Optimal Common Resource in Quantum Theories

• Optimal Common Resource (OCR) is the minimal quantum state from which every target state can be generated using allowed free operations.
• It is constructed using majorization theory, where infimum and supremum operations within a complete lattice determine optimality.
• The framework offers practical methods for bipartite entanglement while revealing challenges in extending OCR constructions to multipartite systems.

An Optimal Common Resource (OCR) is a central concept in quantum resource theories, specifically in the context of resource conversion governed by majorization. The OCR is the minimal or “weakest” resource state from which all members of a prescribed target set can be generated via the allowed free operations of the theory, such as Local Operations and Classical Communication (LOCC) in entanglement theory or incoherent operations in coherence theory. The existence, uniqueness, and explicit construction of OCRs have been established for majorization-based resource theories and for all bipartite pure-state entanglement scenarios, whereas the multipartite case exhibits greater complexity and partial results. The framework is built on the mathematical structure of the majorization lattice, guaranteeing that OCRs exist—even for arbitrary (possibly continuous) target families—and providing a geometric recipe for their determination (Guo et al., 2016, Bosyk et al., 2019).

1. Definitions and Preliminaries

Let $T = \{\rho_i: i \in I\}$ be a collection of target quantum states within a given quantum resource theory (QRT). The conversion rules between states in such theories often induce a preorder $\,\rightarrow\,$ on the set of allowable states, typically determined by comparison of associated vectors (e.g., Schmidt coefficients, eigenvalues, or coefficients in a fixed basis).

Common Resource. A state $\rho$ is a common resource for $T$ if, for every $\sigma \in T$, $\rho \rightarrow \sigma$ (i.e., $\rho$ can be transformed into every $\sigma$ using only free operations).

Optimal Common Resource (OCR). A state $\rho^{\mathrm{ocr}}$ is an OCR for $T$ if:

• (i) It is a common resource for $T$,
• (ii) For any other common resource $\rho'$, either $\rho' \rightarrow \rho^{\mathrm{ocr}}$ or $\rho^{\mathrm{ocr}}$ and $\rho'$ are incomparable.

Equivalently, the OCR is the “least” common resource, meaning that it supplies all target states under the resource preorder, and no strictly weaker common resource exists.

2. Majorization and the Lattice Structure

For quantum resource theories governed by majorization—entanglement (pure states), coherence (pure states), and purity—the resource preorder is encapsulated by the majorization relation on probability vectors in the ordered simplex $\Delta^\downarrow_d$.

Given $x, y \in \Delta^\downarrow_d$, $x \succeq y$ iff for all $k=1, \ldots, d-1$, $\sum_{i=1}^k x_i \geq \sum_{i=1}^k y_i$. This preorder turns the simplex into a complete lattice: any (even uncountable) subset has a unique infimum (meet) and supremum (join).

The existence of OCRs is thus ensured for any collection of target states whose “resource vectors” (e.g., sorted Schmidt coefficients) reside in $\Delta^\downarrow_d$ (Bosyk et al., 2019).

3. Construction of the OCR in Majorization-Based Theories

Fix $$P = \{x^{(i)} : i \in I\} \subset \Delta^\downarrow_d$$, representing the target states. Define the partial sums

$S_k(x) = \sum_{j=1}^k x_j,\quad S_0(x) = 0.$

Construct

$$S_k^{\inf} = \inf_{x\in P} S_k(x), \quad S_k^{\sup} = \sup_{x\in P} S_k(x),\; (k=0,\ldots,d)$$

• The infimum $x^{\inf}$, giving the OCR for theories with conversion $x\rightarrow y$ iff $x\succeq y$ (or reversed—see section 6), is

$x^{\inf}_k = S_k^{\inf} - S_{k-1}^{\inf}.$

This vector is by construction nonincreasing and normalized and satisfies $x^{\inf} \succeq x$ for all $x \in P$.

• The supremum $x^{\sup}$ is constructed similarly, but generally requires a concavification step to ensure nonincreasing order.

For finite $P$, the infimum and supremum reduce to pointwise minima and maxima of the partial sums. For infinite or continuum $P$, the approach extends via the use of Lorenz curves (Bosyk et al., 2019).

4. Optimal Common Resource for Bipartite Pure-State Entanglement

For bipartite pure states $$|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$$, the Schmidt decomposition applies, and Nielsen’s theorem states that $$|\psi\rangle \rightarrow_{\mathrm{LOCC}} |\phi\rangle$$ iff the Schmidt vector $\lambda^\psi$ is majorized by $\lambda^\phi$. For any set $S = \{|\phi_i\rangle\}$ of bipartite pure states (finite or infinite), the unique OCR is explicitly constructible:

• For each $k=1,\dots,d$, let

$m_k = \begin{cases} \min_{i} \sum_{j=1}^k x_j^{(i)}, & \text{if $S$ is finite} \ \inf_{i} \sum_{j=1}^k x_j^{(i)}, & \text{if $S$ is infinite} \end{cases}, \quad m_0 = 0.$

• Set $y_k = m_k - m_{k-1}$ for $k=1,\ldots,d$.
• The resulting vector $(y_1,\ldots,y_d)$ is the Schmidt vector of $|\psi^{\mathrm{ocr}}\rangle$ (Guo et al., 2016).

This construction yields the unique state such that for all $i$, $\lambda^{\mathrm{ocr}} \prec \lambda^{(i)}$ and for any other common resource $|\chi\rangle$, $\lambda^{\chi} \prec \lambda^{\mathrm{ocr}}$.

Computational Complexity: For $N$ target states of Schmidt rank $d$, the OCR is found in $O(Nd)$ time for the minima, $O(d)$ for differences. The example $d=3$, $S = \{(0.7,0.2,0.1),\, (0.6,0.3,0.1)\}$ yields the OCR $(0.6,0.3,0.1)$ (Guo et al., 2016).

5. Multipartite States, Nontriviality, and the GHZ₃ Case

For multipartite entanglement, the absence of a universal Schmidt decomposition results in a much richer structure: the LOCC preorder is not total, and SLOCC equivalence classes proliferate (e.g., W- and GHZ-classes for three qubits). As a result, the systematic construction of OCRs is generally intractable.

Nevertheless, explicit nontrivial common resources have been identified. The three-qutrit GHZ state

$$|GHZ_3\rangle = \frac{1}{\sqrt{3}} (|000\rangle + |111\rangle + |222\rangle)$$

serves as a common resource for all three-qubit pure states ($$|GHZ_3\rangle \rightarrow_{\mathrm{LOCC}} |\phi\rangle$$ for all $$|\phi\rangle \in \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$$) via explicit finite-round LOCC protocols (classification by type of target: W-class and GHZ-class). However, no $3\times 2\times 2$ state can play this role, as demonstrated by an analysis of SLOCC classes and matrix pencil invariants (Guo et al., 2016).

A “trivial” common resource is the maximally entangled state in a $1\otimes (d_1 \cdots d_n)$ system, but this is operationally expensive due to its high-dimensional requirement.

6. Applications and Examples in Quantum Resource Theories

The methodology and existence of OCRs generalize across all majorization-based QRTs:

• Quantum Coherence Theory (Pure States): Under incoherent operations, conversion is governed by reversed majorization; the OCR corresponds to the infimum of the target set, leading to optimal supply of coherent states with minimal resource content (Bosyk et al., 2019).
• Purity Theory: For unital maps, direct majorization applies, and the OCR corresponds to the supremum.
• Continuous or Uncountable Target Sets: The completeness of the majorization lattice ensures that the OCR exists and is explicitly constructible, even for target classes described by continuous parameters.

Examples include:

• Pure states with largest amplitude $\geq \alpha$: OCR is $$\alpha|1\rangle + \sum_{i=2}^d \sqrt{(1-\alpha^2)/(d-1)}\,|i\rangle$$.
• Qubit mixed states with spectra $(p,1-p)$, $p\in[a,b]$: OCR is the mixed state with eigenvalues $(a,1-a)$ (Bosyk et al., 2019).

7. Limits, Generalizations, and Operational Significance

The OCR identifies the minimal resource requirement for achieving a range of state preparations in majorization-based QRTs, with broad implications for quantum communication, thermodynamics, and information theory. For resource theories that lack a majorization structure or in multipartite scenarios, only partial results or nontrivial common resources are generally available.

Approximate majorization and “$\epsilon$-OCRs” have been studied in the thermodynamic context, with the steepest and flattest approximations connected directly to suprema and infima in the majorization lattice, underscoring the geometric and operational power of the lattice formalism (Bosyk et al., 2019).

The interplay between lattice completeness, geometric construction via Lorenz curves, and operational convertibility under free operations provides a unifying perspective and practical mechanism for quantifying minimal resource needs in quantum protocols.