Complete Information Balance in Quantum Measurement

Quantum measurement is a basic tool to manifest intrinsic quantum effects from fundamental tests to quantum information applications. While a measurement is typically performed to gain information on a quantum state, its role in quantum technology is indeed manifold. For instance, quantum measurement is a crucial process element in measurementbased quantum computation. It is also used to detect and correct errors thereby protecting quantum information in error-correcting frameworks. It is therefore important to fully characterize the roles of quantum measurement encompassing information gain, state disturbance and reversibility, together with their fundamental relations. Numerous efforts have been made to obtain the tradeoff between information gain and state disturbance, which becomes a practical basis for secure information processing. However, a complete information balance is necessary to include the reversibility of quantum measurement, which constitutes an integral part of practical quantum information processing. We here establish all pairs of trade-off relations involving information gain, disturbance, and reversibility, and crucially the one among all of them together. By doing so, we show that the reversibility plays a vital role in completing the information balance. Remarkably, our result can be interpreted as an informationconservation law of quantum measurement in a nontrivial form. We completely identify the conditions for optimal measurements that satisfy the conservation for each tradeoff relation with their potential applications. Our work can provide a useful guideline for designing a quantum measurement in accordance with the aims of quantum information processors.


Introduction
Since the early days of quantum mechanics, quantum measurement has been one of the central topics in Seung-Woo Lee: swleego@gmail.com quantum theory [1]. Unlike the measurement in classical world that only passively provides information on a physical state, quantum measurement has a critical role to reveal quantum phenomena from fundamental tests to quantum technology applications. For instance, incompatible quantum measurements are essential to demonstrate quantum nonlocality [2,3]. Quantum measurement is also a crucial tool in quantum information protocols for not only extracting information but also processing computation in one-way quantum computation [4], protecting quantum information by detecting and correcting errors in quantum error-correcting protocols, etc. [5][6][7][8][9][10][11][12]. That is, the role of quantum measurement is manifold in quantum technology, which requires a thorough characterization encompassing all relevant aspects of measurement.
However, there does not exist a unified framework yet to deal with all the information contents together as universal quantities in general quantum measurement [26], which is essential to draw a complete picture on information balance in quantum measurement. This is very important for a broad applicability, e.g. designing a quantum measurement to be adapted to the aim of quantum information processing at hand. Here we establish such a complete information balance in quantum measurement. We derive the full trade-off relations among those three information contents, information gain G, disturbance D, and reversibility R. We particularly show that the reversibility R plays a crucial role in information balance accounting for the gap between G and D. That is, we find that the global trade-off relation (G-D-R) tightens the trade-off between the gain and disturbance (G-D) only [20]. On the other hand, the trade-off between disturbance and reversibility (D-R) compensates the gain and reversibility relation (G-R). Consequently, our result shows that the total information is balanced in an ideal quantum measurement process, which can be interpreted as a conservation of information contents in quantum measurement.
We fully obtain the conditions to saturate all of the trade-off relations and thereby define an optimal quantum measurement providing maximal information contents quantum mechanics fundamentally allows. Our framework can offer useful guidelines for designing measurement-based quantum information protocols in quantum computation [9], teleportation [10], quantum metrology [11], and quantum error corrections [12], etc.. As all the information contents defined here are directly measurable, our results are readily testable and applicable to any quantum information platforms. Our work may contribute to deepening our fundamental understanding of quantum measurement and provide a rigorous practical benchmark to optimize measurement-based quantum information protocols.

General framework
We begin with the general framework for addressing the information changes by a quantum measurement M. Assume that an arbitrary quantum state ρ is prepared to convey information. We perform a quantum measurement M to extract the information, which disturbs and changes the input state ρ to another state. We then apply a subsequent reversing operation R to characterize the reversibility of M (see Fig. 1(a)). The reversing operation is assumed to be chosen to recover the input state ρ, i.e., Without loss of generality, we assume here that the input information is encoded onto pure states ρ = |ψ ψ| in d-dimensional Hilbert space, but the results derived in what follows are valid for any mixed The input, the post-measurement, the estimated and the output state in a single trial, when the outcomes of the measurement and the reversal are r and l, respectively, are denoted by |ψ , |ψr , | ψr , and |ψ r,l , respectively. The closeness of those states on average determines the information contents of a given quantum measurement M.
input states. Consider a quantum measurement described by a set of operators M = {M r |r = 1, . . . , n}, satisfying the completeness relation rM † rMr =1 [6,7]. When the measurement outcome is r, the input state is changed to |ψ r =M r |ψ / p(r, ψ) where p(r, ψ) = ψ|M † rMr |ψ . For each outcome r, we may estimate the input state as | ψ r by a certain estimation strategy. We then apply a subsequent reversing operation described by a set of operators R = {R r,l |l = 1, . . . , m}, satisfying the completeness relation lR † r,lR r,l =1 for each r. The final output state is then given by |ψ r,l =R r,lMr |ψ / p(r, l, ψ), where p(r, l, ψ) = ψ|M † rR † r,lR r,lMr |ψ . The reversing operation R here is assumed to be appropriately chosen according to the measurement outcome r. We note that the whole process cannot be described by a unitary operation as it can be probabilistic and conditional on the result of M and observer's choice of R. Note that our framework differs from the Petz recovery map [45,46].

Information contents
Let us define the information contents within the above general framework. For a given quantum measurement, the changes of information due to the measurement M and the reversing operation R can be defined in terms of the closeness between the input |ψ , the post-measurement |ψ r , the estimated | ψ r , and the output |ψ r,l states. See Fig. 1(b) together with the details of the definitions in Appendix A.
(i) Information Gain: The amount of information obtained by M can be quantified based on the overlap between the input |ψ and the estimated state | ψ r . The estimation fidelity is obtained by averaging | ψ r |ψ | 2 over all input states and possible measurement outcomes, i.e., dψ n r=1 p(r, ψ)| ψ r |ψ | 2 , which has different values depending on the estimation strategies. We define the information gain as the maximum estimation fidelity over all possible strate-gies, which can be obtained as (see Appendix A.2), where λ r 0 is the largest singular value ofM r in the singular value decomposition (Appendix A.1) [20]. The information gain thus lies in the range 1/d ≤ G ≤ 2/(d + 1), where the upper bound is reached by a von Neumann measurement and the lower bound by a unitary operation or a random guess.
(ii) Disturbance: In order to quantify the amount of disturbance by M, we consider the operation fidelity first.
The operation fidelity of M is given by averaging the overlap between the input |ψ and the post-measurement state |ψ r , i.e., dψ n r=1 p(r, ψ)| ψ r |ψ | 2 . We can then evaluate the maximum of the operation fidelity (see Appendix A.3), resulting in (2) In this case, all the singular values λ r i ofM r are involved in determining the average fidelity of the disturbed states. We may then define the disturbance induced by M as the minimum operation infidelity given by Similarly, the operation fidelity of the overall process R • M without postselection can be obtained by evaluating the average fidelity between the input |ψ and output state |ψ r,l , i.e., dψ n r=1 m l=1 P (r, l, ψ)| ψ r,l |ψ | 2 . Its maximum can be obtained as (see Appendix A.4) where the singular values λ r,l i ofR r,l are additionally involved in determining the output fidelity. The operation fidelity lies in the range 2/(d + 1) ≤ F ≤ 1, where the upper bound is reached by a unitary operation and the lower bound by a von Neumann measurement.
(iii) Reversibility: We consider the success event of the reversing operation R, which faithfully recovers the input state, i.e., |ψ r,l ∝ |ψ , to evaluate the reversibility. Assume that the operatorsR r,l for l = 1, . . . , s < m are associated with the success events s.t.R r,lMr |ψ = η r,l |ψ , where |η r,l | 2 is the success probability of the reversing operation when the outcome is r. The reversibility is then obtained as the maximum overall success probability, where λ r d−1 is the smallest singular value ofM r [40]. The reversibility is scaled as 0 ≤ R ≤ 1.
The optimal reversing operator can be defined in the following context. Assume that the measurement operatorM r is represented in the singular value decomposition asM r =V rDr with a unitary opera-torV r and a diagonal matrixD r = d−1 i=0 λ r i |i i| (see Appendix A.1). Its optimal reversing operator can then be written byR r, 1 Here, we set s = 1 without loss of generality). For example, a quantum measurement described byM 1 = √ η|1 1| the measurement outcome r = 2 and the reversal process l = 1, respectively. In this case, the reversibility is given by R = 1 − η from Eq. (5). Note that a unitary operation (no measurement) is deterministically reversible R = 1 while a von Neumann measurement is completely irreversible R = 0. We now have three information contents characterizing a quantum measurement, i.e., information gain G, disturbance D, and reversibility R. These are universal quantities averaged over all input states of a given dimension d and have a clear operational meaning in terms of quantum fidelity, fulfilling the requirements for the information contents in quantum measurement [26]. Note that the information contents evaluated here with pure input states ρ = |ψ ψ| and the accompanying consequences in information balance are generally valid for arbitrary mixed input states ρ, since the maximum averaged in the space of pure states must represent the maximum in the space of convex combination of pure states, i.e. mixed states.

Information balance in quantum measurement
In this section, we now derive the trade-off relations among the information contents of quantum measurement. Before presenting those trade-off relations, we first introduce and prove a useful inequality on the relation between the reversibility R and the overall operation fidelity F(R • M) after the measurement followed by the reversing operation defined in the previous section: We will use Lemma 1 to derive the trade-off relations presented in what follows. Before presenting our main results, we introduce two previously known trade-off relations between information contents, denoted by G-D [20] and G-R [40] as below.
(G-D: Information gain and Disturbance trade-off) The trade-off relation between the information gain G and disturbance D = 1 − F was derived in Ref. [20] as (8) This is the quantitative proof of the heuristic knowledge 'the more information a quantum measurement obtains, the more disturbed the quantum state becomes.' It gives the lower bound of diturbance for a given amount of information that a quantum measurement extracts.
(G-R: Information gain and Reversibility trade-off) The trade-off relation between the information gain G and the reversibility R was derived in Ref. [40] as which was the first information-theoretic approach introducing the role of the reversibility in quantum measurement. It captures the idea that 'the more information a quantum measurement obtains, the less reversible the quantum measurement is.' However, a global trade-off relation including all three information contents has been missing so far. The full quantitative links among the three information contents have also not been completed (see Fig. 2). Let us now derive this global trade-off relation, including all the information contents, G, D, and R, aiming to complete the total information balance as follows. Theorem 1. (G-D-R: Global trade-off relation) The information gain G, the disturbance D and the reversibility R of quantum measurement always satisfy an inequality The inequality in (10) determines the quantitative relation among the three information contents, G, D, and R (see Appendix C for the details of the proof). This relation can be interpreted in various ways. It draws the upper bound of F or equivalently the lower bound of D with respect to both G and R. Or, it indicates the maximum possible reversibility for a given pair of the information amounts G and D.
When d = 2, the inequality in (10) is equivalent to G-D in (8). This is due to the relation between the information gain G and the reversibility R for d = 2, i.e. 2 3 − G = R 6 . It is given by the completeness condition (1) and (5).
On the other hand, for d > 2 in general, the G-D-R inequality in (10) is fundamentally different from G-D in (8) and G-R in (9). Note that G-D-R provides a tighter bound than G-D in the relation between the information gain and the disturbance. This becomes clearer with its saturation condition described below.
(G-D-R saturation condition) The G-D-R inequality (10) is saturated if and only if the quantum measurement satisfies following conditions: all v i for i = 0, · · · , d − 1 are collinear and where is a vector defined with the singular values ofM r . See Appendix C for the detailed description of the condition. We can see that a quantum measurement satisfying the G-D saturation condition (i.e., all v i are collinear and 20]) also satisfies the saturation condition of G-D-R (Table 1), but the converse is not always true. It indicates that the information balance can be more tightly characterized by G-D-R in a broader set of quantum measurements than G-D. In section 5, we will further discuss on the different sets of quantum measurements classified based on the saturation conditions of the trade-off relations. Besides, we also analytically show that the right-hand side of the inequality (10) is always lower than or equal to the right-hand side of the inequality (8) (see Appendix D), which guarantees that G-D-R tightens G-D.
We here introduce and prove another useful inequality on the operation fidelity by a reversing operation as below: See Appendix E for the details of the proof. Note that the equality in (12) holds when M is a unitary operation or a von Neumann measurement. The inequality (12) implies that 'the disturbance in quantum measurement never decreases by any subsequent reversing operation.' We can further generalize this for arbitrary k-times sequential quantum measurements, The inequality in (12) and its extension mentioned above are intuitively plausible by the second law of thermodynamics. These indicates the non-increasing of the average fidelity between the input and output states by reversing operations so that it differs from, but may be fundamentally related to, the data processing inequality [47]. We use (12) to derive a tradeoff relation in what follows.
Let us now derive the quantitative relation between the disturbance and the reversibility of quantum measurement using Lemma 1 and Lemma 2.
in an arbitrary dimension d.
The inequality in (13) complements other trade-off relations, determining the upper bound of R by D. It implies that 'the more disturbing a quantum measurement is, the less reversible it becomes.' Here the equality holds for von Neumann measurements or unitary operations.
We have now completed the full quantitative links among the three information contents, i.e., information gain G, disturbance D, and reversibility R of quantum measurement, as illustrated in Fig. 2. These clearly show how the total information is balanced during the measurement and the reversal process. Suppose that we choose and modify a quantum measurement, intending to extract information maximally from the input quantum state. Such an optimization inevitably increases the disturbance D (i.e., decreases F) from the relation G-D. The operation fidelity F can be interpreted here as the remaining information in the post-measurement state as a part of the initially encoded information into the input state. On the other hand, there exists a hidden part of information, by which we can recover the input state using a subsequent reversing operation on the output state, quantified by R. It turns out that R fills the gap between G and D and tightens further the trade-off relations beyond G-D as proved in this section.
As an example, let us consider a quantum measurement with operatorsM i = √ p|i i| + We clearly see that the bound by G-D-R is tighter than the one by G-D.
In view of the G-D tradeoff, the measurement is not optimal in the sense that it does not maximize the output fidelity F for a given degree of G. However, in fact, it is an optimal measurement saturating the G-D-R tradeoff in Eq. (10). The gap between the solid and the dashed curves is due to the reversibility R, which is also in trade-off with G as plotted in Fig. 3 (right). It thus illustrates the importance of including all information contents G-D-R to characterize a quantum measurement as optimal or non-optimal.
We note that there would exist some missing parts of the information accounted for by none of G, F, or R. Such a missing part should be due to the nonoptimality of quantum measurement or ignorance in the estimation of the input state based on the measurement outcomes. For instance, if we take into account the effect of errors in quantum measurement leading to imperfect reversal, the final output state may not be the same as the input state, i.e., (R • M)(ρ) ∝ ρ = ρ. Specifically, let us consider the case when the reversing operation succeeds but yields the output state ρ(ψ, ) aŝ which deviates from the original input state due to an error parameterized by 0 ≤ ≤ 1. The reversibility can then be evaluated as (see Appendix F) Here F s = dψ 1 0 d p( ) ψ|ρ(ψ, )|ψ is the average fidelity between the input state and the output state of the successful reversal, where p( ) denotes the error probability density. It shows that errors tend to decrease the reversibility in proportion to F s . Note that when F s = 1 without errors the reversibility can reach the maximum value in Eq. (5). Similarly, the amount of information gain can be degraded from the maximum G in Eq. (1) in the presence of errors. Therefore, imperfection and errors result in a missing part of the total information. As a result, the information contents G, D and R of the quantum measurement with errors satisfy but cannot saturate the information trade-off relations.
In this context, we may define the optimal quantum measurement as the one that reaches the upper bounds of trade-off relations without any missing part of the total information. This further generalizes the definition of optimal quantum measurement to reach the bound of G-D with minimal disturbance in [23,41,48]. In a sense, the optimal measurement is the one that conserves the total information in a nontrivial form obeying the trade-off relations. We present how optimal quantum measurements can be classified into different sets according to the considered trade-off relations in the next section.

Classifying optimal quantum measurements
The saturation condition on each trade-off relation determines the criterion for classifying optimal quantum measurements. Let S be the set of all possible quantum measurements. We then denote by S G-D-R , S G-D , S G-R , and S D-R those subsets of quantum measurements saturating the trade-off relations G-D-R, G-D, G-R, and D-R, respectively. The saturation condition of G-R isM † rMr = a r |j r j r | + b r1 where j r ∈ {0, · · · , d−1} and a r and b r are non-negative parameters [40]. It was shown in Ref. [41] that a quantum measurement saturating G-D always satisfies the condition saturating G-R, while the converse is not true, s.t. S G-R ⊃ S G-D . In the previous section from Theorem 1, we have observed that S G-D-R ⊃ S G-D , i.e, a quantum measurement saturating G-D always saturates G-D-R, but the converse is not true. We further find that S G-D-R ∩ S G-R = S G-D from the saturation conditions of G-D-R and G-R ( Table 1). The detailed proof is given in Appendix G. The elements of S D-R are either unitary operators or von Neumann measurements by Theorem 2. Now, the Venn diagram of optimal quantum measurement sets can be constructed as illustrated in Fig. 4, classified based on the optimality of quantum measurement to reach the upper bounds of information contents by the trade-off relations.

All Quantum Measurements
Let us consider some examples of different types of quantum measurements.
(i) Assume that a von Neumann measurementP = |i i| is performed on arbitrary d-dimensional quan-  tum states. It allows one to obtain the maximum information G = 2/(d + 1). Therefore, no significant information remains on the post-measurement state F = 2/(d + 1) nor is recoverable R = 0 (we remind that the range of information gain, operation fidelity, and reversibility are 1/d ≤ G ≤ 2/(d + 1), 2/(d + 1) ≤ F ≤ 1, and 0 ≤ R ≤ 1, respectively). It is straightforward to see that these quantities saturate all the trade-off relations (Fig. 4).

Saturation condition
It becomes a von Neumann measurement when p = 1 and a unitary operator when p = 1/3. It causes a partial collapse of the input state and can be reversed for 1/3 < p < 1. For each outcome i, the optimal reversing operation is defined (non-success). We obtain the information contents as G = (1 + p)/4, F = (3 − p + 2 2p(1 − p))/4, and R = 3(1 − p)/2, which are saturating G-D, G-D-R, and G-R relations, i.e., {M r } ∈ S G-D-R , {M r } ∈ S G-D and {M r } ∈ S G-R .
We have observed that all of the above examples satisfy the trade-off inequalities. The examples (i) -(iv) are optimal quantum measurements that saturate at least one of the trade-off relations. In an optimal quantum measurement, the total information is divided into G, D, and R and balanced by the change of the parameter p. On the other hand, the example (v) is non-optimal which satisfies all the trade-off inequalities but does not saturate any of them. Each of the examples (i) -(v) given above represents each divided region in the diagram of Fig. 4.

Discussion
We have established the complete information balance in quantum measurement by deriving the full quantitative trade-off relations among information gain, disturbance, and reversibility. Under a quantum measurement, the initial information contained in the input state is divided into i) the obtained information G, ii) the remaining information F in the postmeasurement state, and iii) the reversible information R. Our result clearly shows that the three quantities are balanced by the trade-off relations. The reversibility R turns out to play an essential role in completing the information balance, filling the gap between the information gain G and disturbance D = 1 − F. Note that the three information contents are defined to be universal for a given dimension by covering all possible input states and have clear operational meanings, fulfilling a general requirement as an information content in quantum measurement [26].
While all quantum measurements, including noisy or weak measurements, must satisfy the trade-off relations, the conditions to saturate them define optimal quantum measurements. Those optimal measurements may be said to conserve the total information in a nontrivial form according to the trade-off relation. We have classified all quantum measurements into different sets based on their optimality to reach the upper bounds of G, D, and R. We then thoroughly analyzed their relations resulting in the Venn diagram in Fig. 4. These may provide useful guidelines for designing a quantum measurement according to its aim in quantum information protocol. For example, a maximum information gain with minimal disturbance is desirable for estimating or discriminating quantum information [41], whereas a maximum reversibility suits the aim of transmitting [10] or protecting [12] quantum information.
Our results imply that total information does not increase in quantum measurement and reversal process and may hint at the extension of the information conservation law. It should be obviously the case with The output at the bottom yields exactly the same state as the input state with the probability given by R, while the upper estimation part yields an infinite number of approximated copies with the fidelity given by G.
a unitarity process in line with the second law of thermodynamics [49,50]. Extending to the non-unitary quantum measurement and selective processes, our optimal measurements saturating trade-off relations may be considered as defining the information conservation.
Our results can also be interpreted as a quantitative refinement of the no-cloning theorem [51] in the context of quantum measurement. As a direct application, we can consider a universal cloning machine, which has been analyzed so far either as a deterministic process with a lower fidelity [52][53][54][55][56][57] or a probabilistic one for an exact cloning [58,59]. By contrast, our results allow us to optimize further the protocol by compromising the output fidelity and the success probability of the cloning. For example, consider a 1 → N +1 asymmetric, probabilistic, cloning machine yielding |ψ → η r,l |ψ r ⊗N |ψ , where |ψ r ⊗N are the approximate N copies and |ψ a perfect copy. Employing the optimal measurement and the reversal process can realize this cloning machine when N → ∞, as illustrated in Fig. 5, with |η r,l | 2 the success probability, and r and l the outcomes of the measurement and the reversal process, respectively. Compared to the deterministic version [55][56][57], our results provide a method to optimize the cloning process further with enhanced fidelities at the expense of success probability. Note that the trade-off relations among G, D, and R determine the quantitative upper bound of the performance of the cloning machine.
An important path for further study is exploring practical applications, e.g. measurement-based quantum processor, teleportation, metrology, and quantum error correction. Characterizing the information flow in sequential quantum measurements would also be interesting, in which the uncertainty relation between incompatible measurements may be crucial. It may also be valuable to translate our results into the context of quantum thermodynamics [60,61]. As the information contents G, D, and R are defined as mea-surable quantities, the derived trade-off relations are ready to be tested in any quantum information platform, e.g., superconducting [37], ion trap [38], and photonic qubits [35,36,39,41,42]. Our work establishes fundamental criteria for characterizing quantum measurement and offers useful guidelines for designing optimal measurement-based quantum processors.

A.1 Singular value decomposition
By a singular value decomposition, a measurement operatorM r can be represented asM r =V rDrŴr in terms of unitary operatorsŴ r andV r and a diagonal matrixD r = d−1 i=0 λ r i |i i|. Without loss of generality, we set W r =1 and also assume that the singular values are defined here in decreasing order, λ r 0 ≥ λ r 1 ≥ . . . ≥ λ r d−1 ≥ 0. Note that the singular values satisfy the completeness relation Similarly, the operator of reversing operation can be represented asR r,l =V r,lDr,lŴr,l with unitary operatorŝ W r,l andV r,l and a diagonal matrixD r,l = d−1 i=0 λ r,l i |i i|. From the completeness relation lR † r,lR r,l =1, the singular values for the reversing operators satisfy Without loss of generality, we can setV r,l =1 so that the completeness relation can be written by l (λ r,l i ) 2 = 1.

A.3 Disturbance
The fidelity between the input and the output states | ψ r |ψ | 2 , for a given quantum measurement M = {M r }, can be averaged over all possible input states and measurement outcomes r as By the Schur's lemma, we can rewrite it as Since , the maximum operation fidelity is then given as from which the disturbance can also be defined as the minimum operation infidelity D = 1 − F.

A.4 Maximum Operation Fidelity after Reversal
For given M = {M r } and R = {R r,l }, the operation fidelity including all output states without postselection is where the right hand side can be written as the maximum ofF(R • M) is obtained whenŴ r,lVr =1. Therefore,

A.5 Reversibility
We assume thatR r,l for l = 1, 2, ..., s are associated with the success reversal events such that R r,lMr |ψ = η r,l |ψ , with a complex variable η r,l . Since1 − where the equality can be reached when all the vectors v i are collinear. Then, from the inequality of arithmetic and quadratic means, where the equality can be reached when | v 1 | = · · · = | v d−2 |. Here, the right hand side can be rewritten by the completeness relation as (d − 2)(d − g − R). Therefore, we obtain and equivalently . (42) D Proof that G-D-R tightens G-D Assume that we have d − 1 non-negative real numbers x i where i = 1, · · · , d − 1. From the inequality between arithmetic and quadratic mean,

F Reversibility with errors
Using a non-optimal quantum measurement or operation, the reversing process may not be able to exactly reverse the quantum measurement due to errors so that the output state is not exactly the same as the input state, s.t. (R•M)(ρ) ∝ ρ = ρ. Let us consider the case that the reversing operation succeeds after measurement asR r,1Mr |ψ ψ|M † rR † r,1 = |η r,1 | 2 ρ(ψ, ) but the resulting output state ρ(ψ, ) deviates from the original input state. Here, we consider an arbitrary error model parameterized by 0 ≤ ≤ 1 and the error density function p( ), s.t.
By averaging over the whole error region, we obtain where F s = dψ 1 0 d p( ) ψ|ρ(ψ, )|ψ is defined as the average fidelity between the input state and the output state when the reversing operation succeeds.
If it satisfies the saturation condition of G-R, i.e.,M † rMr = a r |j r j r | + b r1 with non-negative a r and b r where j r ∈ {0, . . . , d − 1} [40], we find that λ r 1 = λ r d−1 .
As a result, the quantum measurement satisfies the saturation condition of G-D, i.e., all v i are collinear and [20].