Directional Simulability in Quantum Systems

• Directional simulability is the study of how one set of quantum operations simulates another using free transformations, classical post-processing, and symmetry criteria.
• It employs order theory and linear algebra to classify resource convertibility and operational tasks across quantum, GPT, and channel contexts.
• The framework enables efficient simulation verification by reducing complex commutator closures to manageable convex and algebraic computations.

Directional simulability describes the capacity, within the operational, quantum, or GPT frameworks, for one set of devices, observables, Hamiltonians, or channels to simulate another via prescribed sets of free transformations, classical post-processing, or composition rules. The directional nature refers to the inherent asymmetry present in the simulability relation: that $A$ can simulate $B$ does not, in general, imply the reverse. Directional simulability is stratified through the lens of order theory—most commonly as a preorder—enabling a rigorous classification of physical resources, operational tasks, and information-theoretic phenomena across a diversity of scenarios, including effective Hamiltonian engineering, measurement theory, and quantum channel conversion (Zimborás et al., 2015, Leppäjärvi, 2021, Filippov et al., 2018, 2002.04240). The subject provides both a structural foundation for resource theories and efficient symmetry-based criteria for determining simulation possibilities without requiring direct computation of commutator closures or full extremal decompositions.

1. Formal Definitions Across Frameworks

Directional simulability is defined contextually according to the operational entity of interest:

A. Quantum Interactions/Hamiltonians

Given sets of skew-Hermitian generators $H = \{ iH_1, \ldots, iH_m \}$ (available interactions) and $K = \{ iK_1, \ldots, iK_n \}$ (target interactions), $H$ directionally simulates $K$ ($H \to K$) if the Lie algebra generated by $K$ is contained in that generated by $H$: $$g_K = \operatorname{Lie}(iK_1, \ldots, iK_n) \subseteq g_H = \operatorname{Lie}(iH_1, \ldots, iH_m)$$ (Zimborás et al., 2015).

B. Quantum and GPT Measurements

For observables $A$ and $B$ with outcome spaces $X$ and $Y$ on a state space $S$, $A$ is simulated by $B$ ($A \preceq B$) if there exists a stochastic map $\nu$ such that for all $x\in X$,

$A_x = \sum_{y\in Y} \nu_{y\to x} B_y$

Equivalently, this extends to convex mixtures and post-processings over families of observables, generating a closure under these operations (Leppäjärvi, 2021, Filippov et al., 2018).

C. Quantum Channels

Given two channels $\Phi_1$ and $\Phi_2$, and a convex subcategory $\mathcal{F}$ of free superchannels, $\Phi_1$ directionally F-simulates $\Phi_2$ if there exists a free superchannel $\Lambda \in \mathcal{F}$ such that $\Lambda(\Phi_1) = \Phi_2$, or, for approximate simulability, $$\|\Lambda(\Phi_1) - \Phi_2\|_\diamond \leq \varepsilon$$ (2002.04240).

2. Preorder Structure and Order-Theoretic Properties

The simulability relation is generically a preorder:

• Reflexivity: Every object simulates itself.
• Transitivity: If $C \geq B$ and $B \geq A$, then $C \geq A$.
• Antisymmetry failure: Directional simulability is not necessarily symmetric; $B \geq A$ does not imply $A \geq B$.

The preorder often descends to a partial order on equivalence classes, where two objects are equivalent if each simulates the other (measurement scenario: $A \sim B \Leftrightarrow A \preceq B$ and $B \preceq A$) (Filippov et al., 2018, Leppäjärvi, 2021).

Minimal elements (simulation-irreducible objects) play a key structural role. For measurements in finite-dimensional quantum theory, these are exactly the extreme rank-1 POVMs (up to outcome relabeling) (Filippov et al., 2018).

3. Symmetry Criteria and Efficient Characterization

A. Quantum Hamiltonians

Zimborás et al. established that checking $g_K \subset g_H$ efficiently bypasses explicit commutator closure by analyzing symmetry structures:

• The commutant $C(g_H)$ yields linear symmetries—matrices commuting with all $iH_i$.
• Quadratic symmetries require analyzing $C_2(H)$, the commutant of the symmetric tensor-square Lie algebra.

Directional simulability $H \to K$ holds if and only if:

1. $\dim C_2(H) = \dim C_2(H \cup K)$ (no change in quadratic symmetries on adding targets)
2. $$\operatorname{rank} \widetilde{T} = \operatorname{rank} T$$ (central projections remain unchanged).

These criteria reduce the problem to linear algebraic computations rather than exponential commutator growth (Zimborás et al., 2015).

B. Measurements and Channels

For measurements, simulability is checked via convex geometry and extremality, examining whether one observable's effects lie in the conical hull of another's after classical post-processing (Filippov et al., 2018).

For quantum channels, F-simulability is fully characterized by conditional min-entropy monotones, operationalized through guessing games, and related to classical randomization criteria. The resource-theoretic structure persists for arbitrary families $\mathcal{F}$ of free superchannels (2002.04240).

4. Directional Asymmetry and Operational Consequences

Directional asymmetry is intrinsic:

• In measurements, there exist observables $A, B$ such that $B \preceq A$ but $A \not\preceq B$ (e.g., noisy vs. pure qubit observables, trine POVMs vs. their coarse-grainings) (Filippov et al., 2018, Leppäjärvi, 2021).
• In channels, simulation in one direction via superchannels does not imply the reverse (2002.04240).

This asymmetry underpins resource theories, ranking operational tasks by simulability strength, and underlines the hierarchy of compatibility notions such as $k$-simulability and effective $n$-outcome simulation (Filippov et al., 2018).

5. Connections to Compatibility, Incompatibility, and No-Free-Information

Joint measurability/compatibility of observables is subsumed as a special case of simulability, typically interpreted as being simulable from a single device:

• A collection $\{A^{(i)}\}$ is compatible iff there exists $C$ and post-processings such that each $A^{(i)}$ is a post-processing of $C$ (Leppäjärvi, 2021).
• The intersection of all simulability cones of simulation-irreducible observables contains the fully compatible class.
• In quantum theory, the only measurement compatible with all others (simulable from all irreducibles) is the trivial measurement—a manifestation of the no-free-information (NFI) principle. Some GPTs violate NFI, witnessing nontrivial freely compatible measurements (Leppäjärvi, 2021).

6. Extensions: Resource Theories, GPTs, and Channel Comparison

Directional simulability generalizes naturally to the GPT setting, where simulation relations, operational preorders, and resource monotones retain their formal properties for arbitrary state spaces, observables, and transformations (Filippov et al., 2018, 2002.04240).

In channel theory, F-simulability admits a full resource-theoretic characterization (via F-modified conditional min-entropy), operational interpretation (success probabilities in guessing games), and extensions to bipartite channels and measurement sets. The formulation subsumes Le Cam's classical randomization criterion for statistical experiments (2002.04240).

7. Illustrative Examples and Computational Implications

A. Two-Qubit Hamiltonians

For $H = \{iX_1, iY_1, iX_2, iY_2\}$ and $K = \{iZ_1Z_2\}$, $g_H = su(2)_1\oplus su(2)_2$, $g_{H\cup K}=su(4)$. Quadratic symmetries distinguish their generative capacity: $K$ is not simulable from $H$ (Zimborás et al., 2015).

B. Measurement Simulability

Noisy Pauli observables $B$ can simulate a pure observable $A$ via classical noise; the reverse direction is impossible, indicating strict directional simulability (Filippov et al., 2018).

C. Resource-Theoretic Monotones

Each nontrivial symmetry (element of commutant or quadratic commutant) defines a monotone that cannot increase under allowed simulation, providing a full suite of invariants to decide simulability (Zimborás et al., 2015, 2002.04240).

Efficient computation is enabled in all frameworks by reducing simulation questions to convex or linear-algebraic feasibility, rather than combinatorial enumeration of all possible dynamics or classical processings. This has substantial practical implications for quantum optimal control, device certification, and operationally meaningful resource convertibility.