Monolithic encoding and degenerate error processes

PDF p. 1: Introduction, paragraphs 1–2

A monolithic encoding stores logical information in the multilevel structure of one physical system, such as a bosonic mode, molecule, or atom. In a finite-dimensional description, let $H_L$ and $H_P$ be nonzero finite-dimensional complex inner-product spaces with specified ordered orthonormal bases. Use the finite operator and ket conventions in their coordinates, identifying each space with ${\mathbb{C}}^d$ in its specified basis. An encoding is a linear isometry $V:H_L\to H_P$ from the logical space into the state space of that one system; its image is the code. An encoded superposition is a linear combination of code vectors, expressible in the specified physical orthonormal basis.

A decay path specifies an initial physical basis state and a corresponding decay process. In the article's terminology, two such processes are degenerate when the environmental information available after the decay cannot distinguish which of the two initial basis states decayed. For a precise finite-dimensional realization of this terminology, use the stated $H_P$ and specify a nonzero finite-dimensional complex inner-product space $E$ with a specified ordered orthonormal basis and the same coordinate and ket conventions. The space $E$ contains the entire environmental record available for comparison. Specify also a nonempty finite set $A$ of branch labels. Declare a finite environmental instrument: complex-linear completely positive maps ${\mathcal{I}}_a:\mathcal{L}(H_P)\to \mathcal{L}(E)$ with ${\sum }_{a\in A}\mathrm{tr}({\mathcal{I}}_a(X))=\mathrm{tr}(X)$ for every $X\in \mathcal{L}(H_P)$. The instrument and its physical interpretation are prescribed model data, not inferred from the transition energies.

For path $i=1,2$, choose $\mathrm{\mid }{s}_i⟩$ from the specified physical orthonormal basis and its decay branch $a_i\in A$, and put ${\rho }_i=\mathrm{\mid }{s}_i⟩⟨s_i\mathrm{\mid }$, ${\tau }_i={\mathcal{I}}_{a_i}({\rho }_i)$ and $p_i=\mathrm{tr}({\tau }_i)$. When $p_i>0$, its normalized conditional environmental state is ${\sigma }_i={\tau }_i\mathrm{/}{p}_i$ on the common space $E$; when $p_i=0$, no conditional state is assigned. The comparison requires $p_1,p_2>0$ and concerns the records conditioned on those paths, rather than the incidence rates $p_i$. Preparation and branch indices are bookkeeping labels, not automatically accessible records; if any such label is available to the environment, its register must be included in $E$ and in the declared output states.

In this realization, degeneracy means ${\sigma }_1={\sigma }_2$. An environmental measurement in the adopted finite-outcome model is any finite POVM $(F_b{)}_{b\in B}$ on $E$: $B$ is finite and nonempty, $F_b\ge 0$, and ${\sum }_b{F}_b=I_E$. Adopt the physical outcome law $q_i(b)=\mathrm{tr}({\sigma }_i{F}_b)$. Equal conditional states therefore give the same outcome probabilities for every such measurement. This includes every binary effect test $(F,I_E-F)$ with $0\le F\le I_E$, so it covers every individual environmental event represented by an effect in the adopted model. It neither specifies an infinite-dimensional bath model nor asserts that a particular apparatus realizes these measurements. This pathwise condition alone does not assert that the full environmental record is independent of every encoded superposition, nor that an arbitrary code is exactly correctable.

For levels with energies $E_s$ and $E_f$, call $E_s-E_f$ the transition frequency in units with $\mathrm{\hslash }=1$. Two transitions from equal-energy initial levels to equal-energy final levels have equal frequencies. Adjacent downward transitions in a uniformly spaced ladder also have equal frequencies. These are examples of coincident frequencies, not by themselves a complete specification of the environmental record.

Finite operator and ket conventions

Code restriction and decoded channel types

Complementary error records in a Kraus basis

Bosonic loss, Kerr shifts and finite-code comparison scope

Finite instruments and conditional states

Instrument outcome weights normalize

Positive-probability instrument branches are density operators

Finite POVMs and their outcome law

POVM outcome weights are probabilities

Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes

Given: The stated finite-dimensional encoding, environmental instrument and POVMs, the physical assignments in [A2], and Countable Choice as specified in [A1].

Goal: Justify the conditional-state construction and the scope of indistinguishability and coincident frequencies.

1.1 The isometry in [F1] gives a precise finite-dimensional meaning to an encoded subspace of one physical system; it does not require that every physical platform be finite-dimensional. The specified orthonormal bases and the coordinate and ket conventions in [F6] make the path labels and outer products unambiguous. Each ${\rho }_i=\mathrm{\mid }{s}_i⟩⟨s_i\mathrm{\mid }$ is a density operator: in the specified orthonormal basis its matrix has one diagonal entry $1$ and all other entries $0$, and its quadratic form is $\mathrm{\mid }⟨s_i\mathrm{\mid }x⟩{\mathrm{\mid }}^2\ge 0$. Apply [F4] to the declared instrument and this input. Its branch weight $p_i$ is nonnegative, and when $p_i>0$ the quotient ${\tau }_i\mathrm{/}{p}_i$ is a density operator on $E$. The zero-weight case is left undefined as required by the instrument convention. The probability and conditional-state interpretations are the prescribed assumptions in [A2]. [given, F1, F4, F6, A2]

2.1 For any one of the specified POVMs, [F5] gives a normalized outcome law, with physical interpretation from [A2]. If ${\sigma }_1={\sigma }_2$, substitution gives $q_1(b)=\mathrm{tr}({\sigma }_1{F}_b)=\mathrm{tr}({\sigma }_2{F}_b)=q_2(b)$ for each $b$. This holds for every finite POVM, including the binary family $(F,I_E-F)$ for any effect $F$, since its two members are nonnegative and sum to $I_E$. All accessible registers are already part of $E$ by the stated construction. Thus the conclusion concerns the entire declared conditional record rather than only one measured component. [given, F5, A1, A2, step 1.1]

3.1 The full-record condition in [F2] instead quantifies over all code density operators and includes coherences, so equality for the two specified conditional paths is not used to infer that condition or exact correctability. The model does not assume equal path incidence rates. [F1, F2, step 1.1, step 2.1]

4.1 If $E_{s_1}=E_{s_2}$ and $E_{f_1}=E_{f_2}$, subtraction gives $E_{s_1}-E_{f_1}=E_{s_2}-E_{f_2}$. For a uniformly spaced ladder $E_n=E_0+n\omega$, every adjacent downward gap is $\omega$. In particular, the model in [F3] has this property when $K=0$. Equality of frequencies alone specifies neither all other environmental degrees of freedom nor the instrument or its conditional states; no converse from equal frequencies to path indistinguishability is assumed. [given, F3, A1, step 1.1, step 3.1, algebra] ∎