Majorization–Lattice Theorem

• Majorization–Lattice Theorem is a framework that defines a complete bounded lattice on sorted probability vectors using the majorization preorder, essential for analyzing state convertibility.
• It provides explicit constructions of the meet (infimum) and join (supremum) through cumulative sums and convex envelopes, enabling practical computation in resource theories.
• The lattice structure supports quantum resource operations, such as optimal common resource identification and entanglement transformation, improving protocol efficiency.

The Majorization–Lattice Theorem encapsulates the structure of the set of ordered probability vectors under the majorization preorder, asserting that this set constitutes a complete and bounded lattice. This structure is foundational across majorization-based resource theories, quantum information, and entanglement theory, serving as the mathematical underpinning for state convertibility, optimal resource identification, and the analysis of functional properties such as entropy. Detailed constructions of meet (infimum) and join (supremum) are essential for explicit calculations and for operational tasks involving pure quantum states.

1. Formal Definition and Structure

Let $d\geq 2$, and define the set $\Delta_d^{\downarrow}$ of $d$-dimensional probability vectors with components sorted in non-increasing order: $$\Delta_d^{\downarrow} := \left\{ x = [x_1, ..., x_d]\in\mathbb R^d : x_1 \geq x_2 \geq \cdots \geq x_d \geq 0,\; \sum_{i=1}^d x_i = 1 \right\}.$$ The majorization preorder $\succ$ on $\Delta_d^{\downarrow}$ is defined by

$$x \succ y \iff \sum_{i=1}^{k} x_i \geq \sum_{i=1}^{k} y_i\quad\forall k=1,...,d-1.$$

Under this order, $(\Delta_d^{\downarrow}, \succ)$ exhibits the structure of a complete bounded lattice. The unique top element is $e_d = [1,0,...,0]$ and the bottom element is $u_d = [1/d, ..., 1/d]$ (Bosyk et al., 2019).

For every subset (possibly non-denumerable) $$\mathcal{P} = \{ x^{(i)} \}_{i\in I} \subset \Delta_d^{\downarrow}$$ there exists both a greatest lower bound (meet, $x^{\inf}$) and a least upper bound (join, $x^{\sup}$) within $\Delta_d^{\downarrow}$ (Bosyk et al., 2019, Deside et al., 2023, Bosyk et al., 2016).

2. Explicit Construction of Meet and Join

2.1 Infimum (Meet)

Given $\mathcal{P} = \{x^{(i)}\}_{i\in I}$ with each $x^{(i)} \in \Delta_d^{\downarrow}$, define the set of $k$-th partial sums: $$S_k := \{ S_k(x^{(i)}) \mid i\in I \},\quad S_k(x) := \sum_{j=1}^k x_j,$$ with $S_0 = 0$, $S_d = 1$. The infimum is given by

$s_k^{\inf} := \inf S_k,$

and the components of the meet are

$$(x^{\inf})_k = s_k^{\inf} - s_{k-1}^{\inf},\quad k=1, ..., d.$$

For a pair $x, y \in \Delta_d^{\downarrow}$, this specializes to: $S_k(x \wedge y) = \min\{ S_k(x), S_k(y) \}.$

2.2 Supremum (Join)

Define $t_k := \sup S_k$. Construct the vector $\bar x$ with components $\bar x_k = t_k - t_{k-1}$. In general, $\bar x$ may not satisfy the monotonicity condition required for $\Delta_d^{\downarrow}$. Remedy this by applying the upper convex envelope (smallest concave majorant) to the cumulative sums, resulting in a concave Lorenz curve $\bar L(\omega)$: $$x^{\sup}_k = \bar L(k) - \bar L(k-1),\quad k=1,\dots,d.$$ Alternatively, a “smoothing” algorithm (Cicalese–Vaccaro, 2002) ensures the final vector is non-increasing and normalized (Bosyk et al., 2019, Deside et al., 2023, Bosyk et al., 2016).

3. Proof Structure and Geometric Perspective

Each $x\in\Delta_d^{\downarrow}$ defines a Lorenz curve $L_x: [0, d] \to [0,1]$, with $L_x(k) = S_k(x)$ and linear interpolation elsewhere. $L_x$ is non-decreasing and concave.

• The infimum Lorenz curve is $$L^{\inf}(\omega) := \inf_{i\in I} L_{x^{(i)}}(\omega)$$, which is also non-decreasing and concave, corresponding to a unique $x^{\inf}\in \Delta_d^{\downarrow}$.
• The supremum Lorenz curve is initially $L^*(\omega) := \sup_{i\in I} L_{x^{(i)}}(\omega)$, which may fail concavity. The concave hull $\bar L(\omega)$ yields the unique join.

By construction, for all $i$, $L^{\inf} \leq L_{x^{(i)}} \leq \bar{L}$, establishing the greatestness and leastness of the bounds (Bosyk et al., 2019).

4. Operational Implications in Resource Theories

In quantum resource theories where state convertibility is determined by majorization, the lattice structure is central. For instance:

• In entanglement theory, the conversion of bipartite pure states via LOCC is governed by the majorization relation on their respective Schmidt vector probability distributions (Deside et al., 2023, Bosyk et al., 2016).
• The meet $\lambda_\psi \wedge \lambda_\phi$ for states with Schmidt vectors $\lambda_\psi$ and $\lambda_\phi$ yields the optimal common resource (OCR): the least entangled but still universal state from which both $|\psi\rangle$ and $|\phi\rangle$ can be deterministically reached. The join $\lambda_\psi \vee \lambda_\phi$ gives the optimal common product (OCP): the most entangled state that both $|\psi\rangle$ and $|\phi\rangle$ can produce via LOCC (Deside et al., 2023).
• These constructions generalize to collections of states and underpin protocols for probabilistic and approximate state transformation.

5. Illustrative Examples

Example (d=4):

Let $x=[0.60,0.16,0.16,0.08]$, $y=[0.50,0.30,0.10,0.10]$. Compute cumulative partial sums: $\begin{array}{c|cccc} k & 1 & 2 & 3 & 4 \ \hline S_k(x) & 0.60 & 0.76 & 0.92 & 1.00 \ S_k(y) & 0.50 & 0.80 & 0.90 & 1.00 \ \end{array}$

• Infimum: $$s_k^{\inf} = \min\{S_k(x), S_k(y)\} = [0.50, 0.76, 0.90, 1.00]$$

$x^{\inf} = [0.50, 0.26, 0.14, 0.10]$

• Supremum: $$t_k = \max\{S_k(x), S_k(y)\} = [0.60, 0.80, 0.92, 1.00]$$

$x^{\sup} = [0.60, 0.20, 0.12, 0.08]$

No smaller/larger bounds exist with respect to the majorization preorder (Bosyk et al., 2019).

Example (d=3):

Let $x=(0.5, 0.4, 0.1)$, $y=(0.6, 0.2, 0.2)$.

$\begin{array}{c|cccc} k & 0 & 1 & 2 & 3 \ \hline X_k(x) & 0 & 0.5 & 0.9 & 1 \ X_k(y) & 0 & 0.6 & 0.8 & 1 \ \end{array}$

$$x \wedge y = (0.5, 0.3, 0.2),\quad x \vee y = (0.6, 0.3, 0.1)$$

(Deside et al., 2023, Bosyk et al., 2016).

6. Functional and Algebraic Properties

• Any $d\times d$ doubly stochastic matrix $D$ preserves the lattice order: if $x \prec y$, then $Dx \prec Dy$.
• Schur-convex and Schur-concave functions behave in the standard way with respect to meet and join.
• The Shannon entropy $H(p) = -\sum p_i\log p_i$ is both supermodular and subadditive on $(\Delta_d^{\downarrow}, \wedge, \vee)$:
  • $H(x) + H(y) \leq H(x\wedge y) + H(x\vee y)$ (supermodularity)
  • $H(x) + H(y) \geq H(x\wedge y) + H(x\vee y)$ (subadditivity)
  • (Deside et al., 2023).

7. Approximate Majorization and Contrasts with Fidelity

Approximate transformations, such as in entanglement concentration or protocols requiring only approximate conversion, can employ the supremum in the majorization lattice as an explicit target state. While fidelity-based criteria (such as that of Vidal–Jonathan–Nielsen) select, among all states majorized by a source, the one maximizing fidelity to a given target, this solution is generally not coincident with the lattice supremum when $d\geq 3$. Fidelity is not Schur-monotone in higher dimensions and may not respect the majorization preorder, contrary to the lattice join, which always produces the smallest upper bound in the majorization sense. In the two-dimensional case, fidelity and majorization order are aligned (Bosyk et al., 2016).

• “Optimal common resource in majorization-based resource theories” (Bosyk et al., 2019)
• “Probabilistic pure state conversion on the majorization lattice” (Deside et al., 2023)
• “Approximate transformations of bipartite pure-state entanglement from the majorization lattice” (Bosyk et al., 2016)