Quantum-Reachable Assemblages

• Quantum-reachable assemblages are conditional quantum states and channels defined by positivity, no-signaling, and normalization, underpinning multipartite steering scenarios.
• The framework leverages tools such as the Choi–Jamiołkowski isomorphism and the NPA hierarchy to rigorously characterize achievable quantum behaviors and identify edge assemblages.
• These assemblages drive device-independent protocols in entanglement certification and secure quantum cryptography, with methods like semidefinite programming for robustness testing.

Quantum-reachable assemblages are families of conditional quantum states or channels, arising in multipartite quantum steering and post-quantum information theory, that admit a realization within quantum mechanics and satisfy specified no-signaling constraints. They provide a rigorous mathematical and physical framework that captures the achievable (and non-achievable) behaviors in steering scenarios and quantum channel assemblages under operational, device-independent, and cryptographic paradigms.

1. Mathematical Framework and Definitions

The primary mathematical object is an assemblage—formally, a collection of positive semidefinite operators $\{\sigma_{a_1\ldots a_n|x_1\ldots x_n}\}$ (for state assemblages), or corresponding families of CP maps $\{\Lambda_{a_1\ldots a_n|x_1\ldots x_n}\}$ (for channel assemblages)—indexed by measurement choices $x_i$ and outcomes $a_i$ of $n$ untrusted parties $A_1,\ldots,A_n$ steering the state or channel given to a trusted party $B$ or $C$. These objects are subject to the following constraints:

• Positivity: Each subnormalized quantum state or map is a positive semidefinite operator, or a completely positive map.
• No-signaling: Marginalization over a subset of outcomes does not reveal information about the corresponding measurement settings of other parties. For state assemblages, for every strict subset $I \subset \{1,\ldots,n\}$,

$$\sum_{a_j:j\notin I} \sigma_{a_1\ldots a_n|x_1\ldots x_n} = \sigma_{a_{i_1}\ldots a_{i_s}|x_{i_1}\ldots x_{i_s}},$$

corresponding to marginal consistency conditions. For channel assemblages, analogous constraints apply to the Choi operators (see below).

• Normalization: Global sum or trace normalization, e.g., $$\sum_{a_1,\ldots,a_n} \sigma_{a_1\ldots a_n|x_1\ldots x_n} = \rho_B$$ for some fixed state $\rho_B$.

For channel assemblages, the Choi–Jamiołkowski isomorphism is used to represent each CP map $\Lambda$ by a positive semidefinite operator $J_\Lambda$ acting on $B\otimes A$, where

$$J_\Lambda = (\Lambda\otimes \mathrm{id}_A)(|\Phi^+_{AA}\rangle\langle\Phi^+_{AA}|),$$

and $\Lambda(\rho) = \mathrm{Tr}_A[J_\Lambda(\openone_B\otimes\rho^T)]$.

A quantum-reachable (synonym: realizable or quantum) assemblage is one that admits a constructive realization: $\sigma_{a_1\ldots a_n|x_1\ldots x_n} = \mathrm{Tr}_{A_1\ldots A_n}[(M^{(1)}_{a_1|x_1}\otimes\cdots\otimes M^{(n)}_{a_n|x_n}\otimes\openone)\rho_{A_1\ldots A_n B}],$ for some state and local measurements.

2. No-Signaling vs. Quantum-Reachable Sets

The set of all no-signaling assemblages (state or channel) forms a convex, compact set, denoted $S_2$ (states) or $\mathrm{ns}\Lambda$ (channels). The subset of quantum-reachable assemblages, indicated as $Q$ (states) or $\mathrm{q}\Lambda$ (channels), is proper whenever $n>1$: there exist assemblages that satisfy no-signaling but admit no quantum realization.

This strict inclusion, $$\mathrm{q}\Lambda(n,\dots)\subsetneq\mathrm{ns}\Lambda(n,\dots)$$ for $n>1$, is a manifestation of the phenomenon of "post-quantum steering." The separation is formally established via explicit constructions: for example, the "qubit-CNOT" channel assemblages provide extremal points in $\mathrm{ns}\Lambda(2,2,2,2)$ that are not quantum-reachable, as shown by the Choi isomorphism (Banacki et al., 2022).

The convex geometry is as follows:

Set	Containment	Physical meaning
$S_1$/LHS	$S_1\subset Q\subset S_2$	Admits a local hidden state model
$Q$/$\mathrm{q}\Lambda$	quantum-reachable	Realizable by quantum theory
$S_2$/$\mathrm{ns}\Lambda$	no-signaling	Consistent with no superluminal signaling

Assemblages on the "edge" of $S_2$ (not decomposable as mixtures with a local-hidden-state contribution) embody the most nonclassical steering resources, with necessary rank and determinant-based criteria for extremality (Banacki et al., 2020).

3. Choi–Jamiołkowski Characterization of Channel Assemblages

For channel assemblages, the Choi operator $J(\Lambda_{a_1\ldots a_n|x_1\ldots x_n})$ forms the bridge between no-signaling and quantum sets. The characterization is:

• No-signaling channel assemblage: Operators $J$ are positive, sum to the Choi operator of a CPTP map, and their marginals over any subset are independent of settings outside that subset.
• Quantum channel assemblage: $J$ admits a dilation in the form

$J(\Lambda_{a_1\ldots a_n|x_1\ldots x_n}) = \mathrm{Tr}_{A_1\ldots A_n}[(M_{a_1|x_1}^{(1)}\otimes\cdots\otimes M_{a_n|x_n}^{(n)}\otimes\openone_{C\tilde C})W],$

for global state $W$ and local POVMs.

Channels that are extremal in the no-signaling set correspond (via their Choi operators) to extremal points in the state-assemblage set. Inflexible (pure) assemblages of pure states, where the supports form linearly independent subspaces, provide sufficient conditions for extremality. This carries over directly to the channel scenario (Banacki et al., 2022).

4. Device-Independent and Operational Characterizations

Quantum-reachable assemblages play a critical role in device-independent protocols for entanglement certification and channel witnessing. Robust self-testing frameworks quantify the closeness of a measured assemblage to an ideal reference, either via trace-distance or fidelity. Given an observed Bell violation $I_\text{obs}$, the assemblage-to-assemblage fidelity can be lower-bounded as a function of $I_\text{obs}$ by semidefinite programming (SDP) relaxation (Chen et al., 2020). This allows, for instance, robust certification of all two-qubit entangled states and all non-entanglement-breaking qubit channels, replacing the need for full state or measurement tomography with SDP-based steering analysis.

In the multipartite steering scenario, assemblages reconstructed from tomographically-complete measurements on the trusted party provide a physically transparent and complete operational description of the quantum-reachable set (Rossi et al., 2022). In this framework, almost quantum assemblages arise via the Navascués–Pironio–Acín (NPA) moment-matrix hierarchy and satisfy macroscopic noncontextuality, forming a computationally tractable outer approximation to the set of quantum-reachable assemblages.

5. Extremality, Edge Assemblages, and Witness Construction

The convex geometric theory of assemblages identifies extremal quantum-reachable points via rank and support criteria. An assemblage is on the "edge" of the no-signaling set $S_2$ (with respect to $S_1$) if it is not a nontrivial convex combination with an LHS assemblage. Determinant-based projectors $R_{a|x}$ onto the supports of $\sigma_{a|x}$ provide necessary and sufficient conditions for edge membership: no local deterministic box should correspond to projectors sharing a common $+1$ eigenvector (Banacki et al., 2020). Rank constraints also serve as practical criteria.

For entangled quantum states of rank strictly less than three (in the three-qubit case), such as rank-2 pure states, it is possible to obtain edge assemblages. With rank at least three, no choice of local POVMs yields edge assemblages under post-quantum steering. This operational boundary constrains possible extremal behavior and is currently the subject of ongoing classification efforts.

Witnesses for post-quantum steering and edge assemblages are constructed via Hahn–Banach separation. Block operators $Z_{a|x}=\openone - R_{a|x}$ form the starting point of a family of linear functionals $W$ that are nonnegative on all LHS models but strictly negative on edge assemblages.

6. Asymmetric and Relaxed Steering Scenarios

Relaxing the no-signaling conditions can yield operational settings with partial or asymmetric no-signaling requirements. In the channel context, asymmetric no-signaling channel assemblages require, for example, no signaling only between one party and the joint system of the rest. This structure allows the certification of secure key bits in cryptographic protocols: using extremal quantum assemblies, it is possible to construct correlations $P_{abc|xyz}$ such that, under the full set of no-signaling constraints, an adversary's information is trivialized—thus ensuring security against a general no-signaling eavesdropper (Banacki et al., 2022).

The mathematical formulation reduces, via the Choi isomorphism, to clear linear and trace constraints on the set of conditional Choi operators. These relaxations introduce new operational regimes and invite further paper regarding the dimension and multipartite impact on achievable key rates and security.

7. Open Problems and Frontier Directions

Several theoretical and operational challenges remain open:

• Characterizing necessary and sufficient conditions for extremality of general no-signaling channel assemblages beyond the pure-state (inflexible) regime.
• Mapping the complete boundary $\partial(Q)$ inside the full no-signaling set $S_2$ or $\mathrm{ns}\Lambda$ for multipartite, high-dimensional, or infinite-dimensional scenarios.
• Extending the classification of edge and almost quantum assemblages and their witnesses to settings with many measurement choices or outcomes, or to higher local Hilbert space dimensions.
• Developing efficient (beyond brute-force SDP) algorithms for detecting quantum reachability and edge status, possibly exploiting the algebraic structure of the projectors $R_{a|x}$ and the NPA hierarchy (Rossi et al., 2022).
• Precisely identifying physical or information-theoretic principles (beyond macroscopic noncontextuality) that single out the quantum set as opposed to the almost-quantum or generic no-signaling set.

A related operational question is how extremal or edge assemblages behave in semi-device-independent tasks including randomness certification and cryptographic primitives. The full convex geometry and its boundary phenomena for multipartite and higher-dimensional scenarios remain active fields of research.