Optimal error regions for quantum state estimation

The outcomes Π_k, k = 1,2,...,K, of the POM, with which the data are acquired, are positive (Π_k ⩾ 0) Hilbert-space operators with $\sum _{k=1}^K\Pi _k=1$. If the state ρ describes the system, the probability p_k that the kth detector clicks is

$$\begin{equation} p_k={\mathrm{tr}} \{\Pi_k\rho\} =\langle\Pi_k\rangle\quad \mathrm{(Born \ rule)}. \end{equation} \tag{ 1 }$$

The positivity of ρ and its normalization to unit trace ensure p_k ⩾ 0 for all k and $\sum _{k=1}^Kp_k=1$. Probabilities p = (p₁,p₂,...,p_K) for which there is a state ρ such that (1) holds are permissible probabilities. They make up the probability space. The probability space for a K-outcome POM is usually smaller than that of a tossed K-sided die because not all positive p_k with unit sum are permissible. The quantum nature of the state estimation problem enters only in these additional restrictions on p: quantum state estimation is standard statistical state estimation with constraints of quantum-mechanical origin. The rich methods of statistical inference immediately apply, modified where necessary to account for the restricted probability space.

Whereas p is uniquely determined by ρ through (1), the converse is true only if the POM is informationally complete. In any case, there is always a reconstruction space $\mathcal {R}_0$, a set of ρ that contains exactly one ρ for each permissible p, consistent with the Born rule. If there is more than one reconstruction space, it does not matter which one we choose. As an example, consider a harmonic oscillator with its infinite-dimensional state space. If the POM has two outcomes, with p₁ equal to the probability of finding the oscillator in its ground state and p₂ = 1 − p₁, the reconstruction space is the set of convex combinations of the projector to the ground state and another state with no ground-state component. In this situation, there are very many reconstruction spaces to choose from because any other state serves the purpose, and all one can infer from the data is an estimate of the ground-state probability.

Since the probability space is unique, while there can be many different reconstruction spaces, it is often more convenient to work in the probability space. In particular, the probability space has the desirable property that it is always convex; it is, however, not always possible to find a convex reconstruction space. The primary objective of state estimation is then to find an estimator, or a region of estimators, for the probabilities p. The conversion of p into a state ρ can be performed later, if at all. At this stage, if the POM is not informationally complete, one must invoke additional criteria—beyond what the data tells us—for a unique mapping p → ρ. For example, one could follow Jaynes's guidance [11, 12] and maximize the entropy [13] (see also [14]). However, following the tradition in this topic, we will formally work in a reconstruction space $\mathcal {R}_0$, although all actual calculations are performed in the probability space. Estimators are states in $\mathcal {R}_0$, and regions are sets of states there.

The reconstruction space is an abstract construct that is often not endowed with a self-suggesting unique metric. Instead, a region's prior probability—the quantity that matters most in the present context of statistical inference—offers a natural notion of size. This relieves us of the need to invoke additional, possibly artificial, criteria for the assignment of size, for instance, one that has more to do with a simple parameterization of the state space than the relative importance of different regions in terms of our prior expectations.

We denote by (dρ) the size (≡  prior probability) of the infinitesimal vicinity of state ρ in $\mathcal {R}_0$. The size $S_{\mathcal {R}}$ of a region $\mathcal {R}$ is then

$$\begin{equation} S_{\mathcal{R}}=\int_{\mathcal{R}}(\mathrm{d}\rho)\quad\mbox{with} \displaystyle\int_{\mathcal{R}_0}(\mathrm{d}\rho)=1. \end{equation} \tag{ 2 }$$

A pertinent remark: we exclude pathological cases of improper priors where the prior density (dρ) cannot be normalized; should an improper prior be useful in a particular context, it should come about as the limit of a well-defined sequence of proper priors.

By construction, $S_{\mathcal {R}}$ does not depend on the parameterization used for the numerical representation of (dρ). The primary parameterization is in terms of the probabilities,

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d} p)\,w(p)\quad\mbox{with}\;\; \displaystyle(\mathrm{d} p)=\mathrm{d} p_1\,\mathrm{d} p_2\,\cdots\,\mathrm{d} p_K, \end{equation} \tag{ 3 }$$

where the prior density w(p) is positive for all permissible probabilities and vanishes for non-permissible probabilities. To enforce positivity and normalization of p, w(p) always contains

$$\begin{equation} w_0(p)=\eta(p_1)\,\eta(p_2)\cdots\eta(p_K)\,\delta{\left(\sum_kp_k-1\right)} \end{equation} \tag{ 4 }$$

as a factor, where η() denotes Heaviside's unit step function and δ() is Dirac's delta function. If there are no other constraints, we have the probability space of a K-sided die. For genuine quantum measurements, however, there are additional constraints, some accounted for by more delta-function factors, others by step functions. The delta-function constraints reduce the dimension of the reconstruction space from K − 1 to the number of independent probabilities. Accordingly, there is a factor of constraint w_cstr(p) (containing w₀(p)) that specifies the probability space and appears in all possible priors. In particular, there are two specific priors we will employ as examples below: the primitive prior

$$\begin{equation} (\mathrm{d}\rho)\propto (\mathrm{d} p)\,w_{\mathrm{cstr}}(p) \end{equation} \tag{ 5 }$$

and the Jeffreys prior [15]

$$\begin{equation} (\mathrm{d}\rho)\propto (\mathrm{d} p)\,w_{\mathrm{cstr}}(p)\,\frac{1}{\sqrt{p_1p_2\ldots p_K}}, \end{equation} \tag{ 6 }$$

which is a popular choice of an unprejudiced prior [16].

For the harmonic-oscillator example in section 2.1, which has the same probability space as a tossed coin, the factor w₀(p) selects the line segment with 0 ⩽ p₁ = 1 − p₂ ⩽ 1 in the p₁p₂ plane. If we choose the primitive prior (dρ) = (dp) w₀(p), the subsegment with a ⩽ p₁ ⩽ b has size b − a. For the Jeffreys prior

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d} p)\,w_0(p)\frac{1}{\pi\sqrt{p_1p_2}}, \end{equation} \tag{ 7 }$$

the same subsegment has size $\frac {2}{\pi }[\sin ^{-1}(\sqrt {b})-\sin ^{-1}(\sqrt {a})]$.

In this example, and also in those we use for illustration in section 5 below, it is easy to state quite explicitly the restrictions on the set of permissible probabilities that follow from the Born rule. In other situations, including the examples of section 6, this is more difficult. In yet more complicated situations it could be impossible. It is, however, possible to check numerically if a certain $\tilde {p}=(\tilde {p}_1,\ldots ,\tilde {p}_K)$ is permissible. For example, one calculates a MLE (which can be done efficiently) for relative frequencies $n_k/N=\tilde {p}_k$ (see below), and if the resulting probabilities p are such that $p=\tilde {p}$, then $\tilde {p}$ is permissible; otherwise it is not. For practical reasons, it may be necessary to truncate the full state space—which can be, and often is, infinite-dimensional—to a test space of manageable size. With such a truncation one accepts that not all permissible probabilities are investigated. Therefore, a criterion for judging if the test space is large enough is to verify that the estimated probabilities do not change significantly when the space is enlarged. Examples for the artifacts that result from test spaces that are too small can be found in [17].

The data D consist of a sequence of detector clicks, with n_k clicks in total of the kth detector after measuring N = n₁ + n₂ + ··· + n_K copies of the state⁹. The probability of obtaining D, if ρ is the state, is the point likelihood

$$\begin{equation} L(D|\rho)=p_1^{n_1}p_2^{n_2}\cdots p_K^{n_K}, \end{equation} \tag{ 8 }$$

which attains its maximum value when ρ is the MLE $\widehat {\rho }_{\rm ML}\in \mathcal {R}_0$; the MLE is fully determined by the relative frequencies n_k/N. The joint probability of finding the state ρ in the region $\mathcal {R}$ and obtaining the data D is

$$\begin{equation} \mathrm{prob}(D \wedge \mathcal{R})=\int_{\mathcal{R}}(\mathrm{d}\rho)\,L(D|\rho)\,. \end{equation} \tag{ 9 }$$

For ${\mathcal {R}=\mathcal {R}_0}$, $\mathrm {prob}(D \wedge \mathcal {R}_0)=L(D)$ is the prior likelihood. Since one of the click sequences is surely observed, we have

$$\begin{equation} \sum_DL(D|\rho)=1,\quad\sum_DL(D)=\int_{\mathcal{R}_0}(\mathrm{d}\rho)=1, \end{equation} \tag{ 10 }$$

where the sum is taken over all possible data for N clicks.

We factor the joint probability in two ways

$$\begin{equation} \mathrm{prob}(D \wedge \mathcal{R})=L(D|\mathcal{R})S_{\mathcal{R}}=C_{\mathcal{R}}(D)L(D) \end{equation} \tag{ 11 }$$

and so identify the region likelihood $L(D|\mathcal {R})$ and the credibility $C_{\mathcal {R}}(D)$. These are conditional probabilities: $L(D|\mathcal {R})$ is the probability of obtaining the data D if the actual state is in the region $\mathcal {R}$; $C_{\mathcal {R}}(D)$ is the probability that the actual state is in $\mathcal {R}$ if D were obtained—the posterior probability of $\mathcal {R}$.

For both POMs, the unit disk in the xy plane suggests itself for the reconstruction space $\mathcal {R}_0$. The first POM is the crosshair measurement that combines projective measurements of σ_x and σ_y into a four-outcome POM (K = 4) with probabilities

$$\begin{equation} \begin{array}{@{}l@{}}p_1\\ p_2\end{array}\biggr\} =\frac{1}{4}(1\pm x),\quad \begin{array}{@{}l@{}} p_3\\ p_4\end{array}\biggr\} =\frac{1}{4}(1\pm y)\quad\mbox{with}\;x=\langle\sigma_x\rangle,\,\, y=\langle\sigma_y\rangle\,. \end{equation} \tag{ 22 }$$

The permissible probabilities are identified by

$$\begin{equation} w_{\mathrm{cstr}}(p)\dot{=}\eta(p)\,\delta(p_1+p_2-\textstyle\frac{1}{2})\, \delta(p_3+p_4-\textstyle\frac{1}{2})\,\eta(3-8p^2), \end{equation} \tag{ 23 }$$

where

$$\begin{equation} \eta(p)=\prod_{k=1}^K\eta(p_k)\quad\mbox{and}\quad p^2=\sum_{k=1}^Kp_k^2\,. \end{equation} \tag{ 24 }$$

The dotted equals sign in (23) stands for 'equal up to a multiplicative constant', namely the factor that ensures the unit size of the reconstruction space.

The second POM is the three-outcome trine measurement (K = 3), whose outcomes are subnormalized projectors on the eigenstates of σ_x and ${(-\sigma _x\pm \sqrt {3}\,\sigma _y)/2}$ with eigenvalue +1. It has the probabilities

$$\begin{equation} p_1=\frac{1}{3}(1+x),\quad \begin{array}{@{}l@{}}p_2\\ p_3\end{array}\biggr\} =\frac{1}{6}(2-x\pm\sqrt{3}\,y), \end{equation} \tag{ 25 }$$

for which

$$\begin{eqnarray} &&w_{\mathrm{cstr}}(p)\,\dot{=}\,\eta(p)\,\delta(p_1+p_2+p_3-1)\, \eta(1-2p^2) \end{eqnarray} \tag{ 26 }$$

summarizes the constraints that the permissible values of p₁, p₂, p₃ obey.

Both POMs have the same primitive prior

$$\begin{equation} (\mathrm{d}\rho)=\mathrm{d} x\,\mathrm{d} y\,\frac{1}{\pi}\,\eta(1-x^2-y^2) =\mathrm{d} r^2\,\frac{\mathrm{d}\varphi}{2\pi}, \end{equation} \tag{ 27 }$$

where 0 ⩽ r ⩽ 1 and φ covers any convenient range of 2π. This prior is uniform in x and y, and in r² and φ. The polar-coordinate version, that is $x+\mathrm {i}y=r\,\mathrm {e}^{{\mathrm {i}\varphi }}$, is the more natural parameterization of the unit disk. The Jeffreys prior for the four-outcome POM is

$$\begin{equation} (\mathrm{d}\rho)=\frac{2}{\pi^2}\frac{\mathrm{d} r\,r\,\mathrm{d}\varphi} {\sqrt{1-r^2+{\frac{1}{4}}r^4\sin(2\varphi)^2}}. \end{equation} \tag{ 28 }$$

For the three-outcome POM, we have the Jeffreys prior

$$\begin{equation} (\mathrm{d}\rho)=\frac1{4\pi-24\sin^{-1}\left({\frac{1}{3}}\right)}\frac{\mathrm{d} r\,r\,\mathrm{d}\varphi} {\sqrt{1-{\frac{3}{4}}r^2+{\frac{1}{4}}r^3\cos(3\varphi)}}. \end{equation} \tag{ 29 }$$

Figures 4(a) and (b) show SCRs obtained for simulated experiments in which N = 24 copies of a qubit state are measured. The actual state used for the simulation has (x,y) = (0.6,0.2). Its position in the reconstruction space is indicated by the red star ().

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In figure 4(a), we see the SCRs for the four-outcome POM. Two experiments were simulated, with (n₁,n₂,n₃,n₄) = (8,5,10,1) and (6,3,10,5) clicks of the detectors, respectively, and the triangles (${\triangle}$) show the positions of the corresponding MLEs. For each data, the plot reports the SCRs with credibility c = 0.5 and 0.9, both for the primitive prior of (27) and for the Jeffreys prior of (28). The actual state is inside two of the four SCRs with credibility c = 0.5 and is contained in all four SCRs with credibility c = 0.9.

Not unexpectedly, we get quite different regions for the two rather different sets of detector click counts. Yet, we observe that the choice of prior has little effect on the SCRs, although the total number of measured copies is too small for relying on the consistency of the priors. The same remarks apply to the SCRs for the three-outcome POM in figure 4(b); here we counted (n₁,n₂,n₃) = (15,8,1) and (13,7,4) detector clicks in the simulated experiments.

In section 4 we remarked that the estimator regions are properly communicated by reporting s_λ and c_λ as functions of λ. This is accomplished by the insets in figure 4 for two of the four simulated experiments. The dots give the values obtained by numerical integration that uses a Monte Carlo algorithm. The scatter of these numerical values confirms the expected: the computation of s_λ only requires sampling the probability space in accordance with the prior and determining the fraction of the sample that is in $\mathcal {R}_\lambda$; for the computation of c_λ we need to add the values of L(D|ρ) for the sample points inside $\mathcal {R}_\lambda$; and since L(D|ρ) is a sharply peaked function of the probabilities, the s_λ values are more trustworthy than the c_λ values for the same computational effort. The line fitted to the s_λ values is a Padé approximant (see e.g. [19, section 5.12]) that takes the analytic forms near λ = λ₀ = 0 and λ = 1 into account. The line approximating the c_λ values is then computed in accordance with (20).

In the BB84 scheme, each qubit is measured by the crosshair POM of (22); the resulting two-qubit POM has sixteen outcomes that obey eight constraints that give delta-function factors in w_cstr(p). In the TAT scheme, one qubit is measured by the trine POM of (25) and the other qubit by the antitrine POM that has the signs of x and y reversed in (25); the resulting two-qubit POM has nine outcomes subject to the single delta-function constraint of unit sum. Accordingly, the probability space is eight-dimensional for both schemes¹⁰, and we cannot report the SCRs by showing the optimal error regions in the reconstruction space, as was possible for the two-dimensional probability space in figure 4. Therefore, we employ the strategy of section 4 and report the size s_λ and the credibility c_λ of the respective BLRs as functions of λ.

For the generation of the simulated data, we first add noise to the singlet state by putting it through a random Pauli channel¹¹. The resulting true state has the probabilities for the two-qubit POMs given in the top row of table 1. For example, the '12' entry in the 4 × 4 table for the double-crosshair POM is the probability for outcome $\Pi _1\otimes \Pi _2=\frac {1}{4}(1+\sigma _x)\otimes \frac {1}{4}(1-\sigma _x)$. The '11' entry of the 3 × 3 table for the TAT POM is 16/9 times that number. Note that all marginal probabilities (sums of rows and sums of columns) are equal; this is so because the reduced single-qubit states of the true state are completely mixed. For the same reason, both tables are symmetric and the lower-left and upper-right 2 × 2 subtables of the 4 × 4 table have entries of 1/16 = 0.0625. More generally, there is a one-to-one correspondence between the 16 permissible probabilities in the 4 × 4 table and the nine permissible probabilities in the 3 × 3 table, because all table entries are determined by the expectation values of A⊗B with A,B = 1,σ_x,σ_y.

Table 1. Computer-generated data for the estimation of a two-qubit state from measuring 60 identically prepared copies. The first row gives the joint probabilities of the true state. The broken second row shows the number of detector-click pairs obtained in the simulated experiment (and their expected values) together with the single-qubit marginals. The third row reports the joint probabilities of the MLEs for the data in the second row. In each row, we have a 4 × 4 table on the left for the double-crosshair POM of the BB84 scenario and a 3 × 3 table on the right for the nine-outcome POM of the TAT scheme. The rows of a 4 × 4 table for the double-crosshair POM refer to the four Π_j of the first qubit in the pair and the columns refer to the Π_k of the second qubit; entry 'jk' is the probability for outcome Π_j⊗Π_k. Analogously, entry 'jk' in a 3 × 3 table for the TAT scheme is the expectation value of Π_j⊗Π_k with trine outcome Π_j and antitrine outcome Π_k.

Simulated measurements of 60 qubit pairs in the true state for each POM produced the counts of detector-click pairs in the second row of table 1; expected values are given in parentheses. Owing to the statistical fluctuations, the tables of counts are not symmetric¹² and the marginal counts are not equal.

The third row of table 1 shows the corresponding MLE probabilities. These probabilities are equal to the relative frequencies of the counts for the 9-outcome POM but are different from the relative frequencies for the 16-outcome POM. This tells us that the computer-generated data are not typical for the double-crosshair POM, whereas we have typical data for the TAT POM.

As noted in section 4, the primary task of the data evaluation is the computation of the multi-dimensional integrals that give the size s_λ of the BLRs for the whole range of 0 < λ < 1. For the data in table 1, these are integrals over eight-dimensional regions. We used a random-sampling technique for this purpose.

As a preparation, we generated a random sample of 648 785 permissible sets of probabilities, uniformly distributed in accordance with the primitive prior (see section 6.3 below). In view of the one-to-one correspondence between the permissible probabilities of the 16-outcome POM and the 9-outcome POM, the same random sample can be, and was, used for both POMs.

The actual data processing consists of two steps. In the first step, we determine the size s_λ for the 160 values of λ with −log₁₀ λ = 0.1(0.1)16.0. This requires a simple counting of how many samples are inside the BLR $\mathcal {R}_{\lambda }$ if the primitive prior is used. In the case of the Jeffreys prior, one adds the weights (p₁p₂··· )^−1/2 of the samples inside the BLR. The correct normalization follows from s_λ=0 = 1.

In the second step, the integrals needed in (20) are evaluated, for which a simple linear interpolation between adjacent (λ,s_λ) pairs is sufficiently accurate. Then, c_λ is known as a function of λ and the λ values for which we have 99% or 95% credibility are determined.

We show s_λ and c_λ as functions of λ in figure 5. Table 2 reports the λ values of the 99% and 95% credibility thresholds. We observe that for the 16-outcome POM, the true state is inside the SCRs with 99% credibility for both the primitive prior and the Jeffreys prior, whereas it is inside the 95% SCR only for the primitive prior but not for the Jeffreys prior. This is more evidence that these data are untypical. By contrast, for the 9-outcome POM, the true state is inside all SCRs for both priors and both values of the credibility.

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Table 2. Threshold λ values for 99% and 95% credibility for the data of table 1 and figure 5, and the sizes of the respective BLRs. The true state is inside the $\mathcal {R}_{\lambda }$s with λ < 3.368 × 10⁻³ for the 16-outcome POM (with its untypical data), and inside the BLRs with λ < 0.2486 for the 9-outcome POM.

Typicality, or lack thereof, can also be noticed in figure 5. Since the Jeffreys prior gives more weight to the regions near the boundary of the probability space than the primitive prior, and less weight to regions deep inside, one expects that the values of s_λ for the primitive prior are larger than those for the Jeffreys prior if the data are typical and, accordingly, the MLE is not close to the boundary. This is indeed the case for the TAT data, but not for the double-crosshair data.

The two steps of data evaluation, the computation of the size s_λ and then the credibility c_λ, take a few seconds of central processing unit (CPU) time. The preparation of the random sample of permissible probabilities, which could be done ahead of the data taking, lasts much longer. Our sample of 648 785 probabilities took almost 100 h of CPU time on a standard desktop (Intel i7-870 CPU, using one of the four cores and 8 GB RAM). The procedure we employed was simple and reliable but not optimized for speed. There is clearly much room for improvement (see section 7).

For each potential sample of probabilities we first generate nine random numbers $x_1,x_2,\ldots ,x_9$ uniformly and independently between 0 and 1. Then, the nine probabilities p_k = log(x_k)/log(x₁x₂···x₉) constitute a sample in the eight-dimensional simplex of the classical nine-sided die, and the samples are distributed in accordance with the primitive prior. The sample p = (p₁,p₂,...,p₉) is accepted if it is a permissible set of probabilities for the TAT POM with its nine outcomes. Only 9.27% of the 7 × 10⁶ candidate probabilities generated were accepted.

Whereas the generation of another sample p is fast, the test of permissibility is the part that consumes most of the CPU time. After identifying the candidate p with the relative frequencies of a measurement with the 9-outcome POM, we calculate a MLE for these frequencies. If the probabilities of the MLE are equal to p, this sample probability is accepted, otherwise it is rejected.

The time-honored strategy of choosing a uniform prior on $\mathcal {R}_0$ in which all states are treated equally gets us into a circular argument. Our identification of the size of a region with its prior content amounts to assigning equal probabilities to regions of equal sizes, prior to acquiring any data. But that just means that we now have to declare how we measure the size of a region without prejudice, and we are again faced with the original question about a uniform prior.

In fact, there is no unique meaning of the uniformity of a prior. In the sense that each prior tells us how to quantify the size of a region, each prior is uniform with respect to its induced size measure. To illustrate, reconsider the harmonic-oscillator example of section 2.1. For the primitive prior of (5), the parameterization

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle p_1=\textstyle\frac{1}{2}(v+u),\quad p_2=\textstyle\frac{1}{2}(v-u),\\ \displaystyle \mathrm{d} p_1\,\mathrm{d} p_2=\mathrm{d} u\,\mathrm{d} v\,\textstyle\frac{1}{2} \end{array} \end{eqnarray} \tag{ A.1 }$$

gives

$$\begin{eqnarray} (\mathrm{d}\rho)&=& \mathrm{d} u\,\mathrm{d} v\,\textstyle\frac{1}{2}\eta(v+u)\eta(v-u)\delta(v-1)\nonumber\\ &\to&\mathrm{d} u\,\textstyle\frac{1}{2}\quad \mbox{with}\;\; -\!1\leqslant u\leqslant 1, \end{eqnarray} \tag{ A.2 }$$

where we integrate over v in the last step and so observe that the primitive prior is uniform in u, that is, the size of the region u₁ < u < u₂ is proportional to u₂ − u₁. Likewise, the parameterization

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle p_1=v(\sin\alpha)^2,\quad p_2=v(\cos\alpha)^2,\\ \displaystyle \mathrm{d} p_1\,\mathrm{d} p_2=\mathrm{d}\alpha\,\mathrm{d} v\,v\sin(2\alpha) \end{array} \end{eqnarray} \tag{ A.3 }$$

gives

$$\begin{equation} (\mathrm{d}\rho)\to\mathrm{d}\alpha\,\frac{2}{\pi}\quad \mbox{with}\;\; \displaystyle 0\leqslant\alpha\leqslant\frac{\pi}{2} \end{equation} \tag{ A.4 }$$

for the Jeffreys prior of (6), which is uniform in α. Other priors can be treated analogously, each of them yielding a uniform prior in an appropriate single parameter.

Visualization of the uniformity for qubit priors can be found in figure A.1. Plot (b) shows uniform tiling of the unit disk by tiles of equal size. Here size is measured by the primitive prior of (27), which is uniform in x and y, and in r² and φ (the latter is used for the plot). Plots (c1) and (c2) show uniform tilings of the unit disk for the Jeffreys prior for the four-outcome POM of (28), while plots (d1) and (d2) show those for the three-outcome POM of (29). The crosshair symmetry of the four-outcome POM and the trine symmetry of the three-outcome POM are manifest in their respective uniform tilings.

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The parameterizations in (A.1) and (A.3), and the tilings of figure A.1 exhibit in which explicit sense the primitive prior and the Jeffreys prior are uniform. However, the priors are what they are, irrespective of how they are parameterized. They are explicitly uniform in a particular parameterization and implicitly uniform in all others. Uniformity, it follows, cannot serve as a principle that distinguishes one prior from another.

This ubiquity of uniform priors for a continuous set of infinitesimal probabilities is in marked contrast to situations in which prior probabilities are assigned to a finite number of discrete possibilities, such as the 38 pockets of a double-zero roulette wheel. Uniform probabilities of 1/38 suggest themselves, are meaningful, and clearly distinguished from other priors, all of which have a bias.

Uniformity in a particularly natural parameterization of the probability space might also be meaningful. This, however, invokes a notion of 'natural' that others may not share.

In many applications, estimating the state is not a purpose in itself, but only an intermediate step on the way to determining some particular property of the physical system. The objective is to find the value of a parameter that quantifies the utility of the state.

For example, one could be interested in the fidelity of the actual state with a target state, or in an entanglement measure of a two-partite state, or in another quantity that tells us how useful are the quantum-information carriers for their intended task. In a situation of this kind, one should, if possible, use a prior that is uniform in the utility parameter of interest. In contrast to the situation of the previous section, where requiring uniformity in $\mathcal {R}_0$ may be ill-advised because uniformity is a parameterization-dependent notion, here we specify uniformity for the parameter we are interested in.

To illustrate, consider a single qubit. Suppose the utility parameter is the purity ξ(ρ) = tr{ρ²} of the state ρ. With the Bloch-ball representation of a qubit state, $\rho =\frac {1}{2}(1+\boldsymbol {\varrho }\cdot \boldsymbol {\sigma })$, where ${\boldsymbol {\varrho }={\mathrm {tr}}\{\boldsymbol {\sigma }\rho \}=\langle \boldsymbol {\sigma }\rangle }$ is the Bloch vector and σ is the vector of Pauli matrices, the purity is

$$\begin{equation} \xi(\rho)=\textstyle\frac{1}{2}(1+\varrho^2)\quad\mbox{with} \ \varrho=\boldsymbol{|\varrho|}\,. \end{equation} \tag{ A.5 }$$

A prior uniform in purity induces a prior on the state space according to

$$\begin{equation} (\mathrm{d}\rho)\propto\mathrm{d}\xi\,\mathrm{d}\Omega\propto \varrho\,\mathrm{d} \varrho\,\mathrm{d}\Omega, \end{equation} \tag{ A.6 }$$

where we parameterize the Bloch ball by spherical coordinates (ϱ,θ,ϕ). Here, dΩ is the prior for the angular coordinates; the prior for the radial coordinate ϱ is fixed by our choice of uniformity in ξ. Irrespective of what we choose for dΩ, the marginal prior for ϱ is uniform in ξ.

If one can quantify the utility of an estimator by a cost function, an optimal prior can be selected by a minimax strategy. For each prior in the competition one determines the maximum of the cost function over the states in the reconstruction space, and then chooses the prior for which the maximum cost is minimal. In classical statistics, such minimax strategies are common (see, for instance, [22, chapter 5]); for an example in the context of quantum state estimation, see [23].

Symmetry considerations are often helpful in narrowing the search for the appropriate prior. For a particularly instructive example, see section 12.4.4 in Jaynes's posthumous book [24].

Returning to the uniform-in-purity prior of (A.6), one can invoke rotational symmetry in favor of the usual solid-angle element, dΩ = sin θdθ dϕ, as the choice of angular prior. The reasoning is as follows: the purity of a qubit state does not change under unitary transformations; unitarily equivalent states have the same purity. Now, regions that are turned into each other by a unitary transformation have identical radial content whereas the angular dependences are related by a rotation. Invariance under rotations, in turn, requires that the prior is proportional to the solid angle, hence the identification of dΩ with the differential of the solid angle. Note that the resulting prior element (dρ) is different from the usual Euclidean volume element, ϱ²dϱ sin θdθ dϕ, which would be natural if the Bloch ball were an object in the physical three-dimensional space, but it is not.

Symmetry arguments should be used carefully and not blindly. For a fairly tossed coin, the prior should not be affected if the probabilities for heads and tails are interchanged, w(p₁,p₂) = w(p₂,p₁). However, for the harmonic-oscillator example of section 2.1, which has the same reconstruction space as the coin, there is poor justification for requiring this symmetry because the two probabilities—of finding the oscillator in its ground state, or not—are not on equal footing.

When one speaks of an invariant prior, one does not mean the invariance under a change of parameterization—all priors are invariant in this respect—but rather a form-invariant construction in terms of a quantity that, preferably, has an invariant significance. We consider two particular constructions that make use of the metric induced by the response of the selected function to infinitesimal changes of its variables.

The first construction begins with a quantity F(p) that is a function of all probabilities p = (p₁,...,p_K). We include the square root of the determinant of the dyadic second derivative in the prior density as a factor,

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d} p)\,\Bigg|\det{\left\{{\left( \frac{\partial^2F}{\partial p_j\,\partial p_k} \right)}_{\!\!jk}\right\}}\Bigg|^{1/2} w_{\mathrm{cstr}}(p), \end{equation} \tag{ A.7 }$$

where w_cstr(p) contains all the delta-function and step-function factors of constraint as well as the normalization factor that ensures the unit size of the reconstruction space. The prior defined by (A.7) is invariant in the sense that a change of parameterization, from p to α, say, does not affect its structure

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d}\alpha)\,\Bigg|\det{\left\{{\left( \frac{\partial^2F}{\partial\alpha_j\,\partial\alpha_k} \right)}_{\!\!jk}\right\}}\Bigg|^{1/2} w_{\mathrm{cstr}}\bigl(p(\alpha)\bigr), \end{equation} \tag{ A.8 }$$

because the various Jacobian determinants take care of each other. Since w_cstr(p) enforces all constraints, the p_k are independent variables when F(p) and G(p,ν) are differentiated in (A.7) and (A.9), respectively.

For the second construction, we use a data-dependent function G(p,ν) of the probabilities p and the frequencies ν = (ν₁,ν₂,...,ν_K) with ν_j = n_j/N. Here, the square root of the determinant of the expected value of the dyadic square of the p-gradient of G is a factor in the prior density

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d} p)\,\Bigg|\det{\left\{\overline{{\left( \frac{\partial G}{\partial p_j}\frac{\partial G}{\partial p_k} \right)}_{\!\!jk}}\right\}}\Bigg|^{1/2} w_{\mathrm{cstr}}(p), \end{equation} \tag{ A.9 }$$

where $\overline {\,f(\nu )\,}$ denotes the expected value of f(ν),

$$\begin{equation} \overline{\,f(\nu)\,}=\sum_D L(D|\rho)f(\nu)\,. \end{equation} \tag{ A.10 }$$

We have, in particular, the generating function

$$\begin{equation} \overline{\,\exp{\left(\sum_{k=1}^Ka_k\nu_k\right)}\,} =\left(\sum_{k=1}^K\mathrm{e}^{{a_k/N}}p_k\right)^N \end{equation} \tag{ A.11 }$$

for the expected values of products of the ν_k. The prior defined by (A.9) is form-invariant in the same sense, and for the same reason, as the prior of (A.7).

Table A.1 reports a few examples of '$\sqrt {\det }$' factors constructed by one of these two methods. It is worth noting that the Jeffreys prior can be obtained from the entropy of the probabilities by the first method as well as from the relative entropy between the probabilities and the frequencies by the second method. The latter is a variant of Jeffreys's original derivation [15] in terms of the Fisher information.

Table A.1. Form-invariant priors constructed by one of the two methods described in the text. The '$\sqrt {\det }$' column gives the p-dependent factors only and omits all p-independent constants. The first method of (A.7) proceeds from functions of the probabilities that have extremal values when all probabilities are equal or all vanish save one. The second method of (A.9) uses functions that quantify how similar are the probabilities and the frequencies. The 'hedged prior' is named in analogy to the 'hedged likelihood' [25].

Method	Primary function	$\sqrt {\det }$
1st	$\displaystyle -\sum _kp_k\log p_k$	$\displaystyle \frac {1}{\sqrt {p_1p_2\cdots p_K}}$
	(Shannon entropy)	(Jeffreys prior)
1st	$\displaystyle \sum _kp_k^2$	1
	(purity)	(primitive prior)
2nd	$\displaystyle \sum _k\nu _kp_k$	$\sqrt {p_1p_2\cdots p_K}$
	(inner product)	(hedged prior)
2nd	$\displaystyle \sum _k\nu _k\log (\nu _k/p_k)$	$\displaystyle \frac {1}{\sqrt {p_1p_2\cdots p_K}}$
	(relative entropy)	(Jeffreys prior)

Sometimes there are reasons to expect that the actual state is close to a certain target state with probabilities t = (t₁,t₂,...,t_K). This is the situation, for example, when a source is designed to emit the quantum-information carriers in a particular state. A conjugate prior

$$\begin{equation} (\mathrm{d}\rho)=(\mathrm{d} p)\left(p_1^{t_1}p_2^{t_2}\cdots p_K^{t_K}\right)^{\beta} w_{\mathrm{cstr}}(p) \quad\mbox{with} \ \beta>0 \end{equation} \tag{ A.12 }$$

could then be a natural choice. Such priors are called 'conjugate' in standard statistics literature because the (··· )^β factor has the same structure as the point likelihood: a product of powers of the detection probabilities. The (··· )^β factor is maximal for p = t and the peak is narrower when β is larger.

The conjugate prior can be understood as the 'mock posterior' for the primitive prior that results from pretending that β copies have been measured in the past and data obtained that are most typical for the target state. Therefore, a conjugate prior is quite a natural way of expressing the expectation that the apparatus is functioning well. The posterior content of a region will be data-dominated only if N is much larger than β.

In this context, it may be worth noting that the Bayesian mean state,

$$\begin{equation} \widehat{\rho}_{\rm BM}=\int_{\mathcal{R}_0}(\mathrm{d}\rho)\,\rho, \end{equation} \tag{ A.13 }$$

computed with the conjugate prior above, is usually not the target state unless β is large. One could construct priors for which $\widehat{\rho}_{\rm BM}$ is the target state, but the presence of the w_cstr(p) factor requires a case-by-case construction.

All priors used as examples—the ones in (A.2), (A.4) and (A.12), and in table A.1—have in common that they are defined in terms of the probabilities and, therefore, they refer to the particular POM with which the data are collected. While this takes the significance of the data duly into account, it does not seem to square with the point of view that prior probabilities are solely a property of the physical processes that put the quantum-information carriers into the state that is then diagnosed by the POM.

When adopting this viewpoint, one begins with a prior density defined on the entire state space. In addition to the parameters that specify the reconstruction space (essentially the probabilities p), this full-space prior will depend on parameters whose values are not determined by the data. There could be very many nuisance parameters of this kind. In the harmonic-oscillator example of section 2.1, the data tell us only about the ground-state population but nothing about the population in any specific excited state. For a prior assigned on the formally infinite-dimensional state space, all but the ground-state population are nuisance parameters. Upon integrating the full-space prior over the nuisance parameters, one obtains a marginal prior on the reconstruction space. As a function on the reconstruction space, the marginal prior is naturally parameterized in terms of the probabilities and so fits into the formalism we are using throughout.

We note that the invoking of 'additional criteria' for a unique mapping from p to ρ, as mentioned at the end of section 2.1, is exactly what would be required if one wishes to report estimated values of the nuisance parameters. That, however, goes beyond making statements that are solidly supported by the data and is, therefore, outside the scope of our present discussion.

The symmetric uniform-in-purity prior of sections A.2 and A.3 provides an example for marginalization if the POM only gives information about x = 〈σ_x〉 and y = 〈σ_y〉 but not about z = 〈σ_z〉. We express the full-space prior in Cartesian coordinates, integrate over z, and arrive at

$$\begin{eqnarray} (\mathrm{d}\rho)&=&\mathrm{d} x\,\mathrm{d} y\,\frac{1}{2\pi}\int\nolimits_{-\infty}^{\infty}\mathrm{d} z\, \frac{\eta(1-x^2-y^2-z^2)}{\sqrt{x^2+y^2+z^2}}\nonumber\\ &=&\mathrm{d} x\,\mathrm{d} y\,\frac{1}{\pi}\,\eta(1-x^2-y^2) \cosh^{-1}\frac{1}{\sqrt{x^2+y^2}}\,.\quad \end{eqnarray} \tag{ A.14 }$$

This marginal prior is a function on the unit disk in the xy plane, which is the natural choice of reconstruction space here. When one expresses (dρ) in polar coordinates, one sees that (dρ) is uniform in φ and in ${r^2 \cosh ^{-1}(1/r)-\sqrt {1-r^2}}$, which increases monotonically from −1 to 0 on the way from the center of the disk at r = 0 to the unit circle where r = 1. Plot (a) in figure A.1 illustrates the matter.