Calibrated diagonal lifts and transition phase differences

Let $H={\mathbb{C}}^d$ be a finite operator space with specified orthonormal basis $\{\mathrm{\mid }x⟩{\}}_{x=1}^d$, where $d\ge 1$. For specified real numbers $d_x$, the diagonal lift is the linear operator $D={\sum }_x{d}_x\mathrm{\mid }x⟩⟨x\mathrm{\mid }$. Its phase evolution at real time $t$, in units with $\mathrm{\hslash }=1$, is $U_t=e^{-itD}$, acting on an operator $X$ by $X\mapsto U_{t}X{U}_t^{†}$. In particular, the matrix unit $\mathrm{\mid }x⟩⟨y\mathrm{\mid }$ acquires the phase $e^{-it(d_x-d_y)}$.

If an unperturbed Hamiltonian $H_0$ is diagonal in this same basis with eigenvalues $E_x$, define the directed transition energy for $x\to y$ as the source energy minus the destination energy. Replacing $H_0$ by $H_0+D$ changes this difference from $E_x-E_y$ to $(E_x-E_y)+(d_x-d_y)$; the lift contribution is therefore $d_x-d_y$.

For the calibration criterion, a recovery rule is a supplied parameterized family of quantum channels ${\mathcal{R}}_{a,\tau }:\mathcal{L}(H_R)\to \mathcal{L}(H_L)$, indexed by $(a,\tau )\in A$, where $H_R$ and $H_L$ are specified nonzero finite operator spaces, $\mathcal{L}(K)$ denotes the linear endomorphisms of $K$, and $A\subseteq {\mathbb{R}}^d\times [0,\mathrm{\infty })$ is a specified nonempty set of admissible lift vectors and elapsed intervals. Any auxiliary inputs, such as a recorded outcome, are fixed or supplied separately. The phase data are calibrated for this rule at an admissible pair $(a,\tau )$ with $a=(d_1,\dots ,d_d)$ when the actual vector $a$ and interval $\tau$ are supplied as explicit inputs and the rule evaluates the specified channel ${\mathcal{R}}_{a,\tau }$. Each channel is complex-linear, completely positive and trace-preserving; the word recovery specifies its intended use and supplies no correction guarantee or physical control assumption.

This availability is an assumption about the rule's inputs, not a claim that an unknown jump time has been observed. An unknown jump time remains a variable; averaging over it requires a separately specified probability distribution.

When probabilistic averaging is invoked, the additional inputs are a probability space $(T,\mathcal{T},\mu )$ of allowed real times $T\subseteq \mathbb{R}$ and a specified $\mu$-integrable real or complex scalar function $f$ on $T$. The probability measure $\mu$ is the jump-time distribution, and the average means the expectation ${\mathbb{E}}_{\mu }[f]={\int }_{T}fd\mu$. Neither this distribution nor a choice of $f$ is supplied by calibration alone.

Finite operator and ket conventions

The exponential series of a bounded operator

The complex exponential by its power series

exp(z+w)=exp z exp w, and the complex exponential extends the real exponential

Finite-dimensional quantum channel

Probability measures and probability spaces

Expectation of a nonnegative or integrable random variable

Given: The finite operator space, basis and real diagonal data in the definition, real $t$, and, for calibration, the supplied channel family and admissible input pair. When probabilistic averaging is invoked, also the supplied probability space and integrable scalar function in the definition.

Goal: Verify the phase evolution and transition difference, the typing of the calibration criterion, and the meaning of the averaging caveat.

1.1 By the basis action in [F1], $D\mathrm{\mid }x⟩=d_x\mathrm{\mid }x⟩$ and $D^n\mathrm{\mid }x⟩=d_x^n\mathrm{\mid }x⟩$ for every nonnegative integer $n$. The space $H$, with its standard norm, is Banach and $-iD$ is bounded by [F2], so [F3] applies. Acting with its norm-convergent series on each basis vector and using the scalar series in [F4] gives $U_t\mathrm{\mid }x⟩=e^{-it{d}_x}\mathrm{\mid }x⟩$. [F1, F2, F3, F4, algebra]

1.2 At a supplied admissible pair $(a,\tau )$, the given family assigns a channel ${\mathcal{R}}_{a,\tau }$ with the source and target operator spaces in [F5]. Evaluating this supplied family defines the rule's use of the known lift vector and interval. It introduces no assertion that a corrective channel family exists for an unspecified noise model, that physical control is available, or that an unknown jump time or its distribution has been supplied. [given, F5]

2.1 For $z\in \mathbb{C}$ write $p_N(z)={\sum }_{n=0}^N{z}^n\mathrm{/}n!$. Conjugating the finite sum gives $\overline{p_N(z)}=p_N(\bar{z})$, and $\mathrm{\mid }\overline{p_N(z)}-\overline{e^z}\mathrm{\mid }=\mathrm{\mid }{p}_N(z)-e^z\mathrm{\mid }\to 0$ by [F4]. Thus $\overline{e^z}=e^{\bar{z}}$. Since $t,d_y$ are real, the diagonal matrix from step 1.1 and the conjugate transpose in [F1] give $U_t^{†}\mathrm{\mid }y⟩=e^{it{d}_y}\mathrm{\mid }y⟩$. [F1, F4, step 1.1, algebra]

3.1 For every basis vector $\mathrm{\mid }z⟩$, the three factors in $U_t\mathrm{\mid }x⟩⟨y\mathrm{\mid }{U}_t^{†}$ give $e^{-it{d}_x}{e}^{it{d}_y}\mathrm{\mid }x⟩⟨y\mathrm{\mid }z⟩=e^{-it(d_x-d_y)}\mathrm{\mid }x⟩⟨y\mathrm{\mid }z⟩$ by [F4]; equality on the basis proves the asserted matrix-unit action. When $H_0\mathrm{\mid }x⟩=E_x\mathrm{\mid }x⟩$, $(H_0+D)\mathrm{\mid }x⟩=(E_x+d_x)\mathrm{\mid }x⟩$, and subtracting the destination eigenvalue from the source eigenvalue gives the stated transition shift. [F1, F4, step 1.1, step 2.1, algebra]

4.1 With the additional averaging inputs in the definition, [F6] applies to $f$ on $(T,\mathcal{T},\mu )$ and defines its finite average as ${\int }_{T}fd\mu$. This operation uses the specified measure $\mu$ as well as $f$; neither is an input furnished by the calibration criterion itself. Thus the averaging caveat is a requirement to supply a probability model, not a probability law inferred from the channel family or an observation of the unknown time. [given, F6] ∎