Bi-Pooling Property: Order-Independence in Quantum States

• The bi-pooling property is the order-independence in pooling two quantum state assignments via a semidefinite program, ensuring symmetric extraction of a unique pooled state.
• It quantifies compatibility by maximizing the common positive operator under both density matrices, reflecting the overlap of observers' information.
• Operational success of the method hinges on specific measurement assumptions, thereby framing its practical relevance in quantum state inference and pooling techniques.

The bi-pooling property is the order-independence of a two-observer pooling rule for quantum state assignments: given two density matrices $\rho_A$ and $\rho_B$ on the same Hilbert space, one defines a compatibility measure through a semidefinite program and, from its optimizer, a unique pooled state $\sigma_{AB} = E^*/\Tr E^*$. In the treatment of compatibility and information pooling developed around the Brun–Finkelstein–Mermin notion of compatibility and analyzed via semidefinite programming, bi-pooling denotes the fact that pooling $\rho_A$ with $\rho_B$ yields the same result as pooling $\rho_B$ with $\rho_A$ (Brun et al., 2013). The concept is not merely algebraic: its operational meaning depends on explicit assumptions about how the observers’ information was acquired, and its scope is therefore narrower than any fully assumption-free notion of combining quantum state assignments.

1. Compatibility of two state assignments

Two density matrices $\rho_A$ and $\rho_B$ for one and the same quantum system are compatible in the Brun–Finkelstein–Mermin sense if and only if

$$\operatorname{supp}(\rho_A)\cap \operatorname{supp}(\rho_B)\neq \{0\}.$$

Equivalently, there exists at least one nonzero vector $\rho_B$0 to which both state assignments give nonzero probability. This criterion expresses whether the two assignments could represent observers with differing information about the same underlying system.

The support-overlap condition is a zero-probability consistency requirement. If the supports do not overlap, then no joint information state $\rho_B$1 can respect both observers’ zero-probability assignments. In this sense, compatibility is weaker than equality of state assignments but stronger than mere coexistence as arbitrary descriptions.

The 2013 analysis emphasizes that compatibility measures are only sensible under assumptions about the kind of information used to generate the state assignments. It also stresses that a state assignment does not, in general, represent all of the information possessed by an observer. This restricts the interpretive domain of any pooling rule: a pooling map is not automatically licensed by the abstract density operators alone (Brun et al., 2013).

2. Quantitative compatibility and its semidefinite program

To quantify the degree to which two state assignments can simultaneously be correct, Kitaev suggested the measure

$\rho_B$2

Here the optimization variable $\rho_B$3 is a positive semidefinite operator constrained to lie below both state operators. The objective $\rho_B$4 is linear, so the problem is a standard semidefinite program:

• maximize $\rho_B$5
• subject to $\rho_B$6, $\rho_B$7, and $\rho_B$8.

This SDP asks how large a common positive operator can be embedded simultaneously under both $\rho_B$9 and $\sigma_{AB} = E^*/\Tr E^*$0. The quantity $\sigma_{AB} = E^*/\Tr E^*$1 therefore measures the largest weight of a shared sub-state. In the formulation summarized in the paper, one can show that

$\sigma_{AB} = E^*/\Tr E^*$2

with $\sigma_{AB} = E^*/\Tr E^*$3 exactly when the supports do not meet, and $\sigma_{AB} = E^*/\Tr E^*$4 only if $\sigma_{AB} = E^*/\Tr E^*$5 (Brun et al., 2013).

A useful upper bound is

$\sigma_{AB} = E^*/\Tr E^*$6

where $\sigma_{AB} = E^*/\Tr E^*$7 is the trace distance. This places the compatibility measure in direct relation to a standard distinguishability quantity: larger trace distance constrains the size of any common part. A plausible implication is that the SDP-based measure captures a notion of overlap that is stricter than geometric support intersection but weaker than full identity of states.

3. Construction of the pooled state and the bi-pooling property

Let $\sigma_{AB} = E^*/\Tr E^*$8 denote the optimizer of the compatibility SDP. The pooled state is then defined by

$\sigma_{AB} = E^*/\Tr E^*$9

Equivalently,

$\rho_A$0

In the detailed summary, $\rho_A$1 is described as the unique optimizer, or at least as the unique maximal-trace solution in general. The pooling rule thus extracts the normalized common component of maximal trace.

The bi-pooling property is the symmetry-induced order-independence of this construction. Because the SDP is invariant under interchange of $\rho_A$2 and $\rho_A$3, the optimization

$\rho_A$4

has the same solution as

$\rho_A$5

Accordingly, the same $\rho_A$6 and the same $\rho_A$7 arise whether one speaks of pooling A’s state with B’s or B’s state with A’s. This symmetry and uniqueness are precisely what is identified as the bi-pooling property (Brun et al., 2013).

In operational terms, bi-pooling is not merely commutativity of notation. It asserts that the joint state assignment extracted from the two observers’ information is insensitive to the order in which their reports are combined, provided the pooling rule is the SDP rule above. This suggests that the pooled state is a canonical two-party object for that specific compatibility notion.

4. Measurement-model interpretation and required assumptions

The identification of $\rho_A$8 as the appropriate pooled state depends on a specific information-acquisition model. Two assumptions are singled out:

• the prior state is maximally mixed;
• a single joint positive measurement $\rho_A$9 is performed, with outcomes split so that A learns $\rho_B$0 alone and B learns $\rho_B$1 alone.

Under these assumptions, maximizing the joint-outcome probability

$\rho_B$2

subject to reproducing the marginals $\rho_B$3 and $\rho_B$4 forces $\rho_B$5 to saturate exactly the constraints $\rho_B$6 (Brun et al., 2013). The SDP is therefore not an abstract optimization detached from physical procedure; it arises from a concrete split-outcome POVM scenario.

Within that scenario, the pooled state $\rho_B$7 is operationally the most likely joint-state outcome. The mathematical object selected by the SDP coincides with the state associated with the most probable common event compatible with both observers’ partial information. This interpretation, however, is conditional rather than universal. Without the maximally mixed prior and the single bipartite positive POVM with separately learned outcome labels, the same marginals $\rho_B$8 and $\rho_B$9 can arise from different underlying data structures, leading to different joint states.

The paper therefore treats the pooling rule as unique only relative to the specified assumptions. This is an important limitation: the density operators alone do not determine a unique pooled state absent a model of how the information was generated.

5. Extension beyond two observers

For a family of $\rho_B$0 observers with state assignments $\rho_B$1, the same formal pattern extends through

$\rho_B$2

with pooled state $\rho_B$3. This generalization preserves the SDP structure and the interpretation of compatibility as the size of a common positive operator lying below every $\rho_B$4.

The extension is mathematically straightforward but operationally incomplete. The simple two-party measurement picture does not extend in a unique way to $\rho_B$5. In particular, one loses the interpretation of maximizing a single common-outcome probability unless one imposes a very special symmetrical joint measurement and prior (Brun et al., 2013).

This marks a conceptual boundary for bi-pooling. In the two-party case, symmetry of the SDP directly yields order-independence. For $\rho_B$6 observers, a formally analogous common-part optimization exists, but the operational narrative that justifies the pooled state is no longer canonical. A plausible implication is that multi-observer pooling requires stronger structural assumptions than pairwise pooling if one wants the pooled state to retain a direct measurement-theoretic meaning.

6. Relation to Post–Peierls and Equal-Support compatibility

The SDP treatment places Brun–Finkelstein–Mermin compatibility between two other notions: Post–Peierls (PP) compatibility and Equal-Support (ES) compatibility. PP compatibility requires that for every measurement there is at least one outcome to which all $\rho_B$7 assign nonzero probability. ES compatibility requires that all $\rho_B$8 have exactly the same support.

The corresponding measures can also be written in optimization form. For PP compatibility,

$\rho_B$9

For ES compatibility, the measure is the maximum scalar $\rho_A$0 such that

$\rho_A$1

The hierarchy stated in the summary is

$\rho_A$2

and likewise

$\rho_A$3

Notion	Criterion	Relative strength
ES	all $\rho_A$4 share exactly the same support	strongest
BFM	supports have nontrivial overlap	intermediate
PP	every measurement has an outcome to which all assign nonzero probability	weakest

This ordering clarifies what bi-pooling does and does not capture. It is tied specifically to the BFM notion of overlap and to the maximal common sub-state selected by the SDP. It is therefore neither the only possible pooling principle nor a generic feature of all compatibility notions. Similar SDP-based measures exist for PP and ES compatibility, but the bi-pooling construction in the strict sense refers to the two-party BFM pooling rule (Brun et al., 2013).

7. Uniqueness, scope, and limitations

The BFM pooling rule is unique under two assumptions: all prior ignorance is represented by the maximally mixed state, and the observers’ information arises from a single bipartite positive POVM whose outcome labels are distributed so that each observer learns only one component. Under these conditions, the SDP optimizer $\rho_A$5 determines a unique pooled state and does so symmetrically.

Without those assumptions, uniqueness fails. Different joint measurements or different methods of preparing correlated data can reproduce the same marginals $\rho_A$6 and $\rho_A$7 while yielding different underlying joint states. Thus, the same pair of density matrices need not imply a unique pooled state in general. The bi-pooling property is therefore a statement about a specific reconstruction procedure, not an assumption-free theorem about arbitrary quantum state assignments.

In summary, the two-party SDP

$\rho_A$8

simultaneously defines a compatibility measure,

$\rho_A$9

and a pooled state,

$\rho_A$0

Its order-independence is immediate from symmetry of the feasible set, and that symmetry is what is designated the bi-pooling property. The construction is exact, concise, and technically natural within its intended measurement model, but its interpretive validity depends on that model rather than on the reduced state assignments alone (Brun et al., 2013).