Soundness and completeness of quantum root-mean-square errors

Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of root-mean-square (rms) error to quantum measurements. A straightforward generalization based on the noise-operator was used to reformulate Heisenberg’s uncertainty relation on the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some inaccurate measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is state-dependent, operationally definable, and perfectly characterizes accurate measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable. An improved definition extends the notion of root-mean-square error from classical to quantum measurements. How to define and measure the error of a measurement is one of the basic characteristics of experimental science. The root-mean-square error is a frequently used metric, but extending this notion from classical to quantum measurements is not trivial. Attempts to generalize this error to quantum measurements have been made, but many approaches suffer from anomalies, which unwantedly see the error vanish for certain types of measurements. Masanao Ozawa from Nagoya University now presents an improved definition for a quantum generalization of the classical root-mean-square error, which doesn’t suffer from such limitations.


INTRODUCTION
The notion of the mean error of a measurement of a classical physical quantity was first introduced by Laplace 1 (p. 324) as the mean of the absolute value of the error. Subsequently, the rootmean-square (rms) error was introduced by Gauss 2 (p. 39) as a mathematically more tractable definition to derive the principle of the least square, and has been broadly accepted as the standard definition for the mean error of a measurement. In those approaches the error of a measurement of a quantity Θ is defined as N = Ω − Θ, where Ω is the quantity actually observed, here we call the meter quantity. Then Gauss's rms error is defined as 〈N 2 〉 1/2 , where hÁ Á Ái stands for the mean value, while Laplace's mean error as 〈|N|〉. From the above definition, Gauss's rms error ε G is determined by the joint probability distribution μðθ; ωÞ ¼ PrfΘ ¼ θ; Ω ¼ ωg (1) of Θ and Ω as ε G ðμÞ 2 ¼ X ω;θ ðω À θÞ 2 μðθ; ωÞ; (2) so that ε G (μ) = 〈N 2 〉 1/2 , and it perfectly characterizes accurate measurements: ε G (μ) = 0 if and only if Ω = Θ holds with probability 1, i.e., P fμðθ; ωÞjθ ¼ ωg ¼ 1. A straightforward generalization of Gauss's definition to quantum measurements has been introduced as follows. [3][4][5] Let A be an observable of a system S, described by a Hilbert space H, to be measured by a measuring process M. Let M be an observable representing the meter of the observer in the environment E described by a Hilbert space K. The Hilbert spaces H and K are supposed to be finite dimensional throughout the present paper for simplicity of the presentation, although the arguments supporting the main results are extended to the infinite dimensional case with well-known mathematical methods.
In order to resolve the conflict, here we introduce the following requirements for any sensible error measure generalizing the classical root-mean-square error: (I) the operational definability, (II) the correspondence principle, (III) the soundness, and (IV) the completeness. The operational definability ensures that the error measure is definable by the operational description of the measuring process. The correspondence principle ensures that the error measure is consistent with the classical rms error in the case when the latter is also applicable. The soundness ensures that the error measure vanishes for any accurate measurements, while the completeness ensures that the error measure does not vanish for any inaccurate measurements. As shown later, the noiseoperator based q-rms error ε NO satisfies all the requirements (I)-(III) except (IV), whereas any error measures based on the distance of probability measures, such as the Wasserstein 2-distance, satisfy (I) and (III) but do not satisfy (II) nor (IV). We propose an improved definition for a quantum generalization of the classical rms error, which is still based on the noise operator but satisfies all requirements (I)-(IV). Moreover, it is shown that the new error measure maintains the previously obtained universally valid uncertainty relations [8][9][10][11][12][13][14][15][16][17] and their experimental confirmations [18][19][20][21][22][23][24] without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable. [25][26][27]

Operational definability
The probability distribution of the output x of the measurement is given by where P M(τ) (x) is the spectral projection of M(τ) for x 2 R, i.e., P M(τ) (x) is the projection with range {|Ψ〉 ∈H⊗K| M(τ)|Ψ〉 = x|Ψ〉}. It is fairly well-known that every measuring process has its probability operator-valued measure (POVM) that operationally describes the statistics of the measurement outcome. [28][29][30][31] The POVM Π of the measuring process M is a family Π ¼ fΠðxÞg x2R of positive operators on H defined by and satisfies the generalized Born formula We consider the requirements for any quantum generalization ε of the classical root-mean-square error ε G to quantify the mean error ε (A, M, |ψ〉) of the measurement of an observable A in a state |ψ〉 described by a measuring process M; we shall also write ε (A, M, ρ) if the state is represented by a density operator ρ. The first requirement is formulated using the notion of POVM as follows.
(I) Operational definability. The error measure ε should be definable by the POVM Π of the measuring process M, the observable A to be measured, and the initial state |ψ〉 of the measured system S.
The operational definability determines the mathematical domain of the error measure and requires that the mean error (i.e, the value of the error measure in the given state) should be determined by the operational description of the statistics of measurement outcomes.

Correspondence principle
The second requirement is based on a common practice in generalizing a classical notion to quantum mechanics. Even in quantum mechanics, there are cases where the original classical notions are directly applicable, and in those cases the generalized notions should be consistent with the original ones.
In the problem of generalizing the classical root-mean-square error to quantum mechanics, this principle is applied to the case where A(0) and M(τ) commute as two operators. In this case, the observables A(0) and MðτÞ are jointly measurable and their joint probability distribution μ(x, y) is given by μðx; yÞ ¼ hψ; ξjP Að0Þ ðxÞP MðτÞ ðyÞjψ; ξi: Then we can apply the classical definition of the root-mean-square error to the joint probability distribution μ to obtain the classical root-mean-square error ε G (μ) of this measurement; in this case, the measuring process is classically described as a black-box with the input-output joint probability distribution μ(x, y). Thus, the quantum generalization ε should satisfy εðA; M; jψiÞ ¼ ε G ðμÞ: Thus, we should require that Eq. (14) holds if A(0) and M(τ) commute. However, we should proceed further to avoid possible inconsistencies, since there is a case where a pair of observables commute only on a subspace and they have the joint probability distribution only for states in that subspace as discussed by von Neumann 32 (p. 230). To include such a general situation, we define the notions of commutativity and joint probability distribution in a sate-dependent manner. We say that observables X and Y commute in a state |Ψ〉 if for any x, y. A probability distribution μ(x, y) on R 2 , i.e., μ(x, y) ≥ 0 and P x;y μðx; yÞ ¼ 1, is called a joint probability distribution (JPD) of observables X, Y in |Ψ〉 if hΨjf ðX; YÞjΨi ¼ X x;y f ðx; yÞ μðx; yÞ (16) for any polynomial f(X, Y) of observables X, Y. Then, there exists a JPD of observables X, Y in |ψ〉 if and only if X and Y commute in |ψ〉 as shown in Theorem 1 in Methods. In this case, the JPD μ is uniquely determined by To prevent the inconsistency between the original classical notion and its quantum generalization we pose the following requirement.

Soundness
To discuss the soundness we need to clarify what measuring process M is considered to accurately measure an observable A in a given state |ψ〉. This fundamental problem has, to the best of our knowledge, not been discussed in the literature except for our previous investigations, [33][34][35][36] in which we introduced the following definition. We say that the measuring process M accurately measures an observable A in a state |ψ〉 if A(0) and M(τ) commute in |ψ, ξ〉 and their JPD μ satisfies μ(A(0) = M(τ)) = 1, where μðAð0Þ ¼ MðτÞÞ ¼ P x;y:x¼y μðx; yÞ. We will provide a justification of this definition including its operational accessibility in Methods: state-dependent definition for accurate measurements of quantum observables.
Under the above definition we pose the soundness requirement.
(III) Soundness. The error measure ε should vanish for any accurate measurements. Now, we can see that any error measure ε satisfying the correspondence principle, (II), also satisfies the soundness, (III), since in this case we have the JPD μ of A(0) and M(τ) satisfying ε (A, Since the noise-operator based q-rms error ε NO satisfies the correspondence principle, (II), it also satisfies the Soundness, (III).
Note that if A is accurately measured in |ψ〉, then A and Π are identically distributed in |ψ〉.

Completeness
Now, we introduce the following requirement.
(IV) Completeness. The measurement should be accurate if the error measure ε vanishes.
Busch, Heinonen, and Lahti 37 (p. 263) pointed out that there is a measuring process M such that ε NO (A, M, |ψ〉) = 0 but M does not accurately measure A in |ψ〉. For a simple example, let with Π(y) = P M (y). Then we have ε NO (A, Π, |ψ〉) = 0, but the measurement is not accurate, since A and Π are not identically distributed as 〈ψ|P A (2)|ψ〉 = 1/2 but 〈ψ|Π(2)|ψ〉 = 0. Thus, the noise-operator based q-rms error ε NO does not satisfy the completeness requirement. As shown above, the noiseoperator based q-rms error ε NO satisfies all the requirements (I)-(III) but does not satisfy (IV).
Locally uniform quantum root-mean-square error We call any error measure ε satisfying (I) and (II) a quantum rootmean-square (q-rms) error. A q-rms error ε is said to be sound if it satisfies (III). It is said to be complete if it satisfies (IV). A sound and complete error measure correctly indicates the cases where the measurement is accurate and where not (Busch 25 , p. 1263). A primary purpose of this paper is to find a sound and complete qrms error, and to establish universally valid uncertainty relations based on it.
We shall show that there is a simple method to strengthen the noise-operator based q-rms error to obtain a sound and complete q-rms error. In addition to (I)-(IV), this error measure is shown to have the following two properties.
(VI) Conservation property for dichotomic measurements. The error measure ε coincides with the noise-operator based q-rms error ε NO for dichotomic measurements, i.e., ε NO (A, For any t 2 R, define ε t ðA; Π; jΨiÞ ¼ ε NO ðA; Π; e ÀitA jΨiÞ: We call fε t ðA; Π; jΨiÞg t2R the q-rms error profile for A and Π in |ψ〉. If A(0) and M(τ) commute in the state |ψ, ξ〉, then we have ε t ðA; Π; jψiÞ ¼ ε NO ðA; Π; jψiÞ (22) for all t 2 R. Thus, the q-rms error profile is considered to provide additional information about the error of measurement M in the case where A(0) and M(τ) do not commute in the state |ψ, ξ〉.
To obtain a numerical error measure from fε t ðA; Π; jψiÞg t2R , we define the locally uniform q-rms error by εðA; Π; jψiÞ ¼ sup t2R ε t ðA; Π; jψiÞ: Then ε is a sound and complete q-rms error, satisfying both the dominating property, (V), and the conservation property for dichotomic measurements, (VI), as shown in Theorem 3 in Methods: sound and complete quantum root-mean-square errors, where we introduce other two sorts of q-rms errors to clarify the physical motivation behind the above definition.
The logical relationships among requirements (I)-(IV) are summarized as follows. Under the major premise (I), we have shown that (i) (III) follows from (II), (ii) (II) does not follow from (III), since the Wasserstein 2-distance satisfies (III) but does not satisfies (II), and that (iii) (II) and (IV) are independent, since ε NO satisfies (II) but does not satisfy (IV) and since there exists an error measure ε satisfying (IV) but not satisfying (II), e.g. εðA; Π; jψiÞ ¼ sup jϕi ε NO ðA; Π; jϕiÞ, where |ϕ〉 varies over all the states. Note that there exists an error measure ε satisfying (I), (III), and (IV), but does not satisfy (II), e.g., εðA; Π; jψiÞ ¼ sup jϕi ε NO ðA; Π; jϕiÞ, where |ϕ〉 varies over the cyclic subspace CðA; jψiÞ generated by A and |ψ〉. 38 Universally valid uncertainty relations In what follows, we shall show that all the universally valid measurement uncertainty relations obtained so far 8,10,11,[13][14][15][16] for the noise-operator based q-rms error ε NO are maintained by the locally uniform q-rms error ε with the same forms by property (V) and that their experimental confirmations reported so far [18][19][20][21][22][23][24] for dichotomic measurements are also reinterpreted to confirm the relations for the new error measure ε by property (VI). Moreover, the state-independent formulation based on this notion maintains Heisenberg's original form for the measurement uncertainty relation, whereas the state-dependent formulation violates it. The new error measure ε thus clears a prevailing view that the state-dependent formulation of measurement uncertainty relations is not tenable. [25][26][27] Let A, B be two observables of a quantum system S described by a Hilbert space H. Any simultaneous measurement of A and B in a state |ψ〉 defines a joint POVM Π(x, y) on R 2 for the Hilbert space H, for which the marginal POVM Π A ðxÞ ¼ Πðx; RÞ describes the A-measurement and the marginal POVM Π B ðyÞ ¼ ΠðR; yÞ describes the B-measurement. 12 Then the mean errors of the simultaneous measurement of A and B described by the joint POVM Π(x, y) in the state |ψ〉 are defined as ε(A, Π A , |ψ〉) and ε(B, Π B , |ψ〉), respectively, for a given q-rms error ε. In what follows we abbreviate ε(A) to ε(A, Π A , |ψ〉) and ε(B) to ε(B, Π B , |ψ〉) unless confusion may occur.
The above general formulation includes the error-disturbance relation for the A-measurement error of a measuring process M and the thereby caused disturbance on B, since the B-disturbance is generally defined by the error of the accurate B-measurement following the A-measurement. 8,11,39 This definition of the Bdisturbance is described in the Heisenberg picture as follows. Given a measuring process M, we can make an accurate simultaneous measurement of commuting observables M(τ) and B(τ). Then an approximate simultaneous measurement of A(0) and B(0) is obtained if the measurement of A(0) is replaced by the accurate measurement of M(τ) and the measurement of B(0) is replaced by the accurate measurement of B(τ). This simultaneous measurement is described by the joint POVM Π defined by Πðx; yÞ ¼ hξjP MðτÞ ðxÞP BðτÞ ðyÞjξi: In this case, for a given q-rms error measure ε, we define the mean error ε In this respect, it should be noted that the recent claim by Korzekwa, Jennings, and Rudolph (KJR) 27 of the impossibility of state-dependent error-disturbance relations is unfounded. In fact, KJR admitted that their basic assumption called the operational requirement (RO) should be applied to the notion of disturbance, but cannot be applied to the notion of error (KJR 27 , p. 052108-6); it can be easily seen that if (RO) were to be applied to the error, it would contradict the correspondence principle. However, such a discrimination between the disturbance and the error contradicts the above standard definition of the disturbance as the error of a successive measurement.
Heisenberg's original formulation of the uncertainty principle states that canonically conjugate observables Q, P can be measured simultaneously only with a characteristic constraint (Heisenberg 40 , p. 172) where the unambiguous lower bound h=2 is due to a subsequent elaboration by Kennard 41 (see also ref. 42 ). Heisenberg justified this relation under the repeatability hypothesis or its approximate version, an obsolete assumption on the state change in measurement; see ref. 42 for a detailed discussion. A counter example of Heisenberg's relation (38) was given in ref. 43 in the error-disturbance scenario with ε = ε NO , using a position measuring model originally constructed in ref. 44 to invalidate the standard quantum limit for gravitational-wave detectors with free-mass probe. 45,46 In ref. 47 continuously many linear position measuring processes including the above have been constructed that violate Heisenberg's relation (38) in the error-disturbance scenario for an arbitrary choice of the q-rms error ε. Thus, the violation of Heisenberg's relation (38) is not due to a particular choice of the q-rms error ε.
In contrast to the violation of Eq. (38) in the state-dependent formulation, Appleby 48 holds for ε = ε NO , except for the case where sup jψi ε NO ðQ; Π Q ; jψiÞ ¼ 0 or sup jψi ε NO ðP; Π P ; jψiÞ ¼ 0, where the supremum is taken over all the possible states |ψ〉. An apparent drawback of the above relation is that the state-independent error measures sup jψi ε NO ðQ; Π Q ; jψiÞ and sup jψi ε NO ðP; Π P ; jψiÞ are defined by the q-rms error ε NO that is not complete. However, this drawback turns out to be immediately cleared if one uses the locally uniform q-rms error ε instead, since the relation sup jψi ε NO ðX; Π X ; jψiÞ ¼ sup jψi εðX; Π X ; jψiÞ (40) holds obviously for any observable X. Thus, Eq. (39) holds for ε ¼ ε, one of the sound and complete q-rms errors. It should be noted that in the state-independent formulation as above the error measures sup jψi εðQ; Π Q ; jψiÞ and sup jψi εðP; Π P ; jψiÞ often diverges. 47,48 Even in the original γ-ray thought experiment, the error measure sup jψi εðQ; Π Q ; jψiÞ diverges as the wave packet goes beyond the scope of the microscope. Thus, Heisenberg's original form holds in the state-independent formulation but not due to the tradeoff between the resolution power and the Compton recoil. The notion of the resolution power of a microscope is well-defined only in the case where the object is well-localized in the scope of the microscope, and it cannot be captured by the state-independent formulation. The above remarks are also applied to the recent revival of the stateindependent formulation by Busch, Lahti, and Werner; 49,50 in fact, the Busch-Lahti-Werner formulation in ref. 49 is equivalent to Appleby's formulation 48 for any linear measurements. 47 For detailed discussions, we refer the reader to ref. 47 .
A generalization of Heisenberg's relation (38) to arbitrary pair of observables A and B is obtained by using the noise-operator based rms error ε = ε NO as the relation where C A;B ¼ 1 2 jhψj½A; Bjψij, holding for any joint POVMs with unbiased or independent noise operators 3,4,6,7,10,51,52 (see also refs. 53,54 ). By the dominating property, (V), the above relation also holds for the locally uniform rms error ε ¼ ε.
Using the noise-operator based q-rms error ε = ε NO , the first universally valid relation εðAÞεðBÞ þ εðAÞσðBÞ þ σðAÞεðBÞ ! C A;B (42) was given in 2003 (refs. 8,10 ), which is universally valid for any observables A,B, any system state |ψ〉, and any joint POVM Π, where the standard deviations σ(A), σ(B) are taken in the state |ψ〉. By the dominating property, (V), the above relation also holds for the locally uniform q-rms error ε ¼ ε. Thus, we have a statedependent universally valid uncertainty relation for simultaneous measurements described by a sound and complete q-rms error.
Using the noise-operator based q-rms error ε = ε NO , Branciard 15,16 considerably strengthened the above universally valid relation (42) as well as the relations proposed by Hall 13 and by Weston, Hall, Palsson, and Wiseman 14 in several ways. All those Branciard relations also hold for the locally uniform q-rms error ε ¼ ε by the dominating property, (V); see Branciard 16 [Section IV] for the alternative forms of the above mentioned relations to which the dominating property can directly apply.
Those universally valid relations for the noise-operator based qrms error have already been experimentally confirmed in the error-disturbance scenario for dichotomic measurements (i.e., A(0) 2 = B(0) 2 = M(τ) 2 = B(τ) 2 = I) with observing the violation of Eq. (41). [18][19][20][21][22][23][24] Interestingly, the above experiments were intended to confirm relations for the noise-operator based q-rms error ε = ε NO , but they also can be reinterpreted as confirmations for the corresponding relations and the violation of Eq. (41) with the locally uniform q-rms error ε ¼ ε, one of sound and complete qrms errors, since in those experiments we have ε NO ¼ ε by the conservation property for dichotomic measurements, (VI). Thus, we already have a well-developed theory of state-dependent measurement uncertainty relations based on a sound and complete q-rms error, in contrast to a prevailing claim that the state-dependent formulation of measurement uncertainty relations is not tenable. [25][26][27]

State-dependent commutativity and joint probability distributions
The state-dependent notion of commutativity was originally discussed by von Neumann 32 (p. 230) as follows. Suppose that |Ψ〉 is a superposition of common eigenstates of X and Y, namely, there exists an orthonormal family {|x, y〉} of states such that X|x, y〉 = x|x, y〉, Y|x, y〉 = y|x, y〉, and that jΨi ¼ P x;y jx; yihx; yjΨi. In this case, a measurement of the observable Z ¼ X x;y z x;y jx; yihx; yj (43) with a one-to-one assignment of real values ðx; yÞ7 !z x;y gives a joint measurement of X and Y in the state |Ψ〉 and their joint probability distribution μ(x, y) = Pr{X = x, Y = y} of X and Y is given by In this case, X and Y commute on the subspace M spanned by {|x, y〉} but do not necessarily commute on M ? . Then we have the following theorem. Theorem 1. For any pair of observables X, Y and state |Ψ〉, the following conditions are all equivalent.
(i) The state |Ψ〉 is a superposition of common eigenstates of X and Y.
(ii) The observables X and Y commute in the state |Ψ〉, i.e., Eq. (15) holds for any x,y. (iii) There exists a JPD μ of X and Y in |Ψ〉, i.e., there exists a probability distribution μ(x, y) on R 2 satisfying Eq. (16) for any polynomial f(X, Y) of observables X, Y. (iv) P x;y hΨjP X ðxÞ^P Y ðyÞjΨi ¼ 1, where ∧ stands for the infimum of two projections. In this case, the JPD μ is uniquely determined by Proof. The following proof is obtained by adapting the more general arguments previously given in refs. [34][35][36]55,56 to the case discussed here.
(i) ⇒ (iv): Suppose that |Ψ〉 is a superposition of common eigenstates of X and Y, namely, there exists an orthonormal family of states {|x, y〉} such that X|x, y〉 = x|x, y〉, Y|x, y〉 = y|x, y〉, and that jΨi ¼ P x;y jx; yihx; yjΨi. Then we have X x;y P X ðxÞ^P Y ðyÞjΨi ¼ X x;y jx; yihx; yjΨi ¼ jΨi; and hence (iv) holds.
(iv) ⇒ (ii): Let u,v∈R. It is easy to see that It follows from condition (iv) that By symmetry we obtain Thus, (ii) holds.
(ii) ⇒ (iii): Let Then μ(x, y) ≥ 0 for all x; y 2 R, since Taking f(X), g(Y) as f(X) = P X (x) and g(Y) = P Y (y), we have so that X and Y commute in |Ψ〉. It follows that P X (x)P Y (y)|Ψ〉 is a common eigenvector of X and Y if P X (x)P Y (y)|Ψ〉 ≠ 0. It follows from jΨi ¼ P x;y P X ðxÞP Y ðyÞjΨi that |Ψ〉 is a superposition of common eigenstate of X and Y.
Suppose that (i)-(iv) hold and let μ be a JPD of X, Y in |Ψ〉. Then It follows that hΨjf ðX; YÞjΨi ¼ X State-dependent definition for accurate measurements of quantum observables To characterize accurate measurements of a quantum observable in a given state, here, we take two approaches, one based on classical correlation and the other based on quantum correlation, which will be eventually shown to be equivalent.
Since the WJD is operationally accessible, condition (W) is also operationally accessible. Obviously, (W) is logically weaker than or equivalent to (S). If condition (S) holds, the measurement should be considered an accurate measurement for the consistency with the classical description. On the other hand, if the measurement is accurate, any operational test for the possible error should be passed. Observing the WJD is one of available tests for the accurate measurement, and it is natural to consider that the test is passed if ν (x, y) = 0 for all x, y with x ≠ y and that the test is failed, or the error is witnessed, if νðx; yÞ 6 ¼ 0 for some x, y with x ≠ y; this type of test has been discussed in detail by Mir et al. 58 and Garretson et al. 59 in the context of witnessing momentum transfer in a which-way measurement. Thus, condition (W) should be satisfied by any accurate measurement, since a failure of (W), or a non-zero value of ν (x, y) for a pair (x, y) with x ≠ y, witnesses an error of the measurement.
Therefore, condition (S) is a sufficient condition for the measurement to be accurate, and condition (W) a necessary condition. The following theorem shows that both conditions are actually equivalent so that both of them are necessary and sufficient conditions for the measurement to be accurate.
Theorem 2. For any measuring process M, an observable A, and a state |ψ〉, condition (S) and condition (W) are equivalent.

Sound and complete quantum root-mean-square errors
In addition to the locally uniform q-rms error, here, we introduce the following two sorts of q-rms errors. For any invertible density function f, we define the f-distributed q-rms error ε f by ε f ðA; Π; jψiÞ 2 ¼ Z R ε t ðA; Π; jψiÞ 2 f ðtÞdt: For any invariant mean m on R, 60 define the m-distributed q-rms error ε m by ε m ðA; Π; jψiÞ 2 ¼ m t ½ε t ðA; Π; jψiÞ 2 : Then we have the following theorem. if A ¼ P n a n P A ða n Þ. Proof. It is obvious from definition that ε satisfies the operational definability, (I). From Eq. (22), ε satisfies the correspondence principle, (II), and hence satisfies the soundness, (III). To prove the completeness, (IV), suppose εðA; Π; jψiÞ ¼ 0. Then we have MðτÞe ÀitA jψijξi ¼ Að0Þe ÀitA jψijξi: Since t was arbitrary, we have MðτÞ X j a j e Àitj A jψijξi ¼ Að0Þ X j a j e Àitj A jψijξi for any {a j } and {t j }. By Fourier expansion, the set of operators P j a j e Àitj A includes all functions of A, so that we have for all x ∈ R. Thus, P A (x)|ψ〉|ξ〉 is a common eigenstate of M(τ) and A(0) for a common eigenvalue x, if P A (x)|ψ〉 ≠ 0, and jψijξi ¼ P x P A ðxÞjψijξi is a superposition of those common eigenstates of MðτÞ and A(0) with common eigenvalues. It follows from Theorem 3 in ref. 33 that condition (W) holds. Thus, ε satisfies the completeness requirement (IV). Therefore, we conclude that ε is a sound and complete q-rms error. The proofs for ε f and ε m are similar, and assertion (i) follows.
Assertion (v) follows from Theorem 5.2 in ref. 60 . This completes the proof. QED Consider a quantum system with single degree of freedom described by a pair of canonically conjugate observables Q, P prepared in a state jϕi such that |〈p|ϕ〉| 2 = f(p). By the relation Z R ε t ðA; Π; jψiÞ 2 f ðtÞ dt ¼ ε NO A; Π; Z R e ÀipA jψihψje ipA jhpjϕij 2 dp 2 ; the above definition of ε f is equivalent to making the canonical approximate A-measurement with the Q-meter prepared in the state |ϕ〉 such that |〈p|ϕ〉| 2 = f(p) in the P-basis before evaluating the noise-operator based q-rms error. 61,62 The definition of ε m is also equivalent to making the canonical approximate A-measurement with the Q-meter prepared in the m-Dirac state before evaluating the noise-operator based q-rms error. 61 It is well-known that there is no canonical choice of f or m in general to achieve the ideal measurement of an arbitrary A. 60,61 By Theorem 3 (iv), our definition for ε is equivalent to εðA; Π; jψiÞ ¼ sup f ε f ðA; Π; jψiÞ; where f varies over all the invertible wave functions. Thus, although there is no canonical choice of f in general, the definition of ε can be interpreted as choosing the most error-sensitive f among all the invertible wave functions f.

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