Three-Outcome Partial Bell Measurement

• The measurement is a two-qubit process that resolves only three Bell sectors by using a POVM with two rank-1 projectors and a rank-2 coarse-graining of odd-parity states.
• It enables semi-device-independent self-testing and practical protocols such as quantum Bernoulli factories and rotation sensing by excluding the antisymmetric Bell state.
• Implementations on superconducting hardware and optical setups map Bell states to computational bases, illustrating efficient designs under linear optics constraints.

Searching arXiv for relevant papers on three-outcome partial Bell basis measurements and related Bell-basis measurement implementations. A three-outcome partial Bell basis measurement is a two-qubit measurement that resolves only three Bell sectors rather than performing a full four-outcome Bell-state measurement. In recent work, this structure appears in two closely related forms: as an explicit three-outcome POVM with two rank-1 Bell projectors and one rank-2 coarse-grained Bell sector, and as an effective three-outcome regime of Bell-state analysis in which the antisymmetric singlet state $|\Psi^{-}\rangle$ has zero support because the inputs lie in a symmetric subspace (Bhaumik et al., 28 Aug 2025, Roy, 5 Feb 2026, Bao et al., 22 May 2026). In both forms, the measurement isolates the operationally relevant Bell statistics while avoiding the need for complete Bell discrimination, and it has been used for semi-device-independent self-testing, entanglement-assisted quantum Bernoulli factories, and rotation sensing.

1. Formal definition and Bell-space structure

The standard Bell basis on $\mathbb{C}^2 \otimes \mathbb{C}^2$ is

$$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$

A three-outcome partial Bell basis measurement, in the strict POVM sense analyzed in the self-testing literature, is not a full projective measurement onto four orthogonal Bell states. Instead, it is the POVM

$$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$

with $M_1$ and $M_2$ rank-1 and $M_3$ rank-2; the third outcome coarse-grains the odd-parity subspace spanned by $|\Psi^{+}\rangle$ and $|\Psi^{-}\rangle$ (Bhaumik et al., 28 Aug 2025). The POVM is complete and positive semidefinite, with

$M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$

A distinct but related usage arises when a full Bell measurement is physically implemented but one Bell component is provably absent on the states of interest. For two identical input quoins in the Bernoulli-factory experiment, and for the two-qubit marginals of the symmetric anti-coherent sensor states used in rotation sensing, $\mathbb{C}^2 \otimes \mathbb{C}^2$0 never occurs ideally, so the Bell analysis becomes effectively three-outcome (Roy, 5 Feb 2026, Bao et al., 22 May 2026).

Setting	Three-outcome structure	Mechanism
Semi-device-independent certification	$\mathbb{C}^2 \otimes \mathbb{C}^2$1	Explicit POVM with rank-2 coarse-graining
Quantum Bernoulli factory	$\mathbb{C}^2 \otimes \mathbb{C}^2$2	$\mathbb{C}^2 \otimes \mathbb{C}^2$3 has zero support for identical inputs
Rotation sensing	$\mathbb{C}^2 \otimes \mathbb{C}^2$4	Symmetric two-body marginals exclude the singlet

This comparison shows that “partial Bell basis measurement” can denote either a coarse-grained Bell POVM or a symmetry-reduced Bell analysis. A plausible implication is that the common feature is not the precise grouping of Bell states, but the extraction of all task-relevant Bell information from only three outcomes.

2. Symmetry and the disappearance of the fourth Bell outcome

The key structural fact behind the effective three-outcome regime is the decomposition of two-qubit space into symmetric and antisymmetric sectors. With swap operator $\mathbb{C}^2 \otimes \mathbb{C}^2$5, the Bell states satisfy

$\mathbb{C}^2 \otimes \mathbb{C}^2$6

Thus $\mathbb{C}^2 \otimes \mathbb{C}^2$7 span the symmetric triplet subspace, while $\mathbb{C}^2 \otimes \mathbb{C}^2$8 is the antisymmetric singlet (Bao et al., 22 May 2026).

In the Bernoulli-factory setting, a $\mathbb{C}^2 \otimes \mathbb{C}^2$9-quoin is prepared as

$$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$0

Two identical quoins yield the product state $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$1, which lies entirely in the symmetric subspace. Its Bell-basis expansion contains only $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$2, $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$3, and $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$4, with no $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$5 component, so the outcome probabilities are

$$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$6

in the ideal case (Roy, 5 Feb 2026). Two useful identities follow immediately: $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$7

In rotation sensing, the same triplet restriction follows from a different symmetry principle. The $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$8-photon polarization states are restricted to the totally symmetric, total-spin $$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},\qquad |\Psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}.$$9 subspace, and the collective rotation

$$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$0

commutes with all qubit permutations, hence preserves symmetry exactly for all $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$1 (Bao et al., 22 May 2026). For any pure state in $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$2, every two-qubit reduced density matrix has zero overlap with the antisymmetric projector $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$3, so pairwise Bell projections populate only $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$4 and $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$5 (Bao et al., 22 May 2026).

This symmetry analysis clarifies a frequent misunderstanding. A three-outcome Bell measurement does not necessarily mean that the fourth Bell state cannot be defined or measured; rather, in the symmetric-input regimes studied experimentally and metrologically, the fourth outcome is absent because the state family itself has no support there.

3. Semi-device-independent self-testing of the partial Bell POVM

A semi-device-independent certification protocol for the three-outcome partial Bell basis measurement was given in a communication scenario with two spatially separated senders $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$6, $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$7 and one receiver $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$8 (Bhaumik et al., 28 Aug 2025). Each sender $$M_1=|\Phi^{+}\rangle\!\langle\Phi^{+}|,\qquad M_2=|\Phi^{-}\rangle\!\langle\Phi^{-}|,\qquad M_3=|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|,$$9 receives two bits $M_1$0, prepares a qubit state $M_1$1, and sends it to the receiver. The receiver has input $M_1$2. For $M_1$3, the receiver performs binary prepare-and-measure tasks based on $M_1$4 random access codes with relabeling $M_1$5, $M_1$6. For $M_1$7, the receiver performs the three-outcome POVM $M_1$8 and outputs $M_1$9 (Bhaumik et al., 28 Aug 2025).

The protocol uses two witnesses. The RAC witness is

$M_2$0

with maximum quantum value

$M_2$1

The three-outcome communication witness is

$M_2$2

with maximum quantum value $M_2$3, where

$M_2$4

and

$M_2$5

$M_2$6

$M_2$7

(Bhaumik et al., 28 Aug 2025).

Theorem 4 states that if the observed values reach $M_2$8 and $M_2$9, then there exist single-qubit unitaries $M_3$0 such that the senders’ preparations coincide with the ideal states of Theorem 1 and the receiver’s measurement is, up to $M_3$1,

$M_3$2

This is semi-device-independent in the precise sense used in the paper: the only assumption is a dimension bound $M_3$3 on the communicated systems, and no pre-shared entanglement among distant parties is required (Bhaumik et al., 28 Aug 2025).

The robustness analysis is analytic rather than sample-complexity based. If $M_3$4 and $M_3$5 is arbitrary, Theorem 4 gives explicit lower bounds on the fidelities of $M_3$6 and $M_3$7 with the ideal $M_3$8, using constants

$M_3$9

together with operator inequalities derived from a local self-dual channel $|\Psi^{+}\rangle$0 (Bhaumik et al., 28 Aug 2025). For the bound to be informative, the paper stipulates

$|\Psi^{+}\rangle$1

The paper does not provide explicit finite-sample confidence intervals; the robustness guarantees are asymptotic, assuming well-estimated probabilities (Bhaumik et al., 28 Aug 2025).

4. Physical realizations and measurement circuits

On IBM superconducting hardware, a Bell measurement was realized by mapping Bell states to the computational basis and then measuring both qubits in the $|\Psi^{+}\rangle$2 basis (Roy, 5 Feb 2026). The standard mapping is implemented by applying a CNOT with control qubit 1 and target qubit 2, then a Hadamard on qubit 1, followed by $|\Psi^{+}\rangle$3-basis measurement. Under this sequence,

$|\Psi^{+}\rangle$4

The experiment used IBM’s 127-qubit Eagle r3 device, ibm_sherbrooke, selecting adjacent qubits with CNOT (ECR) error $|\Psi^{+}\rangle$5, readout errors $|\Psi^{+}\rangle$6, and single-qubit gate errors $|\Psi^{+}\rangle$7 (Roy, 5 Feb 2026). The quoin state was prepared from $|\Psi^{+}\rangle$8 by a calibrated $|\Psi^{+}\rangle$9-rotation with angle $|\Psi^{-}\rangle$0. For each $|\Psi^{-}\rangle$1 in a sweep from $|\Psi^{-}\rangle$2 to $|\Psi^{-}\rangle$3, the experiment acquired $|\Psi^{-}\rangle$4 shots, and the mapped Bell distributions were consistent with the ideal three-outcome prediction: $|\Psi^{-}\rangle$5 remained near $|\Psi^{-}\rangle$6 independent of $|\Psi^{-}\rangle$7, $|\Psi^{-}\rangle$8 followed $|\Psi^{-}\rangle$9, $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$0 followed $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$1, and $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$2 remained near zero with small leakage (Roy, 5 Feb 2026).

The optical implementations proposed in the certification and sensing papers use standard polarization-encoded photonic qubits and linear optics (Bhaumik et al., 28 Aug 2025, Bao et al., 22 May 2026). The basic components are a non-polarizing 50:50 beam splitter, polarizing beam splitters, half-wave plates or a polarization phase gate, phase shifters, and single-photon detectors. Hong–Ou–Mandel interference at the beam splitter separates antisymmetric and symmetric sectors: $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$3 gives anti-bunching, while the symmetric Bell states bunch. Additional phase-sensitive interferometry then separates $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$4 from $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$5, while orthogonal-polarization signatures are grouped as the $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$6 sector in the certification setting or identified as $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$7 in the sensing setting because $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$8 has no support there (Bhaumik et al., 28 Aug 2025, Bao et al., 22 May 2026).

These implementations illustrate two experimentally distinct routes to three outcomes. One route begins from a nominally full Bell analysis and becomes partial because the input family forbids one Bell component. The other route implements a genuinely coarse-grained Bell POVM that is matched to what linear optics can realize deterministically without ancillas.

5. Bernoulli-factory primitives from three Bell outcomes

The Bernoulli-factory experiment uses the three nonvanishing Bell outcomes as a randomness-processing primitive (Roy, 5 Feb 2026). Because $M_1+M_2+M_3=I_4,\qquad M_i\succeq 0.$9 independent of $\mathbb{C}^2 \otimes \mathbb{C}^2$00, a single Bell measurement on two identical quoins yields an exact fair coin by the deterministic rule: output 1 if the outcome is $\mathbb{C}^2 \otimes \mathbb{C}^2$01 or $\mathbb{C}^2 \otimes \mathbb{C}^2$02, and output 0 if the outcome is $\mathbb{C}^2 \otimes \mathbb{C}^2$03 or $\mathbb{C}^2 \otimes \mathbb{C}^2$04. In the ideal case this gives

$\mathbb{C}^2 \otimes \mathbb{C}^2$05

and uses exactly two quoins per fair bit (Roy, 5 Feb 2026).

The same Bell-measurement statistics also implement the classically inconstructible function $\mathbb{C}^2 \otimes \mathbb{C}^2$06. The procedure keeps only runs yielding $\mathbb{C}^2 \otimes \mathbb{C}^2$07 or $\mathbb{C}^2 \otimes \mathbb{C}^2$08, discards $\mathbb{C}^2 \otimes \mathbb{C}^2$09 and $\mathbb{C}^2 \otimes \mathbb{C}^2$10, and labels $\mathbb{C}^2 \otimes \mathbb{C}^2$11 as output 1 and $\mathbb{C}^2 \otimes \mathbb{C}^2$12 as output 0. The resulting conditional bias is

$\mathbb{C}^2 \otimes \mathbb{C}^2$13

Because half of the shots fall in $\mathbb{C}^2 \otimes \mathbb{C}^2$14, this construction consumes, on average, four quoins per output bit and is independent of $\mathbb{C}^2 \otimes \mathbb{C}^2$15 (Roy, 5 Feb 2026).

The experiment also constructs the Bernoulli doubling primitive

$\mathbb{C}^2 \otimes \mathbb{C}^2$16

which is classically inconstructible in the sense used by the paper (Roy, 5 Feb 2026). The conditional $\mathbb{C}^2 \otimes \mathbb{C}^2$17 frequency supplies a coin with bias $\mathbb{C}^2 \otimes \mathbb{C}^2$18, and a standard classical Bernoulli factory subroutine then transforms a coin of bias $\mathbb{C}^2 \otimes \mathbb{C}^2$19 into a coin of bias $\mathbb{C}^2 \otimes \mathbb{C}^2$20, using only fair coins generated internally from the same Bell measurement and no external classical randomness source. The paper identifies the $\mathbb{C}^2 \otimes \mathbb{C}^2$21 step as a Catalan-number-based repeat-until-success procedure. In the reported data, the median quoin cost for doubling was $\mathbb{C}^2 \otimes \mathbb{C}^2$22 per output bit, reflecting the intrinsic inefficiency of the $\mathbb{C}^2 \otimes \mathbb{C}^2$23 classical subroutine (Roy, 5 Feb 2026).

The resource profile is correspondingly uneven. Each Bell measurement has shallow depth—one $\mathbb{C}^2 \otimes \mathbb{C}^2$24 rotation per qubit, one CNOT, one Hadamard, and final $\mathbb{C}^2 \otimes \mathbb{C}^2$25-basis measurement—and consumes exactly two quoins. The fair coin has constant cost, the $\mathbb{C}^2 \otimes \mathbb{C}^2$26 coin has constant average cost, and the doubling construction is dominated by classical post-processing overhead (Roy, 5 Feb 2026). This suggests that the practical power of the three-outcome Bell primitive is strongest for functions whose post-processing acts directly on the triplet statistics, rather than through an additional classical Bernoulli factory reduction.

6. Metrological use in rotation sensing and relation to full Bell measurements

In rotation sensing with second-order anti-coherent states, pair-wise Bell state analysis with an additional path degree of freedom yields a three-outcome Bell measurement that is optimal for the sensing task (Bao et al., 22 May 2026). Because the relevant sensor states occupy the totally symmetric subspace and collective rotations preserve that subspace, only the symmetric Bell states $\mathbb{C}^2 \otimes \mathbb{C}^2$27, $\mathbb{C}^2 \otimes \mathbb{C}^2$28, and $\mathbb{C}^2 \otimes \mathbb{C}^2$29 appear in projection, while $\mathbb{C}^2 \otimes \mathbb{C}^2$30 is absent (Bao et al., 22 May 2026). The optical analyzer uses a 50:50 beam splitter, a polarization phase $\mathbb{C}^2 \otimes \mathbb{C}^2$31 gate, PBSs, and detectors; within the symmetric subspace, discrimination among the three triplet Bell states is deterministic conditioned on having a detected pair (Bao et al., 22 May 2026).

For small rotation angle $\mathbb{C}^2 \otimes \mathbb{C}^2$32, the optimal symmetric-subspace projectors give probabilities

$\mathbb{C}^2 \otimes \mathbb{C}^2$33

and the resulting estimator saturates the quantum Cramér–Rao bound in the small-angle regime, with

$\mathbb{C}^2 \otimes \mathbb{C}^2$34

(Bao et al., 22 May 2026). Numerically, the paper gives $\mathbb{C}^2 \otimes \mathbb{C}^2$35 for $\mathbb{C}^2 \otimes \mathbb{C}^2$36 with $\mathbb{C}^2 \otimes \mathbb{C}^2$37 and $\mathbb{C}^2 \otimes \mathbb{C}^2$38 for $\mathbb{C}^2 \otimes \mathbb{C}^2$39 with $\mathbb{C}^2 \otimes \mathbb{C}^2$40, with $\mathbb{C}^2 \otimes \mathbb{C}^2$41 and $\mathbb{C}^2 \otimes \mathbb{C}^2$42, respectively (Bao et al., 22 May 2026).

The relation to full Bell-state measurement is central across all three works. The self-testing paper emphasizes that complete Bell-state measurement is impossible with passive linear optics without ancillary systems, while the chosen three-outcome grouping is easily implementable in an optical setup and compatible with those constraints (Bhaumik et al., 28 Aug 2025). The sensing paper makes the complementary point that the omitted $\mathbb{C}^2 \otimes \mathbb{C}^2$43 outcome carries no Fisher information for the relevant symmetric states, so resolving the three triplet outcomes is sufficient and optimal for the task (Bao et al., 22 May 2026). The Bernoulli-factory experiment reaches a similar operational conclusion on superconducting hardware: for identical quoins, the antisymmetric Bell sector adds no useful information because it has zero ideal support, and the remaining three outcomes already suffice to generate the fair coin, $\mathbb{C}^2 \otimes \mathbb{C}^2$44, and the doubling primitive (Roy, 5 Feb 2026).

A common misconception is that a partial Bell measurement is merely an imperfect approximation to a full Bell measurement. The recent literature gives a more specific picture. In one setting, the three-outcome POVM is the target measurement to be certified; in the others, the three-outcome regime is the exact information-bearing structure induced by symmetry. Under those conditions, the fourth Bell outcome is not a missing resource but an unoccupied or deliberately coarse-grained sector.