Catalytic majorization – Open Quantum Problems

Cite this problem as Problem 4.

Problem

By a theorem of Nielsen [1], we have a completely explicit criterion to decide when a single copy of a pure bipartite state can be converted to another such state using only local quantum operations and classical communication. Using Nielsen’s criterion, one can show the existence of the phenomenon of catalysis [2]: there exist situations when a state $\psi$ cannot be converted to state $\psi'$ , but nevertheless $\psi \otimes \phi$ can be converted to $\psi' \otimes \phi$ for a suitably chosen entangled state $\phi$ , the “catalyst”.

The problem is to give a similarly efficient criterion to decide which pure bipartite states can be converted into each other using a catalyst.

Background

Nielsen’s criterion provides a surprisingly direct rendering of the intuition that a “more entangled” pure state has a “more mixed” reduced state. Thus a bipartite pure state $\psi$ can be converted to $\psi'$ if and only if the eigenvalue vector of the partial trace of $\psi$ is more mixed than that of $\psi'$ , in the sense of majorisation of probability vectors [3]. We say that one probability vector $p=(p_1,\ldots,p_n)$ is majorised by another, $q=(q_1,\ldots,q_n)$ , if one and hence all of the following equivalent statements hold:

$\sum_{i=k}^{n} p^\downarrow_i \le \sum_{i=k}^n q^\downarrow_i \; \forall k \in \{1, \ldots, n\}$ , where $p^\downarrow$ and $q^\downarrow$ denotes the vectors brought into non-increasing order.
There is a doubly stochastic matrix $D$ (positive entries, sum of all rows and all columns = 1) such that $p=D q$ .
For every convex function $f : \mathbb{R}\rightarrow \mathbb{R}$ , we have $\sum_i f(p_i) \le \sum_i f(q_i)$ .

Denoting the above condition as $p \prec q$ , we have that $\psi_{AB}$ can be converted to $\psi'_{AB}$ with LOCC if and only if $\lambda(\psi_A) \prec \lambda(\psi'_A)$ [1], where $\lambda$ denotes the vector of eigenvalues of a given operator.

The problem of existence of catalysts for such a transformation can be rephrased completely in this context of majorisation of classical probability vectors, since tensoring pure bipartite states means again tensoring of probability vectors for the eigenvalues of the reduced density operators. Thus we would like to characterize the order relation of “catalytic majorisation”:

When does there exist a vector $r$ such that $(p\otimes r) \prec (q\otimes r)$ ?

Solution

A complete solution was independently obtained by Klimesh [3, 4] and Turgut [5]. It can be phrased in terms of classical Rényi entropies, which are defined as
$\displaystyle H_\alpha(p) = \begin{cases} \frac{\operatorname{sgn}(\alpha)}{1-\alpha} \log \sum_{i=1}^n p_i^\alpha & \alpha \in \mathbb{R} \setminus \{0,1\}\\ \log \operatorname{rank}(p) & \alpha = 0 \\ -\sum_{i=1}^n p_i \log p_i & \alpha = 1. \end{cases}$
We additionally need the function $f_0 (p) = \sum_{i=1}^n \log p_i.$

The main finding of [4] and [5] is then as follows. Consider vectors $p, q \in \mathbb{R}^n$ such that $p^\downarrow \neq q^\downarrow$ and at least one of $p$ and $q$ has no zero components. Then, the following are equivalent:

There exists a vector $r \in \mathbb{R}^c$ such that $(p\otimes r) \prec (q\otimes r)$ .
$H_\alpha(p) > H_\alpha(q)$ for all $\alpha \in \mathbb{R}\setminus \{0\}$ and $f_0 (p) > f_0 (q)$ .

The condition can be simplified if one only requires that the transformation from $\psi$ to $\psi'$ be realised approximately. In [6] it was then shown that, for all vectors $p, q \in \mathbb{R}^n$ such that $\sum_{i=1}^n p_i = \sum_{i=1}^n q_i$ , the following are equivalent:

For any $\varepsilon > 0$ , there exists a vector $r \in \mathbb{R}^c$ such that $(p\otimes r) \prec (q_\varepsilon \otimes r)$ for some $q_\varepsilon \in \mathbb{R}^n$ with $\|q_\varepsilon - q \| \leq \varepsilon$ .
$H_\alpha(p) \geq H_\alpha(q)$ for all $\alpha \in \mathbb{R}$ .

Note that the dimension of the catalyst, $c$ in the above, is generally unbounded [7].

One should also note that an alternative condition for catalytic transformations that only uses Rényi entropies with $\alpha>1$ was independently established in [8].

References

[1] M. A. Nielsen, Phys. Rev. Lett. 83, 436-439 (1999), quant-ph/9811053.

[2] D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566-3569 (1999), quant-ph/9905071.

[3] M. Klimesh, Entropy Measures and Catalysis of Bipartite Quantum State Transformations, Proc. 2004 International Symposium on Information Theory (ISIT 2004), June 27-July 2, 2004, Page 357.

[4] M. Klimesh, Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation, quant-ph/0709.3680.

[5] S. Turgut, Necessary and Sufficient Conditions for the Trumping Relation, J. Phys. A: Math. Theor. 40 (2007) 12185-12212.

[6] F. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, PNAS 112, 3275 (2015).

[7] S. Daftuar and M. Klimesh, Mathematical structure of entanglement catalysis, Phys. Rev. A 64, 042314 (2001).

[8] G. Aubrun and I. Nechita, Catalytic Majorization and $ℓ_{p}$ Norms, Commun. Math. Phys. 278, 133-144 (2008).