Local State Antimarking (LSAM)

• Local State Antimarking (LSAM) is a sequence-level exclusion task in quantum information where spatially separated parties using LOCC certify that a given ordered sequence was not supplied.
• LSAM extends local state antidistinguishability by leveraging sequence sampling without replacement, revealing sequence activation and distinct nonlocal properties.
• The framework offers both theoretical criteria and explicit LOCC protocols, highlighting its practical implications and contrasts with LSAD, CLSD, and CLSM.

Searching arXiv for the specified paper and closely related context. arXiv search: (Chatterjee et al., 10 May 2026) "Local state antimarking : Nonlocality without entanglement" Local State Antimarking (LSAM) is a sequence-level exclusion task in multipartite quantum information in which spatially separated parties, restricted to LOCC, must certify at least one ordered sequence of states that was not supplied. It is introduced as an extension of local state antidistinguishability (LSAD), itself an exclusion-based analogue of local state discrimination, and is used to study nonlocality without entanglement in a setting that is weaker than identification but still operationally stringent. The central results show that any ensemble of mutually orthogonal multipartite pure states is locally antidistinguishable, while certain product-state ensembles exhibit a sequence-level activation phenomenon: they become globally antidistinguishable when sampled without replacement into sequences, yet remain impossible to antimark locally. The framework also establishes that LSAD, LSAM, conclusive local state discrimination (CLSD), and conclusive local state marking (CLSM) are pairwise incomparable in general (Chatterjee et al., 10 May 2026).

1. Formal framework

The starting point is antidistinguishability. Let $S=\{\rho_1,\ldots,\rho_k\}$ be a finite set of quantum states on a finite-dimensional Hilbert space $H$. The set is strongly antidistinguishable if there exists a POVM $M=\{\Pi_j\}_{j=1,\ldots,k}$ such that each labeled state is perfectly excluded by its corresponding outcome,

$$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$

while every outcome is relevant,

$\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$

and the POVM obeys the usual normalization and positivity conditions,

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$

If only the exclusion condition is imposed and some outcomes are allowed never to click, the notion reduces to weak antidistinguishability. The treatment of LSAM uses strong antidistinguishability throughout (Chatterjee et al., 10 May 2026).

For three pure states $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$, the Caves–Fuchs–Schack criterion gives necessary and sufficient conditions. Writing

$$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$

the set is strongly antidistinguishable iff

$x_{12}+x_{13}+x_{23}<1,$

and

$$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$

The trine states provide a standard example, with antidistinguishing POVM $H$0 (Chatterjee et al., 10 May 2026).

LSAD is the multipartite LOCC-restricted version. For a multipartite ensemble $H$1 on $H$2, LSAD means there exists a finite-round LOCC protocol whose effective POVM elements $H$3 satisfy

$H$4

together with relevance and completeness. The paper works in the LOCC model rather than in the larger class of separable POVMs; its possibility and impossibility results are formulated accordingly (Chatterjee et al., 10 May 2026).

LSAM generalizes from single states to ordered sequences. Let $H$5 be a known set of multipartite states. A referee samples without replacement an ordered sequence

$H$6

of $H$7 distinct indices and distributes the corresponding product state

$H$8

with local tensor factors rearranged into the natural party-wise partition $H$9. The task $M=\{\Pi_j\}_{j=1,\ldots,k}$0-LSAM asks the parties, using LOCC only, to output $M=\{\Pi_j\}_{j=1,\ldots,k}$1 distinct strings

$M=\{\Pi_j\}_{j=1,\ldots,k}$2

where $M=\{\Pi_j\}_{j=1,\ldots,k}$3 is the set of all ordered length-$M=\{\Pi_j\}_{j=1,\ldots,k}$4 tuples of distinct indices from $M=\{\Pi_j\}_{j=1,\ldots,k}$5, such that

$M=\{\Pi_j\}_{j=1,\ldots,k}$6

with unit success probability in every run. The paper adopts the weakest exclusion requirement: excluding at least one constituent product state is already sufficient to exclude the ordered tuple as a whole (Chatterjee et al., 10 May 2026).

2. Structural lemmas and technical criteria

A central sufficient condition is the union-of-antidistinguishable-subsets theorem. If a finite state set $M=\{\Pi_j\}_{j=1,\ldots,k}$7 can be written as

$M=\{\Pi_j\}_{j=1,\ldots,k}$8

with every $M=\{\Pi_j\}_{j=1,\ldots,k}$9 strongly antidistinguishable, then $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$0 itself is strongly antidistinguishable. The proof constructs a single POVM by averaging the antidistinguishing POVMs of the subsets:

$$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$1

Completeness and positivity are immediate, while exclusion and relevance are inherited from the individual subset measurements (Chatterjee et al., 10 May 2026).

LSAM also obeys a monotonicity relation in the sequence length. If a set $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$2 permits $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$3-LSAM, then for any $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$4 with $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$5, the same set permits $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$6-LSAM with

$$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$7

The construction simply applies the successful $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$8-LSAM protocol to the first $$\mathrm{Tr}(\rho_j \Pi_j)=0 \quad \text{for all } j=1,\ldots,k,$$9 positions of a longer sequence. Each excluded ordered $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$0-tuple extends to exactly $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$1 ordered length-$\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$2 sequences by filling the remaining positions from the unused symbols without replacement (Chatterjee et al., 10 May 2026).

For qubit pure-state ensembles, the paper uses another criterion repeatedly: a set $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$3 in $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$4 is strongly antidistinguishable iff there exist positive coefficients $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$5 such that

$\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$6

This criterion is employed to rule out local antidistinguishability of certain single-party factors, which in turn precludes strong LOCC exclusion for the associated product ensembles (Chatterjee et al., 10 May 2026).

The paper’s LOCC impossibility arguments do not rely on separable-POVM impossibility theorems. Instead, they proceed by reducing LSAD or LSAM for product ensembles to antidistinguishability of the local single-system ensembles. In particular, when the local single-system sets fail strong antidistinguishability, strong LOCC antidistinguishability of the corresponding multipartite product ensemble fails as well (Chatterjee et al., 10 May 2026).

3. LSAD for orthogonal multipartite pure states

One of the main structural results is that every ensemble of mutually orthogonal multipartite pure states is locally antidistinguishable. The proof uses Walgate–Short–Hardy–Vedral’s local discrimination theorem for pairs of orthogonal bipartite pure states. For any such pair $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$7, there exists a local basis $\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$8 such that

$\sum_i \mathrm{Tr}(\rho_i \Pi_j)>0,$9

with

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$0

Alice measures in $\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$1, Bob performs the conditional orthogonality-resolving measurement, and the pair is perfectly discriminated by LOCC. The LSAD theorem then partitions the full orthogonal set into pairs, applies the pairwise exclusion subprotocols, and combines them using the union theorem. Shared randomness is used to select one of the pair subprotocols, giving a strong LOCC antidistinguishing measurement in which every state has an outcome that never occurs when that state is present (Chatterjee et al., 10 May 2026).

The Bell basis in $\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$2 yields an explicit example:

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$3

Alice and Bob both measure in the computational basis $\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$4. The joint outcomes exclude Bell states according to

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$5

Each excluded state assigns zero probability to its corresponding outcome, and each outcome occurs with nonzero probability on at least one of the other Bell states, so strong LSAD holds (Chatterjee et al., 10 May 2026).

A second explicit protocol applies to Bennett’s nine orthogonal product states in $\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$6,

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$7

with the unnormalized shorthand $\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$8. Alice measures in the orthogonal basis

$\sum_j \Pi_j=I,\qquad \Pi_j\ge 0.$9

Bob measures in

$|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$0

and the joint outcome excludes one of the nine states. The paper provides the complete mapping table. This shows that the canonical “nonlocality without entanglement” states become local under LSAD, even though they are famous for LOCC discrimination obstructions (Chatterjee et al., 10 May 2026).

4. Sequence activation and LSAM nonlocality without entanglement

The defining LSAM phenomenon is that sequence formation can activate global antidistinguishability while LOCC exclusion remains impossible. The basic example is

$|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$1

with $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$2. In the single-shot setting this set is not globally antidistinguishable: the three-state Caves conditions fail. At sequence length $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$3 without replacement, however, one obtains the six ordered product states

$|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$4

with local partition $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$5 (Chatterjee et al., 10 May 2026).

The paper constructs three triples,

$|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$6

each of which satisfies the Caves conditions and is therefore strongly globally antidistinguishable. By the union theorem, the full six-state sequence ensemble $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$7 is strongly globally antidistinguishable (Chatterjee et al., 10 May 2026).

The same activation does not occur locally. The local factor sets for both parties are

$|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$8

and these fail the Caves conditions. Since neither local party possesses an antidistinguishing measurement on the corresponding local three-state ensemble, one cannot realize a strong LOCC antidistinguishing measurement on the product sequence set. Consequently, $|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle$9 is globally antimarkable at $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$0 but not $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$1-LSAM. This is a direct instance of nonlocality without entanglement in LSAM: the underlying states are product states, yet sequence-level exclusion becomes globally possible and remains LOCC-impossible (Chatterjee et al., 10 May 2026).

This example also clarifies the operational role of sampling without replacement. LSAM is not merely repeated LSAD on independent copies. The exclusion target is an ordered tuple in $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$2, and the combinatorics of distinct indices create new global antidistinguishability structures that are absent in the single-shot ensemble. A plausible implication is that LSAM isolates sequence-level nonlocal effects that are invisible to state-by-state exclusion (Chatterjee et al., 10 May 2026).

5. Incomparability with LSAD, CLSD, and CLSM

The paper formulates CLSD and CLSM as conclusive, error-free tasks with permitted inconclusive outcomes. For CLSD, a LOCC POVM $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$3 must satisfy

$$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$4

with $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$5. For CLSM, for each ordered sequence $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$6 there is a LOCC effect $$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$7 such that

$$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$8

where

$$x_{12}=|\langle \psi_1|\psi_2\rangle|^2,\quad x_{13}=|\langle \psi_1|\psi_3\rangle|^2,\quad x_{23}=|\langle \psi_2|\psi_3\rangle|^2,$$9

and an inconclusive $x_{12}+x_{13}+x_{23}<1,$0 completes the POVM (Chatterjee et al., 10 May 2026).

The general implication structure includes

$x_{12}+x_{13}+x_{23}<1,$1

Thus failure of LSAM implies failure of LSD, LSAD, and LSM. However, the paper shows that no strict hierarchy exists among LSAD, LSAM, CLSD, and CLSM (Chatterjee et al., 10 May 2026).

A first inequivalence comes from Duan’s four-state product ensemble

$x_{12}+x_{13}+x_{23}<1,$2

with

$x_{12}+x_{13}+x_{23}<1,$3

Globally, the subsets $x_{12}+x_{13}+x_{23}<1,$4 and $x_{12}+x_{13}+x_{23}<1,$5 satisfy the Caves conditions, so their union $x_{12}+x_{13}+x_{23}<1,$6 is strongly antidistinguishable. Locally, for the single-system set

$x_{12}+x_{13}+x_{23}<1,$7

there do not exist positive weights $x_{12}+x_{13}+x_{23}<1,$8 such that

$x_{12}+x_{13}+x_{23}<1,$9

This precludes local antidistinguishability of the qubit factor set and therefore rules out LSAD for $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$0. Yet at sequence length $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$1, the local two-copy sets can be partitioned into antidistinguishable triples satisfying the Caves conditions, so a LOCC protocol excludes at least one sequence; $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$2 admits $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$3-LSAM. The set is therefore CLSD-nonlocal and LSAD-nonlocal, while remaining LSAM-local (Chatterjee et al., 10 May 2026).

A second separation is provided by

$$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$4

The single-shot product ensemble fails LSAD locally because its local single-system factors violate the Caves conditions. Nonetheless, the paper shows that the two-copy local sets split into antidistinguishable triples, and the union theorem then implies $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$5-LSAM via LOCC (Chatterjee et al., 10 May 2026).

A sharper LSAM-versus-LSAD separation is the PBR-like ensemble

$$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$6

It is not LSAD, because the local single-system set $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$7 is not strongly antidistinguishable. For $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$8, however, there is an explicit local projective measurement on $$[1-(x_{12}+x_{13}+x_{23})]^2 \ge 4x_{12}x_{13}x_{23}.$$9 and on $H$00 in the orthonormal basis $H$01: $H$02

$H$03

where $H$04. Each outcome excludes at least three sequences, yielding $H$05-LSAM by LOCC (Chatterjee et al., 10 May 2026).

The anti-parallel double-SIC construction gives the opposite direction. Let

$H$06

where $H$07 is a qubit SIC, i.e. four states forming a regular tetrahedron on the Bloch sphere. This set is globally and locally strongly antidistinguishable; for example, Alice may measure $H$08. At the same time, known results show that it is not locally conclusively distinguishable. This establishes LSAD-locality without CLSD-locality (Chatterjee et al., 10 May 2026).

6. Stronger nonlocality, explicit LSAM constructions, and open directions

The paper also presents a tripartite product ensemble showing that LSAM can reveal nonlocality even when CLSD and CLSM succeed. Consider

$H$09

where

$H$10

Global antidistinguishability holds for this three-state set when

$H$11

equivalently for

$H$12

For $H$13-LSAM, the local two-copy factor sets become

$H$14

for Alice and Bob, and

$H$15

for Charlie. These local three-state sets are strongly antidistinguishable only when

$H$16

which holds for

$H$17

Outside that interval, but still inside the globally antidistinguishable regime, $H$18-LSAM fails. In the same parameter range, CLSD and CLSM succeed: Alice and Bob measure in $H$19 and Charlie in $H$20. This establishes that LSAM can exhibit a stronger nonlocal obstruction than CLSD or CLSM (Chatterjee et al., 10 May 2026).

Another explicit sequence-level construction is given for the Manna–Bhowmik $H$21-ensemble

$H$22

For $H$23, the ensemble fails $H$24-LSAM, i.e. LSAD. At $H$25, however, there is an explicit Alice POVM $H$26 with

$H$27

and

$H$28

$H$29

such that

$H$30

$H$31

$H$32

$H$33

The six outcomes exclude the local pattern pairs

$H$34

and when Bob applies the same POVM, at least eight global sequences are excluded, yielding $H$35-LSAM (Chatterjee et al., 10 May 2026).

Operationally, LSAM is the sequence-level counterpart of exclusion: the parties need not identify the true sequence, only guarantee that at least one proposed ordered sequence did not occur. Because exclusion is weaker than discrimination, failure of LSAM signals a particularly strong form of nonlocality. The paper’s examples show both upward propagation with sequence length, via the monotonicity lemma, and nontrivial dependence on $H$36, since $H$37-LSAM may hold even when $H$38-LSAM does not, and may fail even when CLSD and CLSM are possible (Chatterjee et al., 10 May 2026).

All constructive positive results are realized by finite-round LOCC, and for the explicit protocols the number of rounds is one: parties measure locally, communicate once, and map outcomes to excluded states or sequences. Negative results are derived from single-system antidistinguishability criteria on the local factors rather than from separable-measurement impossibility theorems. Two open directions are identified: whether there exist mutually orthogonal mixed-state ensembles that fail LSAD, and whether there exist bipartite ensembles that are globally antidistinguishable yet fail LSAM. Tripartite examples already exist, but the bipartite case remains unresolved (Chatterjee et al., 10 May 2026).