Derivation of Standard Quantum Theory via State Discrimination

It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of General Probabilistic Theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory, which is given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.

The central purpose of studies of GPTs is to derive the model of quantum theory from information theoretical principles.Especially, a bound of performance for some fundamental information tasks is important because it can be regarded as an implicit limitation in our physical experimental setting.Studies of CHSH inequality in GPTs [1,2,3,4,5,6,7,8,9,10] are typical trials of such a derivation from operational bounds.However, it is known that Tsirelson's bound cannot single out the model of quantum theory [4,5,6,7,8,9].In the current situation, it is known that mathematical properties about information theoretical objects, for example, symmetry of perfectly distinguishable pure states, can single out quantum theory [14,15,16,17].Such derivations are suggestive, but their physical meaning is not clear.On the other hand, a derivation from a simple bound of information tasks has a clear meaning but is still open.
In order to single out quantum theory from a bound of information tasks, we need to find an information task to satisfy the following requirement; With respect to this task, the performance under a general model outperforms the performance under the standard quantum theory.Typical known results of such fundamental information tasks are the results of perfect discrimination of non-orthogonal states [18,19].The references [18,19] show that certain classes of beyond-quantum measurements can perfectly discriminate a pair of two non-orthogonal states.However, the analyses in the references [18,19] are insufficient to single out quantum theory for the following reasons.First, they deal only with perfect state discrimination.Second, they only deal with a type of measurement with a parameter.In other words, they do not clarify a precise condition when a measurement improves the performance for state discrimination.The above two reasons prevent us from singling out quantum theory.
This paper aims to resolve the above two weaknesses and to derive standard quantum theory through the performance for state discrimination.Therefore, this paper deals with state discrimination in more general models of GPTs without imposing perfectness, and we give an equivalent condition for a measurement to outperform the state discrimination by measurements in the standard quantum theory.
One of the most important values in state discrimination is the total error probability with two hypotheses ρ 0 , ρ 1 generated with probability p and 1 − p by a measurement M := {M 0 , M 1 }.In standard quantum theory, i.e., by applying a POVM M , a tight bound of Err(ρ 0 , ρ 1 ; p; M ) is given [40,41,42] as Hence, we seek an equivalent condition when the minimization of the error probability Err(ρ 0 ; ρ 1 ; p; M ) in a general model is smaller than the value In order to deal with the trace norm ∥ • ∥ 1 , this paper discusses models whose state can be described as a Hermitian matrix, called a quantum-like model at first.This restriction looks strong, but we show that any model with a condition of its dimension essentially satisfies this requirement (Lemma 4).
In terms of the comparison of the error Err(ρ 0 ; ρ 1 ; p; M ), perfect discrimination of non-orthogonal states in [18,19] corresponds to the compatibility of the relations Err(ρ 0 ; ρ 1 ; p; M ) = 0 and In this sense, the references [18,19] show the possibility of violating the bound of POVMs (2).In this paper, we answer a more general question, i.e., "When does a measurement in a general model violate the bound of POVMs?" First, we give a general bound of Err(ρ 0 ; ρ 1 ; p; M ) with its equality condition (Theorem 5).By using the general bound, we give an equivalent condition for the existence of a tuple of ρ 0 , ρ 1 , M satisfying Err(ρ 0 ; ρ 1 ; p; M ) < 1 2 − 1 2 ∥pρ 0 −(1−p)ρ 1 ∥ 1 in a model in the case of p = 1/2 (Theorem 6).Moreover, this paper answers the first purpose of GPTs, i.e., characterization of standard quantum theory via the quantum bound for state discrimination (2).By applying our equivalent condition, we derive standard quantum theory by the existence embedding of state space satisfying the quantum bound (2) (Theorem 7).Because the quantum bound (2) derives the model of quantum theory, the performance for state discrimination completely characterizes any other properties in standard quantum thery, which is a surprising operational meaning of our derivation.There exist many measures of the performance for information tasks that outperform the limit under the standard quantum theory in certain models of GPTs [1,3,7,18,19,22,23,35,36].However, all such known results do not completely characterize the model of quantum theory, i.e., such measures sometimes behave in the same way as standard quantum theory.Our finding is that the quantum bound of the performance for state discrimination is a complete measure (Figure 1).

The setting of GPTs
As a preliminary, we define a positive cone C and its dual cone C * in finite-dimensional real vector space V.A convex set C ⊂ V is called a positive cone if the following three

The class of all models in GPTs
Classes that preceding studies have characterized by certain bounds QT Derivation of QT via the quantum bound (2) in this paper Figure 1.Preceding studies have considered many bounds for the performance for many information tasks.Some of them have characterized certain classes of models satisfying the same bound as Quantum Theory (QT) [1,3,7,18,22,23,35,36], and others have derived QT out of certain classes of models [19].In contrast to the incompleteness of characterization of QT in such studies, we give a complete characterization of QT via the quantum bound for state discrimination (2).conditions hold: (i) for any x ∈ C and any r ≥ 0, rx ∈ C. (ii) C has a non-empty interior.(iii) C ∩ (−C) = {0}.For a positive cone C, the dual cone C * is defined as A dual cone is also a positive cone, and we call an element u ∈ C * ordered unit if there exists r ≥ 0 such that ru − m ∈ C * for any m ∈ C * .By using the above mathematical objects, a model of GPTs is defined as follows.
Definition 1 (A Model of GPTs).A model of GPTs is defined as a tuple G = (V, C, u), where V, C, and u are a real-vector space, a positive cone, and an order unit of C * , respectively.
For a model of GPTs G, the state space and the measurement space are defined as follows.
Definition 2 (State Space of GPTs).Given a model of GPTs G = (V, C, u), the state space of G is defined as Here, we call an element ρ ∈ S(G) a state of G.

Definition 3 (Measurements of GPTs). Given a model of GPTs
Besides, the index i is called an outcome of the measurement.Here, the set of all measurements is denoted as M(G).
In this setting, the state space and the measurement space are always convex.Also, when a state ρ ∈ S(G) is measured by a measurement {M i } ∈ M(G), the probability p i to get an outcome i is given as Standard quantum theory is a typical example of a model of GPTs, i.e., standard quantum theory is given as the model QT := (L H (H), L + H (H)), Tr), where L H (H) and L + H (H) denote the set of Hermitian matrices on a finite-dimensional Hilbert space H and the set of positive semi-definite matrices on H, respectively.In this model, the state space S(QT ) is equal to the set of density matrices.Also, by considering the correspondence from m ∈ (L + H (H)) * to M ∈ L + H (H)) as m(ρ) = Tr ρM , the measurement space M(QT ) corresponds to the set of Positive-Operator-Valued Measures (POVMs).
Next, we consider isomorphic maps between two models in order to introduce quantum-like models.Let If such an isomorphic map exists, these two models are equivalent up to normalization.Actually, for any state ρ ∈ S(G 2 ) and any measurement In other words, these two models possess the same probabilistic structure.
From now on, we compare the performance of measurements in general models with that of POVMs.In this paper, we choose the trace norm ∥ • ∥ 1 as a measure of the performance for state discrimination.In order to compare the performance of general measurements with the bound of POVMs by the trace norm (2), we need to consider any model where the trace norm ∥ • ∥ 1 can be defined.Therefore, we temporarily restrict our target model to a quantum-like model, i.e., a model G = (L H (H), C, Tr) satisfying C ⊂ L H (H). In quantum-like models, we can define the trace norm straightforwardly.A typical example of quantum-like but not quantum models is SEP defined by the positive cone This model is known as one of composite models of standard quantum systems in GPTs [7,18,19,20,31,32,36,36].Hereinafter, we focus on quantum-like models at once.One can consider this restriction strong, but the following lemma states that the restriction is only the restriction of the dimension.
Lemma 4 is mathematically related to Gleason's-type theorems given in [38,39].However, the statement of Lemma 4 is different from those in [38,39], and therefore, we give the proof of Lemma 4 in Appendix A.1.Here, we remark that the class of quantum-like models includes many important models.For example, non-unique models of quantum composite systems always satisfy the condition dim(V) = d 2 .Also, any model satisfies dim(V) = d 2 by considering the composition with an ancillary classical system.

General bound for state discrimination in GPTs
Now, we consider single-shot state discrimination in a quantum-like model G = (L H (H), C, Tr).For convenience, we denote the element f ∈ (L H (H)) * as the element M ∈ L H (H) satisfying Tr M x = f (x) for any x ∈ V.For example, the ordered unit Tr is denoted as the identity matrix I through the correspondence.Therefore, a measurement in M(G) is given as a family {M i } i∈I such that Tr M i ρ ≥ 0 for any ρ ∈ C and i∈I M i = I.
State discrimination is formulated similarly to the standard quantum theory.Given two hypotheses ρ 0 , ρ 1 for an unknown state ρ in S(G), a player is required to determine which hypothesis is true by applying a one-shot measurement {M 0 , M 1 } ∈ M(G) with two outcomes 0 and 1.The player supports the null-hypothesis ρ = ρ 0 if the outcome 0 is observed and supports the alternative hypothesis ρ = ρ 1 if the outcome 1 is observed.This decision has two types of error probabilities Tr ρ 0 M 1 and Tr ρ 1 M 0 .Here, we assume that the unknown state ρ is prepared as ρ 0 and ρ 1 with probability p and 1 − p, respectively.Therefore, the total error probability of this decision is given as (1).We aim to minimize the value (1).
In the following, the state discrimination under the standard quantum theory is referred to the standard quantum state discrimination.Under the standard quantum state discrimination, the right-hand side of (2) gives the minimum for optimizing the measurement M .The main purpose of this paper is to clarify what kind of general model outperforms the bound (2) of the standard quantum state discrimination.
The references [18,19] clarify the existence of a beyond-quantum measurement M that discriminates non-orthogonal pure states ρ 0 , ρ 1 perfectly.As mentioned in Introduction, such examples mean the violation of the bound (2), i.e., the bound for the standard quantum state discrimination.However, these references do not answer the question of when the bound (2) is violated.
To answer this question, we introduce a powerful tool for a two-outcome measurement M = {M 0 , M 1 } ∈ M(G).We define the difference between the maximum and minimum eigenvalues of M 0 as where λ max (M 0 ) and λ min (M 0 ) are the maximum and minimum eigenvalues of M 0 , respectively.Because M 1 + M 2 = I, the value r(M ) does not change even when M 0 is replaced by M 1 .Also, we define the sum of maximum and minimum eigenvalues of M i as The value r ′ (M , i) depnds on the choise i, and the relation r ′ (M , 0) + r ′ (M , 1) = 2 holds.First, using the values r(M ) and r ′ (M , i), we give a general bound for the error probability of state discrimination as Theorem 5.In the following, we denote the positive or negative part of (ρ 0 − ρ 1 ) by (ρ 0 − ρ 1 ) + and (ρ 0 − ρ 1 ) − , respectively.Theorem 5. Let G = (L H (H), C, Tr) be a quantum-like model.Any pair of two states ρ 0 , ρ 1 ∈ S(G) and any measurement M = {M 0 , M 1 } ∈ M(G) satisfy The equality of (8) holds if and only if the following condition holds.
The proof of Theorem 5 is written in Appendix A.2. Theorem 5 reproducts the bound of POVMs (2) because an optimal POVM M satisfies λ max (M i ) = 1 and λ min (M i ) = 0, i.e., r(M ) = r ′ (M , i) = 1.Especially in the case of p = 1/2, the relation between the inequalities (2) and ( 8) is more clear because the inequality ( 8) is written as and any POVM M satisfies r(M ) ≤ 1.In a model of GPTs, the value r(M ) can be larger than 1 because measurement effect M i can possess negative eigenvalues.Therefore, in a model of GPTs, the performance of state discrimination can be improved over the standard quantum.Also, we remark that the inequality (8) is tight.The following example of the tuple ρ 0 , ρ 1 , p, M satisfies the equality condition.Example in separable cone-We focus on the separable cone SEP 2×2 defined as which is given as the set of unnormalized separable states in the 2 × 2-dimensional quantum system.In the following, under the model SEP = (L H (C 4 ), Tr, SEP 2×2 , I), we construct states ρ 0 , ρ 1 , probability p = 1 2 , and a measurement M that satisfy the following properties: which imply the equality of (8).First, we choose the following matrices ρ 0 , ρ 1 : The states ρ 0 and ρ 1 are separable because ρ 0 can be written as ρ 0 = 7 i=1 |ψ i ⟩⟨ψ i |, where Also, the two matrices ρ 0 , ρ 1 satisfy Tr ρ i = 1 (i = 0, 1).Therefore, the two matrices ρ 0 , ρ 1 belong to the state space S(SEP ).Next, we choose the following family of matrices M = {M 0 , M 1 } as: First, M 0 + M 1 = I holds.Next, because M 1 satisfies Positive Partial Transpose (PPT) condition, M 0 ∈ SEP * 2×2 [18].Also, M 1 ∈ L + H (C 4 ) ⊂ SEP * 2×2 , and therefore, the family M belongs to the measurement space M(SEP ).
Then, the tuple ρ 0 , ρ 1 , p, M satisfies the conditions (11) and the equivalent condition (A) for equality of (8).They are easy to check, but we give a detailed calculation on Appendix A.3 for reader's convenience.
Theorem 5 states that the condition r(M ) > 1 is necessary for outperforming the standard state discrimination in the case of p = 1/2.Moreover, given a measurement M , this condition is also sufficient for the existence of a pair of states whose discrimination can be improved by the measurement M over the standard quantum state discrimination.In other words, an equivalent condition for supremacy over the standard quantum state discrimination is simply given as the condition r(M ) > 1 in the case of p = 1/2.Theorem 6.Let G = (L H (H), C, Tr) be a quantum-like model.Given a measurement M = {M 0 , M 1 } ∈ M(G), the following two conditions are equivalent: (i) There exist two states ρ 0 and ρ 1 in S(G) such that The proof of Theorem 6 is also written in Appendix A.4.As shown in Theorem 6, a measurement M outperforms the standard quantum state discrimination if and only if the range r(M ) is strictly larger than 1.As a typical example of such a superior measurement, a measurement given in [18] distinguishes two non-orthogonal separable states perfectly.This example satisfies the statement 1 in Theorem 6 with p = 1 2 because perfect distinguishability corresponds to the equation Err(ρ 0 ; ρ 1 ; p; M ) = 0 and non-orthogonality corresponds to inequality ∥ 1 2 ρ 0 − 1 2 ρ 1 ∥ 1 < 1.In this way, the statement 2 is a simple equivalent condition for outperforming the standard quantum state discrimination.
Here, we simply remark the relation between the violation of the bound 2 and the set of POVMs, i.e., the set M(QT ).Even if the measurement M does not belong to M(QT ), a measurement M does not always violate the bound 2. In other words, there exists a measurement M satisfying r(M ) ≤ 1 and M ̸ ∈ M(QT ).Theorem 6 also ensures that such a measurement never improves the performance for state discrimination in terms of the value Err(ρ 0 ; ρ 1 ; p; M ).

Derivation of quantum theory and quantum simulability
Now, we go back to the first aim, i.e., the characterization of standard quantum theory by the bound of the performance for an information task.Therefore, we do not restrict a model of GPTs.By applying Theorem 6, we characterize the models isomorphic to standard quantum theory by the quantum bound without the restriction to quantum-like models.
Theorem 7. Let G = (V, C, u) be a model of GPTs.The following conditions are equivalent: (i) There exists an isomorphic map f : V → L H (H) from G to the model of standard quantum theory QT .
(ii) There exists an isomorphic map f : The proof of Theorem 7 is written in Appendix A.5.Here, we remark that there always exists an isomorphic map is spaned by d 2 − 1 linearly independent elements and we can take d 2 − 1 linearly independent elements in L + H (H). Theorem 7 implies that the state space of a model non-isomorphic to the model of standard quantum theory cannot be isometry-embedded in the standard quantum state space with satisfying the quantum bound (2).This statement operationally means that a beyond-quantum model always outperforms state discrimination.In contrast, once we can embed the state space of a model in the standard quantum state space with satisfying the quantum bound (2), the model must be a model of standard quantum theory even though the measurement space is restricted only by the performance for state discrimination.Especially, in the sence of embedding, the quantum bound (2) implies all other properties in standard quantum theory.Remark 8. Here, we emphasize that Theorem 7 is not trivial statement even with an isomorphic embedding of state space.When we consider a isomorphic map f : V → L H (H) from G to a quantum like model G(f ) := (L H (H), f (C), Tr) satisfying the following conditions (A) in Theorem 7, there must exist a measurement M ∈ M( G(f )) that does not belong to M(QT).However, as seen in Theorem 6, the measurement M violates the quantum bound (2) in the case of p = 1/2 if and only if r(M ) > 1, which does not always hold even if a measurement does not belong to M(QT).Moreover, we can consider a sequence of models {G i } i∈N whose state space S(G i ) converges S(QT ) with satisfying S(G i ) ⊊ S(QT ).Theorem 7 ensures that there exists a measurement M ∈ M(G i ) satisfying r(M ) > 1 for any i.This is a non-trivial statement.
Here, we also remark that Thereom 7 is not directly shown by preceding studies about the norm determined by the error probability of state discrimination [11,12,13].In the preceding studies [11,12,13], it is clarified that the value D(ρ 0 , ρ 1 ) defined as follows for two states ρ i ∈ S(G) is a norm on any model of GPTs G.
Besides, the preceding studies [11,12,13] also showed the following error bound in a model G.
Err(ρ, σ; By applying (17), the following relation holds in the embedding model G satisfying S(G) ⊂ S(QT ): However, the relation (18) does not directly show Theorem 7 because the relation (18) does not ensure that the equality (18) never holds for a mode G except for QT .Theorem 7 rather shows the equality condition of (18) as the following corollary.
Corollary 9.For any quantum-like model G satisfying S(G) ⊂ S(QT ), the following conditions are equivalent: (i) G = QT .

Discussion
This paper has dealt with imperfect state discrimination in models of GPTs, and we have compared the performance for state discrimination between general measurements in quantum-like models and POVMs.We have introduced the range and the sum of eigenvalues of two-outcome measurement, and we have given a general tight bound of the error sum by the range and the sum (Theorem 5).Besides, we have given an equivalent condition when a general measurement possesses superior performance to POVMs in the case of p = 1/2 (Theorem 6).As an application of the results of the performance for state discrimination in quantum-like models, we have given a kind of the derivation of standard quantum theory out of all models of GPTs not restricted to quantum-like models (Theorem 7).
Becafhe quantum bound (2) derives the model of quantum theory, the performance for discrimination completely characterizes any other properties in standard quantum thery, which is a surprising operational meaning of our derivation.There exist many measures of the performance for information tasks that outperform the performance under the standard quantum theory in certain models of GPTs [1,7,18,19,35,35,36].However, all such known results do not completely characterize the model of quantum theory, i.e., such measures sometimes behave in the same way as standard quantum theory.Our finding is that the quantum bound of the performance for state discrimination is a completely characterizing measure.In other words, Theorem 7 operationally means that the performance for state discrimination completely characterizes any other properties in standard quantum thery.
Finally, we give two important future directions for this work.In this paper, we have given a derivation of standard quantum theory as the existence of a state embedding isomorphic map satisfying the quantum bound, an isomorphic map satisfying the conditions A and B in Theorem 7.Even if we do not restrict state embedding isomorphic maps, no known example other than standard quantum theory satisfies the quantum bound .Therefore, it can be expected to remove the restriction for isomorphic maps, although such a relaxation is desired from the viewpoint of the quantum foundation.If the relaxed statement is valid, the bound of the performance for state discrimination can be regarded more strongly as an important physical principle.The relaxation of the restriction for isomorphic maps is the first important future direction of this work.
This paper has addressed only single-shot state discrimination.In standard quantum information theory, it is more important to clarify asymptotic behaviors of the n-shot case than the single-shot case.The n-shot case is based on a n-composite system, which is difficult to deal with in GPTs because of the non-uniqueness of composite systems [20,24].Recently, the paper [37] calculated the asymptotic performance for hypothesis testing with post-selection even in GPTs by applying a result in the single shot case in general models.However, standard settings of hypothesis testing in GPTs, for example, the setting of Stein's lemma in GPTs, are still open.This paper also has given a result in the single-shot case in general models as a general bound in Theorem 5.The general bound in Theorem 5 is applicable to the calculation of the performance for n-shot hypothesis testing, even in GPTs.However, this application is still open because it requires various additional calculations.Therefore, the above required analysis is the second important future direction of this work.
By applying Theorem 10, we prove Theorem 7 as follows.
Proof of Theorem 7. If dim(V) ̸ = d 2 , both of the two conditions in Theorem 7 are false, i.e., they are equivalent.Therefore, we need to prove the statement in the case of dim(V) = d 2 .
First, we prove the implication 1 ⇒ 2. Because of the condition 1, there exists an isomorphic map f : V → L H (H) from G to QT .By the map f , the transformed model G(f ) is QT , and therefore, the condition 2 holds as the quantum bound.
Second, we prove the implication 2 ⇒ 1 by the contraposition.Therefore, we assume that there does not exists an isomorphic map from G to QT .We need to show there does not exists an isomorphic map f satisfying both of the conditions A. and B. Without loss of generality, we consider an arbitrary isomorphic map f satisfying the condition A., and we need to show that the map f never satisfies the condition B. Because f is an isomorphic map satisfying S( G(f )) ⊂ S(QT ) and G(f ) is not QT , Theorem 10 ensures that there exists a tuple of states, a probability, and a measurement breaking the quantum bound.In other words, the condition B. does not hold.As a result, the implication 2 ⇒ 1 has been proven, and the proof is finished.