho^{\otimes N}$, ${\cal E}:{\cal B(H)}^{\otimes N}\mapsto\mathds{R}_{+}$. In practice, the quantity ${\cal E}(\varrho^{\otimes N})$ can be a measure or monotone of a quantum resource, or some cost function, for instance. Now, suppose we either know by definition or find out that this function is completely (or approximately) determined by $q$ real numbers, namely, $e_{i_{1}}={\cal E}(\varrho^{\otimes i_{1}})$, $e_{i_{2}}={\cal E}(\varrho^{\otimes i_{2}}),\dots,e_{i_{q}}={\cal E}(\varrho^{\otimes i_{q}})$, $i_{j}<N$, $j=1,\dots,q$. That is, once we know the value of ${\cal E}$ for some smaller number of copies, ${\cal E}(\varrho^{\otimes N})$ is determined:

where, for short, we collect the non-negative numbers $e_{i_{j}}$ as the components of a vector in the positive hyperoctant of $\mathds{R}^{q}$,

One of the main results of scal1 is that the possible functional forms of $E^{(N)}(\vec{e})$ must satisfy the constraint

for $N=a^{n}$ and $K=a^{k}$, with $n,k\in\{0,1,2,\dots\}=\mathds{N}$, $k\leq n$. So $N$ and $K$ take values on

the set of all non-negative integer powers of an arbitrary positive integer $a$.

The restriction embodied by (2) stems from the fact that, while the left-hand side of (1) automatically comply with the structure of tensor products (the domain of ${\cal E}$ is a Hilbert-Schmidt space), for the right-hand side, this compliance must be imposed on the functional form of $E$, which is defined on $\mathds{R}^{q}$. More specifically, ${\cal E}(\varrho^{\otimes N})={\cal E}(\sigma^{\otimes(N/K)})$ is a tautology, whenever

Equation (2) guarantees that $E$ will not violate this rule. A figure of merit ${\cal E}(\varrho^{\otimes N})$ that can be expressed through (1) and satisfying the above consistency condition is referred to as a $q$-scalable ($q$-S) function.

In the simplest case, $q=1$, the constraint relation reads

Note that, while ${\cal E}:{\cal B(H)}^{\otimes N}\mapsto\mathds{R}_{+}$, we have $E:\mathds{R}_{+}^{q}\mapsto\mathds{R}_{+}$, so that, typically, the domain of the latter has a dimension which is much lower than that of the former. In general, the functional dependencies that satisfy the constraint imposed by relation (2), may be nonlinear, the $\ell_{1}$-norm of coherence coh being a simple but nontrivial example of a 1-scalable function scal2 .

Inspired by the additivity property as expressed by $E^{(N)}(e)=Ne$ (which is the simplest 1-S function), we will investigate $q$-scalable functions which, in addition, present a linear dependence on the components of $\vec{e}$:

This augmented notion of additivity, here referred to as $q$-additivity, along with the general constraint (2), give rise to explicit recurrence relations that must be satisfied by the entries of $\vec{N}$, as will be shown in the next subsection, for $\vec{e}=(e_{i_{1}},e_{i_{2}})\in\mathds{R}^{2}$ ($q=2$). For $q=1$ the only possible linear dependence is that of usual additivity, $E^{(N)}(e)=Ne$.

We now set to obtain the general constraints that must be satisfied by $q$-additive functions, for arbitrary $q$, that we express as

In this case, the general scalability constraint (2) reads

Comparing equations (20) and (21) we get

Therefore, expression (20) represents a $q$-additive function whenever (22) is satisfied. In the simplest case we have $q=1$ with $i_{1}=1$ and $\eta_{1}(N)=\eta_{1}(K)\eta_{1}(N/K)$. Again, we remark that to obtain the general solution it suffices to set $K=a$.

Let us simplify the notation using $\eta^{i_{\ell}}(N)=\eta^{\ell}_{n}$ ($N=a^{n}$), then:

Note that, by definition, $\eta^{1}_{\ell}=0$ except for $\ell=q$ (denoted by $x$ in previous section) and $\eta^{2}_{\ell}=0$ except for $\ell=1$ ($a$ copies) and for $\ell=q$ (denoted by $y$ in previous section). Following the same reasoning we may generally write:

This relation states that every coefficient is null except if it satisfies the consistency conditions $E^{(1)}=e_{1}$, $E^{(a)}=e_{2}$, etc, up to $\ell<q$. In addition, for $a^{q}$ copies we define new values $\eta^{1}_{q}(a^{q})=x$, $\eta^{2}_{q}(a^{q})=y$, etc. That is why the Kronecker delta symbols are labeled for an index $\ell<q$ or for $\ell=q$. Substituting (24) into (23) we get:

Expression (24) can be put into the following matrix form:

Note that the first line is null except for the last entry $\eta^{1}_{q}$, while the following lines compose a $(q-1)\times(q-1)$ identity matrix on the left-down block and the last column is given by the numbers $\eta^{m}_{q}$. Let us denote this matrix by $\mathcal{Q}$:

It is instructive to test this matrix approach in the 2-additive case:

which indeed leads to the generalized hybrid Fibonacci polynomials (9).

The advantage of this formalism, especially for larger values of $q$, is that one can diagonalize $\mathcal{Q}$ through the operation $\mathcal{\eta}_{n}=D^{-1}\mathcal{\zeta}_{n}$ ($D$ is $q\times q$): $D^{-1}\zeta_{n}=\mathcal{Q}D^{-1}\zeta_{n-1}$, $DD^{-1}\zeta_{n}=D\mathcal{Q}D^{-1}\zeta_{n-1}$, where $[D\mathcal{Q}D^{-1}]_{kl}=\lambda_{k}\delta_{kl}$ is the diagonalized matrix with $\lambda_{k}$ being the eigenvalues. In this basis the solution of the $q$-additivity problem is multiplicative:

and as $\eta_{n}$ is the result of a matrix operation $D^{-1}$ on the vector $\zeta_{n}$, we can expand it in a sum of coefficients $C^{m}_{k}$. These coefficients will be the solutions of determined systems of equations related to expression (24).

The next simpler case is that of 3-additivity, for which we have:

One can find the three eigenvalues of this matrix, change variables $\nu_{k}=\log_{a}\lambda_{k}$, $k=1,2,3$, and write:

Now we have boundary conditions $\eta^{1}(1)=1,\eta^{1}(a)=\eta^{1}(a^{2})=0$, $\eta^{2}(1)=\eta^{2}(a^{2})=0,\eta^{2}(a)=1$ and $\eta^{3}(1)=\eta^{3}(a^{2})=0,\eta^{3}(a^{2})=1$. So, there are three systems of equations to be solved:

These systems can be easily dealt with through some software capable of symbolic manipulations, like mathematica. The nine coefficients we obtained read:

In these solutions we initially assume that all three exponents should be different from each other $\nu_{1}\neq\nu_{2}\neq\nu_{3}\neq\nu_{1}$. However, as we will see, as in the 2-additive case, the way they are combined to compose the solution, relation (32), leads to $E^{(N)}(\vec{e})=(C^{1}_{1}e_{1}+C^{2}_{1}e_{2}+C^{3}_{1}e_{3})N^{\nu_{1}}+(C^{1}_{2}e_{1}+C^{2}_{2}e_{2}+C^{3}_{2}e_{3})N^{\nu_{2}}+(C^{1}_{3}e_{1}+C^{2}_{3}e_{2}+C^{3}_{3}e_{3})N^{\nu_{3}}$, for which no divergence appears for $\nu_{i}=\nu_{j}$.

The previous results can be sumarized as the following statement.

Proposition: Let $\varrho\in\mathcal{B}\left(\mathcal{H}\right)$ and $N\in\mathds{P}_{a}$. If $\mathcal{E}$ is a 3-additive function such that $\mathcal{E}(\varrho^{\otimes N})=E^{(N)}(\vec{e})=\vec{N}\cdot\vec{e}$, with $\vec{e}=(e_{1},e_{2},e_{3})=(\mathcal{E}(\varrho),\mathcal{E}(\varrho^{\otimes a}),\mathcal{E}(\varrho^{\otimes a^{2}}))$, where $E^{(a^{3})}(\vec{e})=x\text{ }e_{1}+y\text{ }e_{2}+z\text{ }e_{3}$, with $x$, $y$ and $z$ (equivalently $\nu_{1}$, $\nu_{2}$ and $\nu_{3}$) known, then, we have:

where $\nu_{k}=\log_{a}\lambda_{k}$, $k=1,2,3$, with $\lambda_{k}$ being the $k$-th eigenvalue of matrix (26).

The 3-additive function normalized per copy, $E^{(N)}(\vec{e})/N$, is unbounded if any of the $\nu_{j}>1$ and vanishes if $\nu_{1},\nu_{2},\nu_{3}<1$, in the limit of large $N$. Again, we have permutation symmetry throughout, i. e., invariance under $\nu_{1}\leftrightarrow\nu_{2}$, $\nu_{2}\leftrightarrow\nu_{3}$ or $\nu_{1}\leftrightarrow\nu_{3}$. The cases in which two or three $\nu_{j}$ coincide must be dealt with care. For instance, if we make $\nu_{1}=\nu$, $\nu_{2}=\nu+\delta$, and $\nu_{3}=\nu+\epsilon$, take first the limit $\delta\rightarrow 0$ and afterwards the limit $\epsilon\rightarrow 0$, we get: