Collective Protection and Exchange-Summed No-Jump Survival in Dipositronium

1. Introduction

A subspace is decoherence-free only with respect to the noise operators chosen to model the system–environment coupling. If those operators produce the same result on all states in the subspace, up to scalar factors common to that subspace, the modeled environment cannot distinguish those states through those operators [1,2,3,4]. Protection against decoherence is therefore channel-relative; it does not follow that the same subspace is protected under a different physical channel. In spontaneous-emission settings, this distinction is familiar: codes must account not only for emitted-photon jumps, but also for the no-jump evolution between them [7,8,9,10]. Thus the scalarity test for no-jump survival is not the same as the DFS test for the original noise algebra.

This paper gives a finite spin-and-channel example of that distinction. It shows that a modeled collective protected space need not remain scalar for conditional no-jump survival once annihilation is written in the exchange-summed form required by identical-particle symmetry.

Dipositronium is a useful setting because it is the minimal positronium molecule in which the fixed-pair picture becomes inadequate. A picture of two positronium atoms suggests a simple collective-spin protected space. True ${\mathrm{Ps}}_2$, however, contains two identical electrons and two identical positrons, with no permanent electron–positron pairs. Any electron can annihilate with any positron, so the annihilation sink must be exchange-summed over all electron–positron contacts [11]. In the equal-contact spin approximation, this exchange-summed sink is scalar on fixed exchange sectors $(S_e,S_p;F)$, but not generally on the whole collective $F=0$ protected space.

2. Scalarity Criteria

Let P be the projector onto a proposed protected subspace $\mathcal{C}$. For a family of modeled noise operators $\left\{L_{\alpha }\right\}$, a standard subspace DFS condition is

$$L_{\alpha }P=c_{\alpha }P,\mathrm{for}\mathrm{all}\alpha .$$

(2)

For closed unitary evolution within the subspace, the effective Hamiltonian must also preserve $\mathcal{C}$. Equation (2) characterizes only the selected noise algebra, not all channels acting on the same physical system.

For conditional survival, suppose the non-Hermitian effective Hamiltonian contains a positive sink operator K,

$$H_{\mathrm{eff}}=H-{\displaystyle \frac{i}{2}}K,$$

(3)

and, for the moment, ignore the unitary part or assume it preserves the subspace. The no-jump operator generated by the sink is

$$N\left(t\right)=e^{-Kt/2}.$$

(4)

A passive scalarity condition for the no-jump branch is

$$PN{\left(t\right)}^{†}N\left(t\right)P=c\left(t\right)P.$$

(5)

This is the one-Kraus-operator form of the usual error-correction scalarity condition [5,6], and it is the condition that the no-jump branch not change relative norms inside $\mathcal{C}$. If (5) fails, then postselected no-jump evolution is known but it is not scalar on the code.

The distinction tested below is therefore

$$L_{\alpha }P=c_{\alpha }P\cancel{⟹}PN{\left(t\right)}^{†}N\left(t\right)P=c\left(t\right)P.$$

(6)

Dipositronium supplies a finite example in which the two tests can be evaluated explicitly.

3. A Fixed-Pair Control Model

For one positronium atom, the spin singlet and triplet states are

$$\begin{array}{cc}\hfill \left|S\right.〉& ={\displaystyle \frac{1}{\sqrt{2}}}(\left|{\uparrow }_e{\downarrow }_p\right.〉-\left|{\downarrow }_e{\uparrow }_p\right.〉),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \left|T_{+}\right.〉& =\left|{\uparrow }_e{\uparrow }_p\right.〉,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \left|T_0\right.〉& ={\displaystyle \frac{1}{\sqrt{2}}}(\left|{\uparrow }_e{\downarrow }_p\right.〉+\left|{\downarrow }_e{\uparrow }_p\right.〉),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \left|T_{-}\right.〉& =\left|{\downarrow }_e{\downarrow }_p\right.〉.\hfill \end{array}$$

(10)

For the idealized collective-dephasing operator

$$L_z={\sigma }_z^{\left(e\right)}+{\sigma }_z^{\left(p\right)},$$

(11)

one has

$$L_z\left|S\right.〉=0,L_z\left|T_0\right.〉=0.$$

(12)

Thus $Span\{\left|S\right.〉,\left|T_0\right.〉\}$ is protected for that selected dephasing operator. This is a result for collective dephasing only; it does not imply that the singlet alone is the two-spin DFS. Under full collective $SU\left(2\right)$ noise, the two-spin singlet is the one-dimensional DFS, while the triplet is not a noiseless subsystem for two spins because there is no multiplicity space [2,3].

Consider two labeled electron–positron pairs, treated as distinguishable positronium atoms. They are used here as a pedagogical control model, not as a molecular identity structure for true ${\mathrm{Ps}}_2$. Define

$$\left|0_L\right.〉=\left|SS\right.〉,\left|1_L\right.〉=\left|TT;J=0\right.〉={\displaystyle \frac{1}{\sqrt{3}}}(\left|T_{+}{T}_{-}\right.〉-\left|T_0{T}_0\right.〉+\left|T_{-}{T}_{+}\right.〉).$$

(13)

The modeled collective generators are

$$J_k^{\left(2\right)}={\displaystyle \sum _{a=1}^2}(s_{e_a,k}+s_{p_a,k}),k=x,y,z.$$

(14)

Both states are total singlets of these generators:

$$J_k^{\left(2\right)}\left|0_L\right.〉=J_k^{\left(2\right)}\left|1_L\right.〉=0,k=x,y,z.$$

(15)

Accordingly, $Span\{\left|0_L\right.〉,\left|1_L\right.〉\}$ is a two-dimensional protected space for this fixed-pair collective model.

A fixed-pair annihilation sink already separates the two logical states. If

$$K_{\mathrm{pair}}^{\left(2\right)}={\displaystyle \sum _{a=1}^2}({\mathrm{\Gamma }}_S{P}_S^{\left(a\right)}+{\mathrm{\Gamma }}_T{P}_T^{\left(a\right)}),$$

(16)

then

$$K_{\mathrm{pair}}^{\left(2\right)}\left|0_L\right.〉=2{\mathrm{\Gamma }}_S\left|0_L\right.〉,K_{\mathrm{pair}}^{\left(2\right)}\left|1_L\right.〉=2{\mathrm{\Gamma }}_T\left|1_L\right.〉.$$

(17)

Thus even the fixed-pair control model distinguishes collective protection from survival scalarity. It does not, however, yet impose the identical-particle structure of molecular dipositronium.

4. Exchange-Summed Annihilation in ${\mathrm{Ps}}_2$

The labels $e_1,e_2,p_1,p_2$ identify the two electrons and two positrons before exchange constraints are imposed. They do not define two permanent positronium atoms. A physical ${\mathrm{Ps}}_2$ state also requires a spatial wavefunction with the correct electron and positron exchange symmetries [11]. The spin calculation below is performed before selecting a full molecular spatial state; it isolates the scalarity of the modeled survival channel.

For an electron–positron pair $(e_i,p_j)$, define

$${\mathrm{\Pi }}_{e_i{p}_j}^{\left(S\right)}={\displaystyle \frac{1}{4}}-{\mathbf{s}}_{e_i}·{\mathbf{s}}_{p_j},{\mathrm{\Pi }}_{e_i{p}_j}^{\left(T\right)}={\displaystyle \frac{3}{4}}+{\mathbf{s}}_{e_i}·{\mathbf{s}}_{p_j}.$$

(18)

These are the singlet and triplet projectors because ${\mathbf{s}}_{e_i}·{\mathbf{s}}_{p_j}$ has eigenvalue $-3/4$ on a singlet pair and $+1/4$ on a triplet pair.

In the conditional no-jump description, the spin-resolved annihilation sinks are represented schematically by

$$K_{2\gamma }={\mathrm{\Gamma }}_2{\displaystyle \sum _{i,j=1}^2}{C}_{ij}{\mathrm{\Pi }}_{e_i{p}_j}^{\left(S\right)},K_{3\gamma }={\mathrm{\Gamma }}_3{\displaystyle \sum _{i,j=1}^2}{C}_{ij}{\mathrm{\Pi }}_{e_i{p}_j}^{\left(T\right)},$$

(19)

with

$$C_{ij}={\delta }^{\left(3\right)}({\mathbf{r}}_{e_i}-{\mathbf{r}}_{p_j}).$$

(20)

Here $C_{ij}$ denotes electron–positron contact. Equation (19) is an effective channel representation, not a first-principles derivation of bound-state QED. It mirrors the contact-operator form used in annihilation-rate calculations [12,13,14], while keeping only the necessary spin and channel algebra. The total sink is

$$K_{\mathrm{ann}}=K_{2\gamma }+K_{3\gamma }.$$

(21)

To isolate the exchange spin algebra, set $C_{ij}=C$. With

$${\mathbf{S}}_e={\mathbf{s}}_{e_1}+{\mathbf{s}}_{e_2},{\mathbf{S}}_p={\mathbf{s}}_{p_1}+{\mathbf{s}}_{p_2},$$

(22)

the sums of projectors become

$$\begin{array}{cc}\hfill \sum _{i,j}{\mathrm{\Pi }}_{e_i{p}_j}^{\left(S\right)}& =\sum _{i,j}\left({\displaystyle \frac{1}{4}}-{\mathbf{s}}_{e_i}·{\mathbf{s}}_{p_j}\right)=1-{\mathbf{S}}_e·{\mathbf{S}}_p,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \sum _{i,j}{\mathrm{\Pi }}_{e_i{p}_j}^{\left(T\right)}& =\sum _{i,j}\left({\displaystyle \frac{3}{4}}+{\mathbf{s}}_{e_i}·{\mathbf{s}}_{p_j}\right)=3+{\mathbf{S}}_e·{\mathbf{S}}_p.\hfill \end{array}$$

(24)

Therefore

$$K_{\mathrm{ann}}=C\left[{\mathrm{\Gamma }}_2+3{\mathrm{\Gamma }}_3+({\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2){\mathbf{S}}_e·{\mathbf{S}}_p\right].$$

(25)

The fixed-pair para/ortho label has disappeared. The exchange-invariant operator ${\mathbf{S}}_e·{\mathbf{S}}_p$ is what governs the spin dependence of the equal-contact no-jump sink.

Proposition 1 

(Exchange-summed scalarity). In the equal-contact spin approximation for ${\mathrm{Ps}}_2$, the annihilation sink $K_{\mathrm{ann}}$ is scalar on each fixed $(S_e,S_p;F)$ sector, where $\mathbf{F}={\mathbf{S}}_e+{\mathbf{S}}_p$. Its eigenvalue is

$${\kappa }_{S_e,S_p,F}=C\left[{\mathrm{\Gamma }}_2+3{\mathrm{\Gamma }}_3+{\displaystyle \frac{{\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_2}{2}}\{F(F+1)-S_e(S_e+1)-S_p(S_p+1)\}\right].$$

(26)

Proof. 

Use

$${\mathbf{S}}_e·{\mathbf{S}}_p={\displaystyle \frac{1}{2}}\left[{\mathbf{F}}^2-{\mathbf{S}}_e^2-{\mathbf{S}}_p^2\right].$$

(27)

On a fixed $(S_e,S_p;F)$ sector, this operator has eigenvalue

$${\displaystyle \frac{1}{2}}\{F(F+1)-S_e(S_e+1)-S_p(S_p+1)\}.$$

(28)

Substitution into (25) gives (26). □

For two electrons and two positrons, $S_e,S_p\in \{0,1\}$. The sector values of

$$Q_S=1-{\mathbf{S}}_e·{\mathbf{S}}_p,Q_T=3+{\mathbf{S}}_e·{\mathbf{S}}_p$$

(29)

are as follows:

$S_e$	$S_p$	F	$Q_S$	$Q_T$
0	0	0	1	3
1	0	1	1	3
0	1	1	1	3
1	1	0	3	1
1	1	1	2	2
1	1	2	0	4

The final line means that, in this equal-contact spin model, the $(1,1;2)$ sector has zero exchange-summed singlet-contact weight.

5. Collective Protection Versus No-Jump Scalarity

The collective spin generators for the four-particle system are the components of

$$\mathbf{F}={\mathbf{S}}_e+{\mathbf{S}}_p.$$

(30)

They vanish on total $F=0$ sectors. The two-dimensional $F=0$ spin space used here is a spin-sector decomposition prior to selection of a full molecular spatial state. The scalarity test is therefore applied at the level of spin sectors. That spin space contains two such sectors:

$$\left|a\right.〉\equiv \left|S_e=0,S_p=0;F=0\right.〉,\left|b\right.〉\equiv \left|S_e=1,S_p=1;F=0\right.〉.$$

(31)

Let

$$P_{F=0}=\left|a\right.〉\left.〈a\right|+\left|b\right.〉\left.〈b\right|.$$

(32)

Both basis states are collective singlets:

$$F_k\left|a\right.〉=F_k\left|b\right.〉=0,k=x,y,z.$$

(33)

Thus the two-dimensional $F=0$ space is protected for the modeled collective generators.

The exchange-summed annihilation sink acts differently. From (26),

$${\kappa }_a\equiv {\kappa }_{0,0,0}=C({\mathrm{\Gamma }}_2+3{\mathrm{\Gamma }}_3),{\kappa }_b\equiv {\kappa }_{1,1,0}=C(3{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3).$$

(34)

Consequently

$$P_{F=0}{K}_{\mathrm{ann}}{P}_{F=0}={\kappa }_a\left|a\right.〉\left.〈a\right|+{\kappa }_b\left|b\right.〉\left.〈b\right|.$$

(35)

This is proportional to $P_{F=0}$ only if ${\kappa }_a={\kappa }_b$, equivalently if ${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_3$, within this equal-contact model. Positronium annihilation does not satisfy that equality.

Corollary 1.  

Failure of scalar survival on the collective protected space.

The collective $F=0$ protected space is generally not scalar for the equal-contact exchange-summed annihilation sink:

$$F_k{P}_{F=0}=0(k=x,y,z),P_{F=0}{K}_{\mathrm{ann}}{P}_{F=0}\cancel{\propto }{P}_{F=0}$$

(36)

unless the singlet and triplet annihilation coefficients are accidentally equal.

This is the central calculation. The collective algebra groups the two $F=0$ sectors together. The no-jump annihilation sink separates them.

6. No-Jump Branch and Distortion Measure

Let

$$N\left(t\right)=e^{-K_{\mathrm{ann}}t/2}.$$

(37)

On a fixed exchange sector,

$$N\left(t\right)P_{S_e,S_p,F}=e^{-{\kappa }_{S_e,S_p,F}t/2}{P}_{S_e,S_p,F},$$

(38)

and hence

$$P_{S_e,S_p,F}N{\left(t\right)}^{†}N\left(t\right)P_{S_e,S_p,F}=e^{-{\kappa }_{S_e,S_p,F}t}{P}_{S_e,S_p,F}.$$

(39)

The no-jump branch is therefore scalar on each fixed exchange-annihilation sector.

On the full collective $F=0$ protected space,

$$P_{F=0}N{\left(t\right)}^{†}N\left(t\right)P_{F=0}=e^{-{\kappa }_{a}t}\left|a\right.〉\left.〈a\right|+e^{-{\kappa }_{b}t}\left|b\right.〉\left.〈b\right|.$$

(40)

This is proportional to $P_{F=0}$ only when ${\kappa }_a={\kappa }_b$. A simple dimensionless measure of the non-scalar survival bias is

$${\mathrm{\Delta }}_{\mathrm{surv}}\left(t\right)={\displaystyle \frac{|e^{-{\kappa }_{a}t}-e^{-{\kappa }_{b}t}|}{e^{-{\kappa }_{a}t}+e^{-{\kappa }_{b}t}}}.$$

(41)

If ${\kappa }_b>{\kappa }_a$, then

$${\mathrm{\Delta }}_{\mathrm{surv}}\left(t\right)=tanh\left[{\displaystyle \frac{({\kappa }_b-{\kappa }_a)t}{2}}\right].$$

(42)

This quantity measures the failure of no-jump scalarity across the two exchange sectors. It is not a full measure of error-correction performance, because emitted-photon jumps and recovery operations are outside the model.

7. Scope of the Result

The result concerns the conditional no-jump operator and its scalarity on spin sectors [7,8,9,10]. The calculation is channel-theoretic, not molecular: it neither computes molecular annihilation rates nor provides a recovery protocol after emitted-photon jumps. Those tasks require the full spatial ${\mathrm{Ps}}_2$ wavefunction, electron–positron contact densities, and explicit decay products [12,13,14].

The equal-contact approximation suppresses spatial structure in order to isolate the exchange spin algebra. Unequal contacts would carry additional molecular information into $K_{\mathrm{ann}}$ and would generally make scalarity on broad collective spaces even less likely. The equal-contact model is therefore the simplest setting in which the exchange-summed annihilation operator can be compared directly with the modeled collective generators.

The comparison can be summarized as follows:

Criterion	Operator	Scalarity condition
Collective protection	$F_k$ or modeled collective generators	$F_{k}P=0$ on collective-singlet sectors
Conditional survival	$K_{\mathrm{ann}}$ or $N\left(t\right)$	$P{K}_{\mathrm{ann}}P\propto P$, equivalently $P{N}^{†}NP\propto P$
Molecular rate calculation	full contact and decay dynamics	requires spatial ${\mathrm{Ps}}_2$ wavefunction and contact expectation values

The first two criteria are channel-level scalarity tests, but they test different operators. The third is a molecular QED problem and is not addressed here.

8. Conclusions

A fixed-pair picture of two positronium atoms suggests a two-dimensional collective-spin protected sector, but true dipositronium requires exchange-summed annihilation over all electron–positron contacts. In the equal-contact spin approximation this gives

$$F_k{P}_{F=0}=0\mathrm{but}\mathrm{generally}{P}_{F=0}{K}_{\mathrm{ann}}{P}_{F=0}\cancel{\propto }{P}_{F=0}.$$

(43)

The collective generators group the two $F=0$ spin sectors together; the conditional no-jump annihilation sink separates them. Thus collective protection and undistorted no-jump survival are different scalarity statements. Dipositronium thereby provides a compact example in which the code space defined by modeled collective generators is operationally too coarse for the exchange-summed no-jump survival channel.