---
title: Three-Outcome Partial Bell Measurement
url: https://www.emergentmind.com/topics/three-outcome-partial-bell-basis-measurement
type: topic
---

# Three-Outcome Partial Bell Measurement

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 [2508.21178] [2602.06193] [2605.24108]. 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 \(|\Psi^{-}\rangle\) [2508.21178]. 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, \(|\Psi^{-}\rangle\) never occurs ideally, so the Bell analysis becomes effectively three-outcome [2602.06193] [2605.24108].

| Setting | Three-outcome structure | Mechanism |
|---|---|---|
| Semi-device-independent certification | \(\{|\Phi^{+}\rangle,|\Phi^{-}\rangle,\{|\Psi^{+}\rangle,|\Psi^{-}\rangle\}\}\) | Explicit POVM with rank-2 coarse-graining |
| Quantum Bernoulli factory | \(\{|\Phi^{+}\rangle,|\Phi^{-}\rangle,|\Psi^{+}\rangle\}\) | \(|\Psi^{-}\rangle\) has zero support for identical inputs |
| Rotation sensing | \(\{|\Phi^{+}\rangle,|\Phi^{-}\rangle,|\Psi^{+}\rangle\}\) | 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 \(\mathsf{S}\vert ab\rangle=\vert ba\rangle\), the Bell states satisfy
\[
\mathsf{S}|\Phi^{\pm}\rangle=+|\Phi^{\pm}\rangle,\qquad
\mathsf{S}|\Psi^{+}\rangle=+|\Psi^{+}\rangle,\qquad
\mathsf{S}|\Psi^{-}\rangle=-|\Psi^{-}\rangle.
\]
Thus \(\{|\Phi^{+}\rangle,|\Phi^{-}\rangle,|\Psi^{+}\rangle\}\) span the symmetric triplet subspace, while \(|\Psi^{-}\rangle\) is the antisymmetric singlet [2605.24108].

In the Bernoulli-factory setting, a \(p\)-quoin is prepared as
\[
|\psi_p\rangle=\sqrt{1-p}\,|0\rangle+\sqrt{p}\,|1\rangle.
\]
Two identical quoins yield the product state \( |\psi_p\rangle\otimes|\psi_p\rangle \), which lies entirely in the symmetric subspace. Its Bell-basis expansion contains only \(|\Phi^{+}\rangle\), \(|\Phi^{-}\rangle\), and \(|\Psi^{+}\rangle\), with no \(|\Psi^{-}\rangle\) component, so the outcome probabilities are
\[
P(\Phi^{+})=\frac{1}{2},\qquad
P(\Phi^{-})=\frac{(1-2p)^2}{2},\qquad
P(\Psi^{+})=2p(1-p),\qquad
P(\Psi^{-})=0
\]
in the ideal case [2602.06193]. Two useful identities follow immediately:
\[
P(\Phi^{+}\cup\Psi^{-})=\tfrac12,\qquad
P(\Phi^{-}\cup\Psi^{+})=\tfrac12.
\]

In rotation sensing, the same triplet restriction follows from a different symmetry principle. The \(N\)-photon polarization states are restricted to the totally symmetric, total-spin \(J=N/2\) subspace, and the collective rotation
\[
\hat{\mathcal U}(\hat n,\theta)=\left[e^{-i\theta\,\hat n\cdot\boldsymbol{\sigma}}\right]^{\otimes N}
=e^{-i\theta\,\hat n\cdot\boldsymbol{\hat J}}
\]
commutes with all qubit permutations, hence preserves symmetry exactly for all \(\theta\) [2605.24108]. For any pure state in \(\mathrm{Sym}^N(\mathbb{C}^2)\), every two-qubit reduced density matrix has zero overlap with the antisymmetric projector \(P_{\mathrm{asym}}=(I-\mathsf S)/2\), so pairwise Bell projections populate only \(\Pi_{\Phi^{\pm}}\) and \(\Pi_{\Psi^{+}}\) [2605.24108].

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 \(A^{(1)}\), \(A^{(2)}\) and one receiver \(R\) [2508.21178]. Each sender \(j\in\{1,2\}\) receives two bits \((x_j,a_j)\in\{0,1\}\times\{0,1\}\), prepares a qubit state \(\rho^{(j)}_{a_j|x_j}\in\mathbb C^2\), and sends it to the receiver. The receiver has input \(k\in\{1,2,3\}\). For \(k=1,2\), the receiver performs binary prepare-and-measure tasks based on \(2\to1\) random access codes with relabeling \(x'_1=x_1\oplus a_1\), \(a'_1=a_1\). For \(k=3\), the receiver performs the three-outcome POVM \(\{M^3_1,M^3_2,M^3_3\}\) and outputs \(i\in\{1,2,3\}\) [2508.21178].

The protocol uses two witnesses. The RAC witness is
\[
S^{RAC}=\sum_{a'_1,x'_1}\frac14\left[p(a'_1|a'_1,x'_1,1)+p(x'_1|a'_1,x'_1,2)\right],
\]
with maximum quantum value
\[
S^{RAC}_Q=\frac{1+1/\sqrt2}{2}\approx 0.85.
\]
The three-outcome communication witness is
\[
S^{Comm}=\frac{1}{8\sqrt2}(W'_1+W'_2+W'_3),
\]
with maximum quantum value \(S^{Comm}_Q=1\), where
\[
A^{(j)}_{x_j}=\rho^{(j)}_{0|x_j}-\rho^{(j)}_{1|x_j},
\]
and
\[
W'_1=(A^{(1)}_0+A^{(1)}_1)\otimes A^{(2)}_0+(A^{(1)}_0-A^{(1)}_1)\otimes A^{(2)}_1,
\]
\[
W'_2=-(A^{(1)}_0+A^{(1)}_1)\otimes A^{(2)}_0+(A^{(1)}_0-A^{(1)}_1)\otimes A^{(2)}_1,
\]
\[
W'_3=-2(A^{(1)}_0-A^{(1)}_1)\otimes A^{(2)}_1
\]
[2508.21178].

Theorem 4 states that if the observed values reach \(S^{RAC}=(1+1/\sqrt2)/2\) and \(S^{Comm}=1\), then there exist single-qubit unitaries \(U_1,U_2\) such that the senders’ preparations coincide with the ideal states of Theorem 1 and the receiver’s measurement is, up to \(U_1\otimes U_2\),
\[
|\Phi^{+}\rangle\!\langle\Phi^{+}|,\quad
|\Phi^{-}\rangle\!\langle\Phi^{-}|,\quad
|\Psi^{+}\rangle\!\langle\Psi^{+}|+|\Psi^{-}\rangle\!\langle\Psi^{-}|.
\]
This is semi-device-independent in the precise sense used in the paper: the only assumption is a dimension bound \(d=2\) on the communicated systems, and no pre-shared entanglement among distant parties is required [2508.21178].

The robustness analysis is analytic rather than sample-complexity based. If \(S^{Comm}\ge 1-\epsilon\) and \(S^{RAC}\) is arbitrary, Theorem 4 gives explicit lower bounds on the fidelities of \(M^3_1\) and \(M^3_2\) with the ideal \(|\Phi^{\pm}\rangle\!\langle\Phi^{\pm}|\), using constants
\[
r=\frac{4+5\sqrt2}{16},\qquad
\mu=-\frac{1+2\sqrt2}{4},\qquad
2\sqrt2\,r+\mu=1,
\]
together with operator inequalities derived from a local self-dual channel \(\Lambda=\Lambda^{(1)}\otimes\Lambda^{(2)}\) [2508.21178]. For the bound to be informative, the paper stipulates
\[
\epsilon\le \frac{3}{12+8\sqrt2}.
\]
The paper does not provide explicit finite-sample confidence intervals; the robustness guarantees are asymptotic, assuming well-estimated probabilities [2508.21178].

## 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 \(Z\) basis [2602.06193]. 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 \(Z\)-basis measurement. Under this sequence,
\[
00\leftrightarrow \Phi^{+},\qquad
01\leftrightarrow \Phi^{-},\qquad
10\leftrightarrow \Psi^{+},\qquad
11\leftrightarrow \Psi^{-}.
\]
The experiment used IBM’s 127-qubit Eagle r3 device, ibm\_sherbrooke, selecting adjacent qubits with CNOT (ECR) error \(\sim 7.0\times10^{-3}\), readout errors \(\sim 10^{-2}\), and single-qubit gate errors \(<3.4\times10^{-4}\) [2602.06193]. The quoin state was prepared from \(|0\rangle\) by a calibrated \(Y\)-rotation with angle \(\theta=2\sin^{-1}\sqrt p\). For each \(p\) in a sweep from \(0\) to \(1\), the experiment acquired \(5\times10^4\) shots, and the mapped Bell distributions were consistent with the ideal three-outcome prediction: \(\Phi^{+}\) remained near \(1/2\) independent of \(p\), \(\Phi^{-}\) followed \((1-2p)^2/2\), \(\Psi^{+}\) followed \(2p(1-p)\), and \(\Psi^{-}\) remained near zero with small leakage [2602.06193].

The optical implementations proposed in the certification and sensing papers use standard polarization-encoded photonic qubits and linear optics [2508.21178] [2605.24108]. 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: \(|\Psi^{-}\rangle\) gives anti-bunching, while the symmetric Bell states bunch. Additional phase-sensitive interferometry then separates \(|\Phi^{+}\rangle\) from \(|\Phi^{-}\rangle\), while orthogonal-polarization signatures are grouped as the \(|\Psi^{\pm}\rangle\) sector in the certification setting or identified as \(|\Psi^{+}\rangle\) in the sensing setting because \(|\Psi^{-}\rangle\) has no support there [2508.21178] [2605.24108].

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 [2602.06193]. Because \(P(\Phi^{+})=1/2\) independent of \(p\), a single Bell measurement on two identical quoins yields an exact fair coin by the deterministic rule: output 1 if the outcome is \(\Phi^{+}\) or \(\Psi^{-}\), and output 0 if the outcome is \(\Phi^{-}\) or \(\Psi^{+}\). In the ideal case this gives
\[
P(\Phi^{+}\cup\Psi^{-})=P(\Phi^{-}\cup\Psi^{+})=\frac12,
\]
and uses exactly two quoins per fair bit [2602.06193].

The same Bell-measurement statistics also implement the classically inconstructible function \(f(p)=4p(1-p)\). The procedure keeps only runs yielding \(\Phi^{-}\) or \(\Psi^{+}\), discards \(\Phi^{+}\) and \(\Psi^{-}\), and labels \(\Psi^{+}\) as output 1 and \(\Phi^{-}\) as output 0. The resulting conditional bias is
\[
\mathbb P(\Psi^{+}\mid \Phi^{-}\cup\Psi^{+})=4p(1-p).
\]
Because half of the shots fall in \(\Phi^{-}\cup\Psi^{+}\), this construction consumes, on average, four quoins per output bit and is independent of \(p\) [2602.06193].

The experiment also constructs the Bernoulli doubling primitive
\[
f_{\wedge}(p)=
\begin{cases}
2p,& p\le \tfrac12,\\[2mm]
2(1-p),& p>\tfrac12,
\end{cases}
\]
which is classically inconstructible in the sense used by the paper [2602.06193]. The conditional \(\Phi^{-}\) frequency supplies a coin with bias \((1-2p)^2\), and a standard classical Bernoulli factory subroutine then transforms a coin of bias \(x=(1-2p)^2\) into a coin of bias \(\sqrt{x}=|1-2p|\), using only fair coins generated internally from the same Bell measurement and no external classical randomness source. The paper identifies the \(\sqrt{\cdot}\) step as a Catalan-number-based repeat-until-success procedure. In the reported data, the median quoin cost for doubling was \(303\) per output bit, reflecting the intrinsic inefficiency of the \(\sqrt{\cdot}\) classical subroutine [2602.06193].

The resource profile is correspondingly uneven. Each Bell measurement has shallow depth—one \(Y\) rotation per qubit, one CNOT, one Hadamard, and final \(Z\)-basis measurement—and consumes exactly two quoins. The fair coin has constant cost, the \(4p(1-p)\) coin has constant average cost, and the doubling construction is dominated by classical post-processing overhead [2602.06193]. 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 [2605.24108]. Because the relevant sensor states occupy the totally symmetric subspace and collective rotations preserve that subspace, only the symmetric Bell states \(\Phi^{+}\), \(\Phi^{-}\), and \(\Psi^{+}\) appear in projection, while \(\Psi^{-}\) is absent [2605.24108]. The optical analyzer uses a 50:50 beam splitter, a polarization phase \(S\) gate, PBSs, and detectors; within the symmetric subspace, discrimination among the three triplet Bell states is deterministic conditioned on having a detected pair [2605.24108].

For small rotation angle \(\theta\), the optimal symmetric-subspace projectors give probabilities
\[
P_0=1-\theta^2\frac{J(J+1)}{3},\qquad
P_i=\theta^2\frac{J(J+1)}{3}n_i^2,\quad i=1,2,3,
\]
and the resulting estimator saturates the quantum Cramér–Rao bound in the small-angle regime, with
\[
F_C=\frac{4J(J+1)}{3}=F_Q
\]
[2605.24108]. Numerically, the paper gives \(F_Q=8\) for \(N=4\) with \(J=2\) and \(F_Q=16\) for \(N=6\) with \(J=3\), with \(\sigma_{\theta}=1/(2\sqrt{2n})\) and \(\sigma_{\theta}=1/(4\sqrt n)\), respectively [2605.24108].

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 [2508.21178]. The sensing paper makes the complementary point that the omitted \(\Psi^{-}\) outcome carries no Fisher information for the relevant symmetric states, so resolving the three triplet outcomes is sufficient and optimal for the task [2605.24108]. 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, \(4p(1-p)\), and the doubling primitive [2602.06193].

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.

Source: https://www.emergentmind.com/topics/three-outcome-partial-bell-basis-measurement