We point out that the Gaussian wavepacket formalism can serve as a concrete realization of the joint measurement of position and momentum which is an essential element in understanding Heisenbergs original philosophy of the uncertainty principle in line with the universal framework of error disturbance and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space being a Positive OperatorValued Measure POVM measurement smoothly interpolates between the projective measurements of position and momentum. We for the first time have obtained the LeeTsutsui LT error and the refined Lee error for the positionmomentum measurement. We find that the LT uncertainty relation becomes trivial 00 in the limiting case of projective measurement of either position or momentum. Remarkably in contrast to the LT relation the refined Lee uncertainty relation which assesses errors for local representability provides a constant lower bound unaffected by these limits and is invariably saturated for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenbergs value.
['High Energy Physics - Phenomenology', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Quantum Physics']
Gaussian Formalism Concrete Realization of Joint Measurement for Heisenbergs Uncertainty Relation for Errors
{"Concrete Realization of Joint Measurement for Heisenberg's Uncertainty Relation for Errors": "Kin-ya Oda ∗ and Naoya Ogawa †  \n∗ Department of Information and Mathematical Sciences, Tokyo Woman's Christian University, Tokyo 167-8585, Japan † Department of Complex Systems Science, Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan  \nMarch 29, 2024", 'Abstract': "We point out that the Gaussian wave-packet formalism can serve as a concrete realization of the joint measurement of position and momentum, which is an essential element in understanding Heisenberg's original philosophy of the uncertainty principle, in line with the universal framework of error, disturbance, and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space, being a Positive Operator-Valued Measure (POVM) measurement, smoothly interpolates between the projective measurements of position and momentum. We, for the first time, have obtained the Lee-Tsutsui (LT) error and the refined Lee error for the positionmomentum measurement. We find that the LT uncertainty relation becomes trivial, 0 = 0, in the limiting case of projective measurement of either position or momentum. Remarkably, in contrast to the LT relation, the refined Lee uncertainty relation, which assesses errors for local representability, provides a constant lower bound unaffected by these limits and is invariably saturated, for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenberg's value.", '1 Introduction': "Heisenberg's Uncertainty Principle, introduced in 1927, is a cornerstone of quantum mechanics, fundamentally altering our understanding of measurement precision and predictability at the quantum level [1]. Originally, this principle concerned the errors and disturbances in the measurement of physical observables. This pivotal concept was then mathematically refined by Kennard, enhancing the principle's precision [2], and later generalized by Robertson to apply to any pair of observables [3]. The result of these developments is now known as the Kennard-Robertson (KR) inequality; see also Ref. [4] for Schrodinger's subsequent contribution, now known as the Schrodinger inequality. Unlike the original Heisenberg's principle, the KR and Schrodinger inequalities exclusively concern the fluctuations of physical observables, and are completely decoupled from the errors and disturbances arising from the measurements.  \nBuilding upon this foundational research, the contributions of Arthurs, Kelly, and Goodman led to the formulation of what is now known as the AKG inequality, significantly advancing our understanding of errors and the costs of measurements in quantum mechanics [5, 6]. Ozawa further expanded upon these concepts, resulting in the development of the Ozawa inequality through his studies on error and error-disturbance relations [7, 8]. Estimation theory, as highlighted in the works of Yuen-Lax [9] and further developed by Watanabe-Sagawa-Ueda, culminated in the WSU inequality [10, 11], offering alternative perspectives. These collective advancements have significantly enhanced our comprehension of quantum uncertainty.  \nLee and Tsutsui have provided an operationally tangible uncertainty relation, which we call the Lee-Tsutsui (LT) inequality, marking a significant advancement in the field of quantum uncertainty. The LT inequality offers a universal and geometric perspective on quantum measurement errors and disturbances, unifying the KR, Schrodinger, AKG, Ozawa, and WSU inequalities into a comprehensive, operationally interpretable framework [12]. This framework was subsequently applied to a concrete two-state system [13]. Lee further expanded this approach by introducing the concepts of local representability and joint measurability, leading to the establishment of what we call the Lee inequality [14, 15]. However, a challenge of this elegant framework is the lack of a concrete realization for the abstractly constructed mathematical objects, beyond the simple two-state system presented in Ref. [13]. The current paper aims to fill this gap, by providing a concrete realization of the key object in Lee's inequality, the representability for a joint measurement, in the case of Heisenberg's positionmomentum uncertainty.  \nRecent developments in the Gaussian wave-packet formalism have opened up new possibilities in quantum field theory [16, 17, 18, 19, 20]. Particularly notable is the rigorous proof of the emergence of time-boundary effect, initially claimed in Ref. [21], which has been recently established [22]. Gaussian wave packets are significant as they form an (over)complete set that spans the position-momentum phase space, including that of the free one-particle subspace, which underpins ordinary non-relativistic quantum mechanics [16]; see also Refs. [23, 24, 25] for a viewpoint that the Gaussian basis can be regarded as a complete set of coherent states in position-momentum space. Furthermore, the Gaussian basis can be generalized to a complete set of Lorentz-invariant wave-packet basis [23, 24, 25], and further to Lorentz-covariant wave-packet basis for spinor fields, incorporating spin degrees of freedom [26].  \nThe importance of the phase space is also emphasized in Refs. [27, 28, 29], particularly in the context of providing a rigorous examination and proof of Heisenberg-type inequalities, with a focus on the precision-disturbance trade-off in quantum measurements.  \nTable 1: Reminder table for the spaces. A typical element is also given for each space.  \n/owner  \n/owner  \n̂  \n̂  \nIn this paper, we posit that a Positive Operator-Valued Measure (POVM) measurement onto the Gaussian wave-packet basis naturally serves as the joint measurement for position and momentum, a key element in the discussion on Heisenberg's uncertainty in Ref. [15]. We provide concrete expressions for the pullback and pushforward, which remained abstract in the original construction [12, 13, 15]. We will demonstrate that the Gaussian basis naturally interpolates between the position-space basis and the momentum-space basis in the limits of infinitely narrow and wide widths, respectively.  \nThe organization of this paper is as follows: In Sec. 2, we briefly review the Lee-Tsutsui (LT) formalism to clarify our notation, focusing on the parts relevant to our discussion. In Sec. 3, we review the Gaussian formalism and show that the Gaussian wave packet smoothly interpolates between the position and momentum bases. In Sec. 4, we study the separate measurement of position or momentum of a Gaussian initial state in the LT formalism. We show the concrete forms of the pullback and pushforward for the position and momentum, which were previously only abstract constructs. In Sec. 5, we present our main findings: that the POVM measurement onto the Gaussian basis can be regarded as the joint measurement of position and momentum. Again we show the concrete form of the pullback and pushforward for the position and momentum for the joint measurement of the Gaussian initial state. In Sec. 6, we summarize our results and discuss future directions.", '2 Lee-Tsutsui formalism': 'We briefly review the formalism of Lee and Tsutsui [12, 13] and of Lee [14, 15], which we collectively call the LT formalism hereafter, focusing on the part relevant to our discussion.', '2.1 Basic quantum mechanics': "A physical observable is represented by a self-adjoint operator ̂ A that satisfies ̂ A † = ̂ A , where ̂ A † is the adjoint of ̂ A , on a Hilbert space H . We write the linear space of all the self-adjoint operators as S ( H ); see Table 1. A physical state is represented by a density operator ̂ ρ that is self-adjoint, ρ † = ρ , has a trace of unity, Tr[ ̂ ρ ] = 1, and has a spectrum that is entirely nonnegative, ̂ ρ ≥ 0. We write the state space Z ( H ); see Table 1. Typically in our discussion, one may assume the state to be a pure state given as a projection operator ̂ ρ = | ψ 〉 〈 ψ | , where | ψ 〉 ∈ H (and 〈 ψ | is its dual). With this case in mind, we also call a vector | ψ 〉 ∈ H the state.  \nAn expectation value of (not necessarily self-adjoint) operator A on ρ is given by  \n̂ ̂ 〈 ̂ A 〉 ̂ ρ := Tr [ ̂ A ̂ ρ ] , (1)  \nand a semi-inner product of any (not necessarily self-adjoint) operators A and B by  \n̂ where the commutator and anticommutator are [ ̂ A, ̂ B ] := ̂ A ̂ B -̂ B ̂ A and { ̂ A, ̂ B } := ̂ A ̂ B + ̂ B ̂ A , respectively. The semi-inner product naturally induces the semi-norm and the standard deviation for any (not necessarily self-adjoint) operator A on ρ : 1  \n̂ ̂ 〈 ̂ A, ̂ B 〉 ̂ ρ := 〈 { ̂ A, ̂ B } 2 〉 ρ , (2)  \n̂ ̂ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ ̂ ρ := √ 〈 ̂ A † , ̂ A 〉 ̂ ρ , σ ̂ ρ [ ̂ A ] := √ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -〈 ̂ A 〉 2 ̂ ρ . (3)  \nIf ̂ A is self-adjoint, the standard deviation reduces to σ 2 ̂ ρ [ ̂ A ] = 〈 ̂ A 2 〉 ̂ ρ - 〈 ̂ A 〉 2 ̂ ρ . Further if ̂ ρ is a pure state ̂ ρ = | ψ 〉 〈 ψ | , the semi-norm reduces to the norm of the state after the operation, ‖ ̂ A ‖ 2 | ψ 〉〈 ψ | = ‖ ̂ A | ψ 〉 ‖ 2 , and accordingly the expectation value (1) reduces to 〈 ̂ A 〉 | ψ 〉〈 ψ | = 〈 ψ | ̂ A | ψ 〉 . 2.2 Measurement process  \nA measurement M is an affine map ̂ ρ ↦→ p in the sense that, for any states ̂ ρ  , ̂ ρ  and ∀ λ ∈ [0 , 1],  \nM ( λ ρ  +(1 -λ ) ρ  ) = λ M ρ  +(1 -λ ) M ρ  , (4)  \n̂ Two comments are in order: First, the affine-ness (4) is the least requirement to allow the interpretation of probability mixture but is powerful enough to derive most of the results in the LT formalism. An affine map between any pair of state spaces, in the right of Table 1, is in general called a process . The LT formalism can handle all the processes on equal foot. Especially when applied to a quantum process Θ : Z  ( H 1 ) → Z  ( H 2 ), its disturbance can be described in quite a parallel manner to various M 's errors given below. For notational simplicity, we do not exploit this beauty of the formalism, and will stick to a measurement M as a process (except in Sec. 2.6). Second, for most of our purpose, one may assume a projective measurement that gives, ∀ ρ ∈ Z ( H ), 2  \n̂ ̂ ̂ ̂ where p is a probability density function (PDF) that satisfies ∫ Ω d ω p ( ω ) = 1 and, ∀ ω ∈ Ω , p ( ω ) ≥ 0, in which Ω is a sample space and d ω its measure. We write the convex set of all the PDFs as W ( Ω ); see Table 1. Hereafter, we write M ̂ ρ := M ̂ ρ as a reminder to reinforce that M ρ is a PDF. As a whole, M : Z ( H ) → W ( Ω ) is defined.  \n̂ [ M ρ ]( ω ) = Tr [ | ω 〉 〈 ω | ρ ] . (5)  \n〈 f 〉 p := ∫ Ω d ωf ( ω ) p ( ω ) , 〈 f, g 〉 p := ∫ Ω d ωf ∗ ( ω ) g ( ω ) p ( ω ) , (6) f ‖ p := √ 〈 f, f 〉 p = √ ∫ Ω d ω | f ( ω ) | 2 p ( ω ) , σ p [ f ] := √ ‖ f ‖ 2 p -∣ ∣ ∣ 〈 f 〉 p ∣ ∣ ∣ 2 , (7)  \n̂ ̂ On the sample space Ω , we define the classical expectation value, semi-inner product, semi-norm, and standard deviation in accordance with their quantum counterparts:  \n‖  \nwhere f, g are arbitrary real functions on Ω (but we have retained the complex conjugate, denoted by ∗ per physics community notation, to accommodate possible generalizations). More concretely for the real function f , we have σ 2 p [ f ] = ∫ Ω d ωf 2 ( ω ) p ( ω ) -(∫ Ω d ωf ( ω ) p ( ω ) ) 2 .", '2.3 LT adjoint, pullback, and pushforward': 'Given a measurement M , we define (what we call) the Lee-Tsutsui (LT) adjoint operator ̂ M /star f of a real function f by, ∀ ̂ ρ ∈ Z ( H ),  \nWealso call this the pullback operator of f . 3 As a whole, we define the LT adjoint M /star : R ( Ω ) → S ( H ), where R ( Ω ) is the space of all the real functions on Ω ; see Table 1.  \n〈 ̂ M /star f 〉 ̂ ρ = 〈 f 〉 M ̂ ρ . (8)  \nIt is important that the following inequality follows from the Kadison-Schwarz inequality:  \n‖ f ‖ M ̂ ρ ≥ ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ ̂ ρ . (9)  \nBy definitions (3), (7), and (8), we may rewrite inequality (9) into that of the standard deviations:  \nσ M ̂ ρ [ f ] ≥ σ ̂ ρ [ ̂ M /star f ] . (10)  \nTo quote [15], the operational cost of acquiring the expectation value of a quantum observable through measurements can never break the quantum limit imposed by the said observable.  \nOnce the LT adjoint is obtained, we define the pushforward function M /star ̂ A of a given self-adjoint operator ̂ A (over a state ρ ) by, ∀ f ∈ R ( Ω ),  \nor more concretely by, ∀ f ∈ R ( Ω ),  \n̂ 〈 ̂ A, ̂ M /star f 〉 ̂ ρ = 〈 M /star ̂ A, f 〉 M ̂ ρ , (11)  \nTr   { ̂ A, ̂ M /star f } 2 ̂ ρ   = ∫ Ω d ω [ M /star ̂ A ] ( ω ) f ( ω ) [ M ̂ ρ ]( ω ) . (12)  \nBy construction, the pushforward function is obtained locally , namely obtained for each given ̂ ρ , and should be more properly written as M ̂ ρ/star ̂ A . However, this dependence on ̂ ρ is trivially understood for most of our purposes, and we will use the shorthand notation to avoid clutter. As a whole, we define the pushforward M /star : S ( H ) → R ( Ω ) (for each given ̂ ρ as mentioned).  \n̂ A ̂ ρ ∼ ̂ B ⇐⇒ ∥ ∥ ∥ ̂ A -̂ B ∥ ∥ ∥ ̂ ρ = 0 , f p ∼ g ⇐⇒‖ f -g ‖ p = 0 .  \nFor our purpose, this step is not necessary and we neglect it, though can be important for other purposes.  \nFinally, we note that for the case of projective measurement (5), its LT adjoint also becomes projective, in the sense that given any function f ∈ R ( Ω ),  \n̂ M /star f = ∫ Ω d ωf ( ω ) | ω 〉 〈 ω | . (13)  \nAlso, the pushforward for the projective measurement of ̂ A becomes the identity M /star ̂ A = id; see Eq. (50) below for a concrete realization.', '2.4 LT error': 'We define (what we call) the LT error [12]:  \nε ̂ ρ [ ̂ A ; M ] := √ ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -∥ ∥ ∥ M /star ̂ A ∥ ∥ ∥ 2 M ̂ ρ = √ Tr [ ρ ̂ A 2 ] -∫ Ω d ω [ M /star ̂ A ] 2 ( ω ) [ M ̂ ρ ]( ω ) , (14)  \nwhere [ M /star ̂ A ] 2 ( ω ) := ([ M /star ̂ A ] ( ω ) ) 2 as above. The meaning of LT error is well explained in Ref. [12], which we will briefly review hereafter. Given an estimator function f ∈ R ( Ω ), we introduce an error with respect to f ( abbr. f -error):  \nε ̂ ρ [ ̂ A ; M , f ] := √ ∥ ∥ ∥ ̂ A -̂ M /star f ∥ ∥ ∥ 2 ̂ ρ + ( ‖ f ‖ 2 M ̂ ρ -∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ ) . (15)  \nThe first term in the square root is the goodness of the fit of ̂ A by the pullback operator of the estimator f , while the second is the increase in the variance from that of the quantum pullback operator ̂ M /star f to that of the classical estimator f :  \n‖ f ‖ 2 M ̂ ρ -∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ = ( ‖ f ‖ 2 M ̂ ρ -〈 f 〉 2 M ̂ ρ ) -( ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ -〈 ̂ M /star f 〉 2 ̂ ρ ) = σ 2 M ̂ ρ [ f ] -σ 2 ̂ ρ [ ̂ M /star f ] , (16)  \nwhere we used the definition of LT adjoint (8).  \nThe first term in the square root in Eq. (15) reads  \n∥ ∥ ∥ ̂ A -̂ M /star f ∥ ∥ ∥ 2 ̂ ρ = 〈 ̂ A, ̂ A 〉 ̂ ρ -2 〈 ̂ A, ̂ M /star f 〉 ̂ ρ + 〈 ̂ M /star f, ̂ M /star f 〉 ̂ ρ = ∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ -2 〈 M /star ̂ A, f 〉 M ̂ ρ + ∥ ∥ ∥ ̂ M /star f ∥ ∥ ∥ 2 ̂ ρ , (17) where we used the definition of pushforward (11). Putting Eq. (17) into Eq. (15), we obtain  \nε 2 ̂ ρ [ ̂ A ; M , f ] = ε 2 ̂ ρ [ ̂ A ; M ] + ∥ ∥ ∥ M /star ̂ A -f ∥ ∥ ∥ 2 M ̂ ρ . (18)  \nThe f -error is minimized by choosing f to be the optimal pushforward function M /star ̂ A , and the LT error (14) gives the smallest error achieved by this choice. The LT error satisfies (what we call) the LT inequality:  \nε ̂ ρ [ ̂ A ; M ] ε ̂ ρ [ ̂ B ; M ] ≥ √ I 2 ̂ ρ [ ̂ A, ̂ B ; M ] + R 2 ̂ ρ [ ̂ A, ̂ B ; M ] , (19)  \nwhere  \nI ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 [ ̂ A, ̂ B ] 2 i 〉 ρ -〈 [ ̂ M /star M /star ̂ A, ̂ B ] 2 i 〉 ρ -〈 [ ̂ A, ̂ M /star M /star ̂ B ] 2 i 〉 ρ , (20)  \nThe imaginary parts I ̂ ρ [ ̂ A, ̂ B ; M ] represents the quantum contribution to the lower bound, while the real parts R ̂ ρ [ ̂ A, ̂ B ; M ] the semi-classical contribution. 2.5 Lee error  \n̂ ̂ ̂ R ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 { ̂ A, ̂ B } 2 〉 ̂ ρ -〈 M /star ̂ A, M /star ̂ B 〉 M ̂ ρ . (21)  \nFollowing Ref. [15], we define the error for local representability , which we call the Lee error hereafter:  \n˜ ε ̂ ρ [ ̂ A ; M ] := √ ∥ ∥ ∥ M /star -1 ̂ A ∥ ∥ ∥ 2 M ̂ ρ -∥ ∥ ∥ ̂ A ∥ ∥ ∥ 2 ̂ ρ , (22)  \nwhere M /star -1 ̂ A := ( M /star ) -1 A . 4 More explicitly,  \nThe Lee inequality reads  \n̂ ˜ ε 2 ̂ ρ [ ̂ A ; M ] = ∫ Ω d ω [ M /star -1 ̂ A ] 2 ( ω ) [ M ̂ ρ ]( ω ) -Tr [ ̂ A 2 ̂ ρ ] . (23)  \nwhere the contributors to the lower bound are  \n˜ ε ̂ ρ [ ̂ A ; M ] ˜ ε ̂ ρ [ ̂ B ; M ] ≥ √ I 2 0 ̂ ρ [ ̂ A, ̂ B ] + ˜ R 2 ̂ ρ [ ̂ A, ̂ B ; M ] , (24)  \nI 0 ̂ ρ [ ̂ A, ̂ B ] := 〈 [ ̂ A, ̂ B ] 2 i 〉 ρ , (25)  \nAgain the imaginary and real parts represent quantum and semi-classical contributions, respectively.  \n̂ ˜ R ̂ ρ [ ̂ A, ̂ B ; M ] := 〈 { ̂ A, ̂ B } 2 〉 ̂ ρ -〈 M /star -1 ̂ A, M /star -1 ̂ B 〉 M ̂ ρ . (26)', '2.6 Joint measurement and classical projection of marginalization': "Following Lee's construction [15], we consider a joint measurement J : Z ( H ) → W ( Ω ), whose sample space (also called the outcome space) is separated into Ω = Ω 1 × Ω 2 so that the outcome PDF is a function of ω i ∈ Ω i ( i = 1 , 2), namely, p ∈ W ( Ω ) is written as p ( ω 1 , ω 2 ).  \nNow we introduce the following classical projection processes π i : W ( Ω 1 × Ω 2 ) → W ( Ω i ) ( i = 1 , 2), defined by the projection to marginal:  \n[ π  p ]( ω 1 ) := ∫ Ω 2 d ω 2 p ( ω 1 , ω 2 ) , [ π  p ]( ω 2 ) := ∫ Ω 1 d ω 1 p ( ω 1 , ω 2 ) . (27)  \nIf two measurements M i : Z ( H ) → W ( Ω i ) ( i = 1 , 2) can be written as M i = π i · J , they are said to admit the joint measurement J .  \nThe LT adjoint π /star i is given in parallel to that of the measurement (8): Given a function f ∈ R ( Ω i ), its LT adjoint function π /star i f ∈ R ( Ω 1 × Ω 2 ) is defined by, ∀ p ∈ W ( Ω 1 × Ω 2 ),  \n〈 π /star i f 〉 p = 〈 f 〉 π i p , ( i = 1 , 2) (28)  \nor more concretely, ∀ p ∈ W ( Ω 1 × Ω 2 ),  \n∫ Ω 1 d ω 1 ∫ Ω 2 d ω 2 [ π /star i f ]( ω 1 , ω 2 ) p ( ω 1 , ω 2 ) = ∫ Ω i d ω i f ( ω i ) [ π i p ]( ω i ) . ( i = 1 , 2) (29)  \nWe see that, by the definition (27), the LT adjoint of f is trivially  \n[ π /star i f ]( ω 1 , ω 2 ) = f ( ω i ) . ( i = 1 , 2) (30)", '3 Gaussian formalism': 'We review the Gaussian wave-packet formalism within a free one-particle subspace, decoupled from the spin degrees of freedom. In this paper, we restrict our attention to the 1D position-momentum space since the generalization to higher dimensions is straightforward. Throughout the paper, we restrict ourselves to the consideration of a fixed time slice and do not consider the time evolution/translation; see Refs. [16, 17, 18, 19] for a Gaussian formalism applied to a relativistic particle, Refs. [23, 24, 25] for insights into the free one-particle subspace of the relativistic particle and generalization to a Lorentz-invariant complete basis, and Ref. [26] for an account on the transformation of the space-like hyperplane under the Lorentz transformation and for a generalization to include the relativistic spin degrees of freedom into a Lorentz-covariant complete basis.', '3.1 Plane-wave basis': "Let { | p 〉 } p ∈ R be the momentum-space basis that spans the 1D free one-particle subspace, which we write H hereafter, without spin degrees of freedom. We employ the normalization  \n〈 p ∣ ∣ p ' 〉 = δ ( p -p ' ) , (31)  \nwhere the right-hand side is the Dirac delta function (distribution), resulting in the standard normalization for the completeness relation (resolution of identity):  \n∫ R d p | p 〉 〈 p | = ̂ 1 , (32)  \nin which ̂ 1 is the identity operator on H . Here and hereafter, the integral region of momentum (and position) is always from -∞ to ∞ unless otherwise stated. 5 The dual position-space basis { | x 〉 } x ∈ R is defined by  \n〈 x | p 〉 := e ipx √ 2 π . (33)  \nHere and hereafter, we employ the natural unit /planckover2pi1 = 1, unless otherwise stated, so that momentum and wavenumber have the same dimension (unit). The definition (33) leads to the completeness in the position space:  \n∫ R d x | x 〉 〈 x | = ̂ 1 . (34)  \nWe identify this position space as an arbitrarily chosen equal-time hyperplane. It is worth reiterating that the position-space basis can be used to span any fixed time slice, even for a relativistic particle, both in quantum mechanics (see e.g. Ref. [25]) and in quantum field theory [16, 17, 18] (this viewpoint is particularly emphasized in Refs. [25, 26]).  \nOn H , we define the position and momentum operators: 6  \nIt follows that  \nand  \n〈 x | ̂ p = -i ∂ ∂x 〈 x | , ̂ p | x 〉 = | x 〉 i ←-∂ ∂x , 〈 p | ̂ x = i ∂ ∂p 〈 p | , ̂ x | p 〉 = | p 〉 ( -i ←-∂ ∂p ) , (36)  \n[ ̂ x, ̂ p ] = i ̂ 1 , [ ̂ x, ̂ x ] = [ ̂ p, ̂ p ] = 0 . (37) The projective measurement for position M pos and that for momentum M mom are defined by, ∀ ̂ ρ ∈ Z ( H ),  \nEspecially when ̂ ρ is a pure state ̂ ρ = | ψ 〉 〈 ψ | , we recover the familiar PDFs: [ M pos | ψ 〉 〈 ψ | ] ( x ) = |〈 x | ψ 〉| 2 , [ M mom | ψ 〉 〈 ψ | ] ( p ) = |〈 p | ψ 〉| 2 . (39)  \n̂ x | x 〉 = x | x 〉 , ̂ p | p 〉 = p | p 〉 . (35)  \n[ M pos ̂ ρ ]( x ) = Tr [ | x 〉 〈 x | ̂ ρ ] , [ M mom ̂ ρ ]( p ) = Tr [ | p 〉 〈 p | ̂ ρ ] . (38)", '3.2 Gaussian basis': "We define the Gaussian wave-packet state | X,P ; σ 〉 on the momentum-space basis [16, 17, 18] (see also Ref. [25] for a historical account):  \n〈 p | X,P ; σ 〉 := ( σ π ) 1 4 exp ( -ipX -σ 2 ( p -P ) 2 ) , (40)  \nwhere we have normalized such that ∥ ∥ | X,P ; σ 〉 ∥ ∥ 2 = 1. Physically, P and X give the centers of the wave packet in the position and momentum spaces, respectively, and σ ( σ -1 ) roughly gives the spatial (momentum) width-squared. The corresponding wave function becomes  \n〈 x | X,P ; σ 〉 = ∫ R d p 〈 x | p 〉 〈 p | X,P ; σ 〉 = 1 ( πσ ) 1 4 exp ( iP ( x -X ) -( x -X ) 2 2 σ ) . (41)  \nIt is important that the Gaussian wave-packet states form a complete basis that spans H : For any fixed σ ,  \n∫ R 2 d X d P 2 π | X,P ; σ 〉 〈 X,P ; σ | = ̂ 1 , (42)  \nwhich can be verified by sandwiching both-hand sides by 〈 p | and | p ' 〉 (or equally by 〈 x | and | x ' 〉 ). 7 It is somewhat beautiful that the phase space measure d X d P 2 π emerges automatically. Note that the spatial width-squared σ is fixed and is not summed in the completeness (42).  \nIt is also noteworthy that the Gaussian basis states are not mutually orthogonal:  \n〈 X,P ; σ ∣ ∣ X ' , P ' ; σ 〉 = ∫ R d p 〈 X,P ; σ | p 〉 〈 p ∣ ∣ X ' , P ' ; σ 〉 = exp ( i P + P ' 2 ( X -X ' ) -( X -X ' ) 2 4 σ -σ 4 ( P -P ' ) 2 ) . (43)  \nIn this sense, the relation (42) is sometimes called the overcompleteness. The relation (42) tells that the Gaussian basis provides a non-projective POVM: { | X,P ; σ 〉 〈 X,P ; σ | } ( X,P ) ∈ R 2 for any given fixed σ .", '4 Separate measurement of position or momentum': 'For the rest of this paper (except in Eq. (86)), we will exclusively focus on measurements of an initial Gaussian pure state:  \n̂ ρ in = | X in , P in ; σ in 〉 〈 X in , P in ; σ in | . (44) In this section, we separately measure either its position by M pos or its momentum by M mom , as defined in Eq. (38). Though we list both the results of position and momentum measurements hereafter for conciseness, it should be kept in mind that either one of the measurements is performed, independently from the other.  \nGiven the initial state (44), its measured PDFs in the position and momentum spaces are  \n[ M pos ̂ ρ in ]( x ) = |〈 x | X in , P in ; σ in 〉| 2 = 1 √ πσ in exp ( -( x -X in ) 2 σ in ) , (45)  \nThey are centered around x = X in and p = P in , with the widths-squared roughly σ in and σ -1 in , respectively.  \n[ M mom ̂ ρ in ]( p ) = |〈 p | X in , P in ; σ in 〉| 2 = √ σ in π exp ( -σ in ( p -P in ) 2 ) . (46)', '4.1 Expectation and variance for separate measurements': "We obtain the expectation value, the squared semi-norm, and the variance (given in Eqs. (1), (3), (6), and (7)) for M pos of x (left) and for M mom of p (right):  \n̂ ̂ ‖ x ‖ 2 ̂ ρ in = ‖ id ‖ 2 M pos ̂ ρ in = X 2 in + σ in 2 , ‖ p ‖ 2 ̂ ρ in = ‖ id ‖ 2 M mom ̂ ρ in = P 2 in + 1 2 σ in , (48)  \n̂ ̂ 〈 x 〉 ̂ ρ in = 〈 id 〉 M pos ̂ ρ in = X in , 〈 p 〉 ̂ ρ in = 〈 id 〉 M mom ̂ ρ in = P in , (47)  \nwhere id is the identity function  \n̂ ̂ σ 2 ̂ ρ in [ ̂ x ] = σ 2 M pos ̂ ρ in [id] = σ in 2 , σ 2 ̂ ρ in [ ̂ p ] = σ 2 M mom ̂ ρ in [id] = 1 2 σ in , (49)  \nid( x ) = x, id( p ) = p ; (50)  \n̂ ̂ ̂ ̂ We also have the following results, which are independent of measurements:  \n〈 [ ̂ x, ̂ p ] 2 i 〉 ̂ ρ in = 1 2 , 〈 { ̂ x, ̂ p } 2 〉 ̂ ρ in = X in P in . (51)  \nrecall that ‖ ̂ x ‖ 2 ρ in := 〈 ̂ x 2 〉 ρ in , ‖ id ‖ 2 M pos ρ in := 〈 id 2 〉 M pos ρ in , etc.  \nFor some readers' ease, we list some matrix elements as well:  \n〈 p | ̂ x | X in , P in ; σ in 〉 = ( X in -iσ in ( p -P in ) ) 〈 p | X in , P in ; σ in 〉 , (52) 〈 x | ̂ p | X in , P in ; σ in 〉 = ( P in + i x -X in σ in ) 〈 x | X in , P in ; σ in 〉 , (53)  \nwhich follow from the derivative representation (36) and may be useful to derive expressions below.", '4.2 Pullback and pushforward for separate measurements': "Having completed our review, we now turn to new developments from here on. R  \n̂ M /star pos f = ∫ R d xf ( x ) | x 〉 〈 x | , ̂ M /star mom f = ∫ R d p f ( p ) | p 〉 〈 p | , (54)  \nGiven a function f ∈ R ( ), we find its LT adjoint:  \nIt is straightforward to check that they satisfy, ∀ ρ ∈ Z ( H ),  \nby expanding the general state as  \n̂ 〈 ̂ M /star pos f 〉 ̂ ρ = 〈 f 〉 M pos ̂ ρ , 〈 ̂ M /star mom f 〉 ̂ ρ = 〈 f 〉 M mom ̂ ρ , (55)  \n̂ ρ = ∫ R d x ∫ R d x ' ρ ( x, x ' ) | x 〉 〈 x ' ∣ ∣ = ∫ R d p ∫ R d p ' ˜ ρ ( p, p ' ) | p 〉 〈 p ' ∣ ∣ , (56)  \nwhere ρ ( x, x ' ) := 〈 x | ̂ ρ | x ' 〉 , with ρ ∗ ( x, x ' ) = ρ ( x ' , x ) and ∫ R d x ρ ( x, x ) = 1 (in particular, it follows that [ M pos ̂ ρ ]( x ) = ρ ( x, x )); and similarly for the momentum space. The result (54) is indeed trivial because of the generality (13), but we spelled it out anyway for concreteness.  \nEquipped with the definite representation (40), yielding the matrix elements (52) and (53), we can explicitly compute the pushforward of x ,  \n[ (57)  \nand the pushforward of p ,  \n̂ M pos /star ̂ x ]( x ) = x, [ M mom /star ̂ x ]( p ) = X in ,  \n̂ ̂ One may verify these pushforwards to satisfy the defining relation (11), or more concretely (12): As an illustration, we show the derivation of M pos /star ̂ p given in the left of Eq. (58). All other computations in this paper can be performed quite similarly. The defining relation (11) reads, ∀ f ∈ R ( R ),  \n̂ [ M pos /star p ]( x ) = P in , [ M mom /star p ]( p ) = p. (58)  \n〈 ̂ p, ̂ M /star pos f 〉 ̂ ρ in = 〈 M pos /star ̂ p, f 〉 M pos ̂ ρ in , (59)  \nTr   { ̂ p, ̂ M /star pos f } 2 ̂ ρ in   = ∫ R d x [ M pos /star ̂ p ]( x ) f ( x ) [ M pos ̂ ρ in ]( x ) . (60)  \nor more concretely from Eq. (12), ∀ f ∈ R ( R ),  \nPutting the pullback (54) into the left-hand side, we obtain  \nl.h.s. = ∫ R d xf ( x ) Tr [ ̂ p | x 〉 〈 x | + | x 〉 〈 x | ̂ p 2 | X in , P in ; σ in 〉 〈 X in , P in ; σ in | ] = ∫ R d xf ( x ) /Rfractur ( 〈 X in , P in ; σ in | x 〉 〈 x | ̂ p | X in , P in ; σ in 〉 ︸ ︷︷ ︸ ( P in + i x -X in σ in ) 〈 x | X in ,P in ; σ in 〉 ) = ∫ R d xf ( x ) P in ∣ ∣ 〈 x | X in , P in ; σ in 〉 ∣ ∣ 2 ︸ ︷︷ ︸ [ M pos ̂ ρ in ]( x ) , (61)  \nwhere /Rfractur denotes the real part and we used Eq. (53) in the second last line. Comparing with the right-hand side of Eq. (60), we obtain the constant solution [ M pos /star ̂ p ]( x ) = P in . 4.3 LT error for separate measurements  \nHaving obtained the pushforward, we can now compute the classical semi-norm for ̂ x , ‖ M pos /star ̂ x ‖ 2 M pos ̂ ρ in = X 2 in + σ in 2 , ‖ M mom /star ̂ x ‖ 2 M pos ̂ ρ in = X 2 in , (62) and the classical semi-norm for ̂ p , ‖ M pos /star ̂ p ‖ 2 Mmom ̂ ρ in = P 2 in , ‖ M mom /star ̂ p ‖ 2 Mmom ̂ ρ in = P 2 in + 1 2 σ in . (63) Combining with the quantum semi-norm (48), the resultant LT error (14) is, for x ,  \n̂ ε 2 ̂ ρ in [ ̂ x ; M pos ] = 0 , ε 2 ̂ ρ in [ ̂ x ; M mom ] = σ in 2 , (64)  \nand, for ̂ p ,  \nWe see that for both the position and momentum measurements, the left-hand side of the LT inequality (19) vanishes:  \nε 2 ̂ ρ in [ ̂ p ; M pos ] = 1 2 σ in , ε 2 ̂ ρ in [ ̂ p ; M mom ] = 0 . (65)  \nLet us see if the right-hand side vanishes consistently.  \nε ̂ ρ in [ ̂ x ; M pos ] ε ̂ ρ in [ ̂ p ; M pos ] = 0 , ε ̂ ρ in [ ̂ x ; M mom ] ε ̂ ρ in [ ̂ p ; M mom ] = 0 . (66)", '4.4 LT inequality for separate measurements': 'The pullback of pushforward is, for x ,  \nand for ̂ p ,  \n̂ ̂ M /star pos M pos /star ̂ x = ̂ x, ̂ M /star mom M mom /star ̂ x = X in ̂ 1 , (67)  \nTherefore for the position measurement,  \n̂ M /star pos M pos /star ̂ p = P in ̂ 1 , ̂ M /star mom M mom /star ̂ p = ̂ p. (68)  \nand for the momentum measurement,  \n[ ̂ M /star pos M pos /star ̂ x, ̂ p ] = i ̂ 1 , [ ̂ x, ̂ M /star pos M pos /star ̂ p ] = 0 , (69)  \n[ ̂ M /star mom M mom /star ̂ x, ̂ p ] = 0 , [ ̂ x, ̂ M /star mom M mom /star ̂ p ] = i ̂ 1 . (70)  \nCombining with the measurement-independent part (51), we see that the quantum contribution to the LT lower bound (20) vanishes:  \nWe can compute  \nI ̂ ρ in [ ̂ x, ̂ p ; M pos ] = 0 , I ̂ ρ in [ ̂ x, ̂ p ; M mom ] = 0 . (71)  \n〈 M pos /star ̂ x, M pos /star ̂ p 〉 M pos ̂ ρ in = 〈 id 〉 M pos ̂ ρ in P in = X in P in , (72) 〈 M mom /star x, M mom /star p 〉 M mom ̂ ρ in = X in 〈 id 〉 M mom ̂ ρ in = X in P in , (73)  \n̂ ̂ where we used Eq. (47). Combining with the measurement-independent part (51), we see that the semi-classical contribution to the LT lower bound (21) also vanishes:  \nThe LT inequality turned out to be 0 = 0!  \nR ̂ ρ in [ ̂ x, ̂ p ; M pos ] = 0 , R ̂ ρ in [ ̂ x, ̂ p ; M mom ] = 0 . (74)', '4.5 Lee error for separate measurement?': "To compute the Lee error (22), we need the inverse of the pullback (54). It is trivial for (so to say) the 'diagonal' measurement:  \nM /star -1 pos ̂ x ( x ) = x, M /star -1 mom ̂ p ( p ) = p. (75)  \nIt is straightforward to check that they satisfy, ∀ ρ ∈ Z ( H ),  \nby expanding the general state as in Eq. (56).  \n̂ 〈 ̂ x 〉 ̂ ρ = 〈 M /star -1 pos ̂ x 〉 M pos ̂ ρ , 〈 ̂ p 〉 ̂ ρ = 〈 M /star -1 mom ̂ p 〉 M mom ̂ ρ , (76)  \nHowever, it is hard to obtain the inverse of pullback for the 'off-diagonal' measurements M /star -1 mom ̂ x ( p ) and M /star -1 pos ̂ p ( x ). For example for the latter, we need to find f ∈ R ( R ) that satisfies ̂ p = ∫ R d xf ( x ) | x 〉 〈 x | to match Eq. (54). However, the derivative representation (36) leads to  \nFormally, the solution (if it exists) can be expressed as f ( x ) = ∫ R d x ' ( -i ∂ ∂x δ ( x -x ' ) ) . 8 This expression evidently requires regularization due to the derivative of the Dirac delta function. For example, using the regularization δ σ ( x -x ' ) = ( πσ ) -1 / 2 exp ( -( x -x ' ) 2 /σ ) leads to a vanishing result for f ( x ), indicating the need for further elaboration. Below, we will demonstrate that a joint measurement in the Gaussian phase space yields a naturally regularized expression, as will be detailed in Eq. (99).  \n̂ p = ∫ R d x ∫ R d x ' | x 〉 〈 x ∣ ∣ ̂ p ∣ ∣ x ' 〉 〈 x ' ∣ ∣ = ∫ R d x ∫ R d x ' ( -i ∂ ∂x δ ( x -x ' ) ) | x 〉 〈 x ' ∣ ∣ . (77)", '5 Joint measurement in phase space': "Below, we first review known facts about the Gaussian matrix element, then restart with our main new development from Sec. 5.1.  \nThe completeness (42) implies that the Gaussian basis can expand any function, including another Gaussian wave packet with a different width-squared: | X in , P in ; σ in 〉 . The expansion coefficient can be computed as [18]  \n〈 X,P ; σ | X in , P in ; σ in 〉 = ∫ R d p 〈 X,P ; σ | p 〉 〈 p | X in , P in ; σ in 〉 = √ 2 ( σ red σ sum ) 1 4 exp ( iP ( X -X in ) -( X -X in ) 2 2 σ sum -σ red 2 ( P -P in ) 2 ) , (78)  \nwhere 9  \n1 σσ (79)  \nσ sum := σ + σ in , σ red := 1 σ + 1 σ in = in σ + σ in .  \nPhysically, the inner product (78) can be understood as the transition amplitude from the initial state | X in , P in ; σ in 〉 to the final state | X,P ; σ 〉 , with the transition probability being  \n|〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 2 π ( 1 √ πσ sum e -1 σ sum ( X -X in ) 2 )(√ σ red π e -σ red ( P -P in ) 2 ) . (80)  \nWe see that the detection probability of the particle is centered around X = X in and P = P in , with the width-squared roughly σ sum and σ -1 red , respectively.  \nWhen we do not observe the momentum, we integrate the probability (80) over all the possible momenta:  \n∫ R d P 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 1 √ πσ sum exp ( -( X -X in ) 2 σ sum ) . (81)  \nIn the limit of detector spatial resolution finer than the initial wave packet size, σ /lessmuch σ in , we obtain σ sum → σ in , recovering the position-space PDF: [ M pos ̂ ρ in ]( X ) = |〈 X | X in , P in ; σ in 〉| 2 ; see Eq. (45). In the opposite limit σ /greatermuch σ in , the probability is governed by the detector spatial resolution, 1 √ πσ exp( -1 σ ( X -X in ) 2 ), which reduces to δ ( X -X in ) in the limit σ → 0 (while keeping σ /greatermuch σ in ), whereas in the opposite (physically more reasonable) limit σ → ∞ , the position dispersion becomes infinitely large, and the information of X in is also lost.  \nWhen we do not observe the position, we integrate the probability (80) over all the possible positions:  \n∫ R d X 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = √ σ red π exp ( -σ red ( P -P in ) 2 ) . (82)  \nIn the limit of detector spatial resolution coarser than the initial wave packet size, σ /greatermuch σ in , we obtain σ red → σ in , recovering the momentum-space PDF: [ M mom ̂ ρ in ]( P ) = |〈 P | X in , P in ; σ in 〉| 2 ; see Eq. (46). In the opposite limit σ /lessmuch σ in , the probability is governed by the detector resolution, √ σ π exp( -σ ( P -P in ) 2 ), which reduces to δ ( P -P in ) in the limit σ → ∞ (while keeping σ /lessmuch σ in ), whereas in the opposite (physically more reasonable) limit σ → 0, the momentum dispersion becomes infinitely large, and the information of P in is also lost.  \nFor reader's ease, we list some matrix elements, which may be useful to derive expressions below: Using the derivative representation (36), we obtain  \n〈 X,P ; σ | x | X in , P in , σ in 〉 = ( X -iσ red ( P -P in ) ) 〈 X,P ; σ | X in , P in , σ in 〉 , (83)  \nX := σ red ( X σ + X in σ in ) , P := σP + σ in P in σ + σ in . (85)  \n̂ 〈 X,P ; σ | ̂ p | X in , P in , σ in 〉 = ( P + i X -X in σ sum ) 〈 X,P ; σ | X in , P in ; σ in 〉 , (84)  \nwhere", '5.1 Projective measurement onto phase space': "We assert that the transition probability (80) can be regarded in the LT formalism as the result of the POVM measurement onto phase space M ph defined by, ∀ ρ ∈ Z ( H ), 10  \n̂ [ M ph ρ ]( X,P ) = Tr [ | X,P ; σ 〉 〈 X,P ; σ | ρ ] . (86)  \nGiven an initial state | X in , P in ; σ in 〉 , or more precisely the initial pure state (44), we recover the PDF (80):  \nFor some reader's ease, we list some results of the integrals:  \n[ M ph ̂ ρ in ]( X,P ) = |〈 X,P ; σ | X in , P in ; σ in 〉| 2 . (87)  \n〈 X 〉 M ph ̂ ρ in = X in , 〈 P 〉 M ph ̂ ρ in = P in , 〈 XP 〉 M ph ̂ ρ in = X in P in , 〈 X 〉 M ph ̂ ρ in = X in , 〈 P 〉 M ph ̂ ρ in = P in , 〈 XP 〉 M ph ̂ ρ in = X in P in , 〈 X 2 〉 M ph ̂ ρ in = X 2 in + σ sum 2 , 〈 P 2 〉 M ph ̂ ρ in = P 2 in + 1 2 σ red , 〈 X 2 〉 M ph ̂ ρ in = X 2 in + σ 2 in 2 σ sum , 〈 P 2 〉 M ph ̂ ρ in = P 2 in + σ 2 σ in σ sum , (88)  \nwhere X ( X,P ) := X and P ( X,P ) := P as well as X ( X,P ) := X and P ( X,P ) := P , with X and P being given in Eq. (85), respectively.  \nNow we consider the marginalization (27). We define the classical projection (marginalization) processes π pos and π mom by, ∀ p ∈ W ( R 2 ) ,  \n[ π pos p ]( X ) := ∫ R d P 2 π p ( X,P ) , [ π mom p ]( P ) := ∫ R d X 2 π p ( X,P ) , (89)  \nwith their trivial LT adjoints (30): ∀ f ∈ R ( R ),  \nMore concretely,  \n[ π /star pos f ] ( X,P ) = f ( X ) , [ π /star mom f ]( X,P ) = f ( P ) . (90)  \n[ π pos M ph ̂ ρ in ]( X ) = ∫ R d P 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = 1 √ πσ sum exp ( -( X -X in ) 2 σ sum ) , (91)  \n[ π mom M ph ̂ ρ in ]( P ) = ∫ R d X 2 π |〈 X,P ; σ | X in , P in ; σ in 〉| 2 = √ σ red π exp ( -σ red ( P -P in ) 2 ) . (92) For the Gaussian pure state (44), the joint PDF (87) has become separable:  \n[ M ph ̂ ρ in ]( X,P ) = 2 π [ π pos M ph ̂ ρ in ]( X ) [ π mom M ph ̂ ρ in ]( P ) . (93) We see that the Gaussian measurement smoothly interpolates the two limiting measurements in the momentum space and in the position space:  \n- · In the limit of detector spatial resolution finer than the original wave-packet size, σ /lessmuch σ in , we obtain σ sum → σ in and σ red → σ , and hence  \n[ π mom M ph ̂ ρ in ]( P ) → √ σ π exp ( -σ ( P -P in ) 2 ) . (95)  \n[ π pos M ph ̂ ρ in ]( X ) → 1 √ πσ in exp ( -( X -X in ) 2 σ in ) = [ M pos ̂ ρ in ]( X ) , (94)  \nWe have recovered the original PDF in the position space: |〈 X | X in , P in ; σ in 〉| 2 (see Eq. (45)), whereas the information of the original wave-packet size is lost in the momentum space. Further limit of spatially coarse detector σ →∞ (while keeping σ /lessmuch σ in ) gives [ π mom M ph ̂ ρ in ]( P ) → δ ( P -P in ), whereas in the opposite (physically more reasonable) limit of spatially fine detector σ → 0 makes the momentum dispersion infinitely large, leading to the loss of information of P in .  \n- · In the limit of detector spatial resolution coarser than the original wave-packet size, σ /greatermuch σ in , we obtain σ sum → σ and σ red → σ in , and hence  \n[ π mom M ph ̂ ρ in ]( P ) → √ σ in π exp ( -σ in ( P -P in ) 2 ) = [ M mom ̂ ρ in ]( P ) . (97)  \n[ π pos M ph ̂ ρ in ]( X ) → 1 √ πσ exp ( -1 σ ( X -X in ) 2 ) , (96)  \nWe have recovered the original PDF in the momentum space |〈 P | X in , P in ; σ in 〉| 2 (see Eq. (46)), whereas the information of the original wave-packet size is lost in the position space. Further limit of spatially fine detector σ → 0 (while keeping σ /greatermuch σ in ) gives [ π pos M ph ̂ ρ in ]( X ) → δ ( X -X in ), whereas in the opposite (physically more reasonable) limit of spatially coarse detector σ →∞ makes the position dispersion infinitely large, leading to the loss of information of X in .", '5.2 Pullback and pushforward for joint measurement': "One might find the LT adjoint M /star ph slightly less trivial because of the non-orthogonality of the basis (43); see also footnote 10. However, we find that the result is diagonal too: For a given function f ∈ R ( R 2 ) , its LT adjoint is  \n̂ M /star ph f = ∫ R 2 d X d P 2 π f ( X,P ) | X,P ; σ 〉 〈 X,P ; σ | , (98)  \nwhich can be verified by inserting the expansion (56) into the defining relation 〈 ̂ M /star ph f 〉 ̂ ρ = 〈 f 〉 M ph ̂ ρ . This shows that the LT adjoint maintains its diagonal nature even in the context of the POVM measurement (86).  \nThe position and momentum operators have the following decomposition in the Gaussian phase space:  \n̂ x = ∫ R 2 d X d P 2 π X | X,P ; σ 〉 〈 X,P ; σ | , ̂ p = ∫ R 2 d X d P 2 π P | X,P ; σ 〉 〈 X,P ; σ | , (99)  \nwhich can be verified by sandwiching with 〈 x | and | x ' 〉 for ̂ x , and with 〈 p | and | p ' 〉 for ̂ p . We can also confirm this by sandwiching with 〈 X ' , P ' ; σ | and | X '' , P '' ; σ 〉 , and then using Eq. (43), (83), and (84). Comparing with Eq. (98) for the general LT adjoint/pullback, we obtain its inverse:  \nFrom Eq. (88), we obtain their classical semi-norms,  \n∥ ∥ ∥ M /star -1 ph ̂ x ∥ ∥ ∥ 2 M ph ̂ ρ in = X 2 in + σ sum 2 , ∥ ∥ ∥ M /star -1 ph ̂ p ∥ ∥ ∥ 2 M ph ̂ ρ in = P 2 in + 1 2 σ red , (101)  \n[ M /star -1 ph ̂ x ] ( X,P ) = X, [ M /star -1 ph ̂ p ] ( X,P ) = P. (100)  \nand their classical semi-inner product,  \n〈 M /star -1 ph ̂ x, M /star -1 ph ̂ p 〉 M ph ̂ ρ in = X in P in . (102)  \nSimilarly, we obtain the pushforward of x and p (over ρ ):  \n̂ ̂ where X and P are given in Eq. (85). From Eq. (88), we obtain  \nand  \n̂ ̂ ̂ [ M ph /star x ]( X,P ) = X, [ M ph /star p ]( X,P ) = P, (103)  \n‖ M ph /star ̂ x ‖ 2 M ph ̂ ρ in = X 2 in + σ 2 in 2 σ sum , ‖ M ph /star ̂ p ‖ 2 M ph ̂ ρ in = P 2 in + σ 2 σ in σ sum , (104)  \nAccordingly, the pullback of pushforward is  \n〈 M ph /star ̂ x, M ph /star ̂ p 〉 M ph ̂ ρ in = X in P in . (105)  \n̂ M /star ph M ph /star x = ∫ R 2 d X d P 2 π X | X,P ; σ 〉 〈 X,P ; σ | , (106)", '5.3 LT error for joint measurement': '̂ ̂ M /star ph M ph /star ̂ p = ∫ R 2 d X d P 2 π P | X,P ; σ 〉 〈 X,P ; σ | . (107)  \nSubtracting the classical semi-norm squared of pushforward (104) from the quantum one of the original operator (48), we obtain the resultant LT errors (14) for x and p :  \nWe see that the left-hand side of the LT inequality (19) becomes  \n̂ ̂ ε 2 ̂ ρ in [ ̂ x ; M ph ] = σ red 2 , ε 2 ̂ ρ in [ ̂ p ; M ph ] = 1 2 σ sum . (108)  \nε ̂ ρ in [ ̂ x ; M ph ] ε ̂ ρ in [ ̂ p ; M ph ] = 1 2 √ σ red σ sum . (109)  \nε 2 ̂ ρ in [ ̂ x ; M ph ] → { 0 ( σ → 0) , σ in 2 ( σ →∞ ) , ε 2 ̂ ρ in [ ̂ p ; M ph ] → { 1 2 σ in ( σ → 0) , 0 ( σ →∞ ) . (110)  \nIn the limits of spatially fine and coarse detectors σ → 0 and ∞ , respectively, we obtain  \nThe product of LT errors (109) goes to zero in both the limits σ → 0 and ∞ , while it takes the maximum value 1 / 4 when σ = σ in .', '5.4 LT inequality for joint measurement': 'Using Eqs. (106) and (107), and then Eqs. (83) and (84), we obtain  \n〈 [ ̂ M /star ph M ph /star ̂ x, ̂ p ] 2 i 〉 ρ in = /Ifractur 〈 X ( P + i X -X in σ sum )〉 M ph ̂ ρ in = σ in 2 σ sum , (111)  \n̂ 〈 [ ̂ x, ̂ M /star ph M ph /star ̂ p ] 2 i 〉 ̂ ρ in = /Ifractur 〈 ( X -iσ red ( P -P in ) ) ∗ P 〉 M ph ̂ ρ in = σ 2 σ sum , (112)  \nwhere /Ifractur denotes the imaginary part and we used Eq. (88) in the last step of each. Subtracting the above two from the measurement-independent part in the former of Eq. (51), we see that the quantum contribution to the LT lower bound (20) vanishes:  \nI ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (113) For the semi-classical contribution to the LT lower bound (21), we subtract the measurementdependent part (102) from the independent part in the latter of Eq. (51). As a result, we find it to vanish:  \nR ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (114) It is important that the LT inequality gives no lower bound for the joint measurement of position and momentum. The LT inequality (19) now becomes  \n1 2 √ σ red σ sum ≥ 0 , (115)  \nwhich is saturated in the limits σ → 0 and ∞ . The left-hand side takes the maximum value 1 / 4 at σ = σ in .', '5.5 Lee error for joint measurement': 'Subtracting the quantum semi-norm (48) from the classical (101), we obtain the Lee errors (22) for ̂ x and p :  \nIt is remarkable that, unlike the LT error, the Lee error is solely determined from the detector resolution, independent of the width of the measured state.  \n̂ ˜ ε 2 ̂ ρ in [ ̂ x ; M ph ] = σ 2 , ˜ ε 2 ̂ ρ in [ ̂ p ; M ph ] = 1 2 σ . (116)  \nTheir product gives the left-hand side of the Lee inequality:  \n˜ ε ̂ ρ in [ ̂ x ; M ph ] ˜ ε ̂ ρ in [ ̂ p ; M ph ] = 1 2 . (117)', '5.6 Lee inequality for joint measurement': "The quantum contribution to the Lee lower bound (25) is identical to that of Heisenberg's, arising from the measurement-independent part, in the former of Eq. (51):  \nI 0 ̂ ρ in [ ̂ x, ̂ p ] = 1 2 . (118)  \nOn the other hand, subtracting the measurement-dependent part (88) from the independent (51), we find that the semi-classical contribution to the Lee lower bound (26) vanishes:  \n˜ R ̂ ρ in [ ̂ x, ̂ p ; M ph ] = 0 . (119) Unlike the LT inequality, the Lee inequality provides the meaningful lower bound, which has turned out to be identical to Heisenberg's, and is invariably saturated for the pure Gaussian initial state.", '6 Summary and discussion': "We have demonstrated that the Gaussian wave-packet formalism can be a practical realization of the joint measurement of position and momentum, shedding light on Heisenberg's original philosophy of the uncertainty principle within the framework of Lee and Tsutsui's error, disturbance, and their uncertainty relations. We have successfully obtained the Lee-Tsutsui (LT) error and the refined Lee error in the context of position-momentum measurement for the first time. Our findings indicate that the LT uncertainty relation, in the limiting case of projective measurement of either position or momentum, becomes trivial: 0 = 0.  \nIn contrast, the refined Lee uncertainty relation, focusing on errors for local representability, provides a constant lower bound that remains unaffected by such limits and is invariably saturated for a pure Gaussian initial state. This lower bound aligns with Heisenberg's value. Our study delves into the Gaussian measurement process in the phase space, demonstrating its capability to interpolate smoothly between the projective measurements of position and momentum.  \nBuilding on the foundational research of Heisenberg; Kennard and Robertson (KR); and Schrodinger, and the advancements by Arthurs, Kelly, and Goodman (AKG); Ozawa; and Watanabe, Sagawa, and Ueda (WSU), our research deepens the understanding of quantum uncertainty. We leverage the Gaussian wave-packet formalism to provide a comprehensive and tangible approach to the abstract constructs of the LT formalism. This paper advocates for the POVM measurement onto the Gaussian wave-packet basis as a natural method for joint position and momentum measurement, offering concrete expressions for previously abstract concepts like pullback and pushforward within the LT formalism. Our work bridges the theoretical and practical aspects of quantum mechanics, setting the stage for further research and applications in quantum uncertainty.  \nIt has been shown that the Gaussian basis used in our current study on a fixed time slice can be extended to the whole spacetime [16, 17, 18, 22]; see Ref. [25] for the complete basis of Lorentz invariant wave packets and Ref. [26] for that of Lorentz covariant ones including the spin degrees of freedom. It would be fascinating to explore the time-energy uncertainty relation on this ground. It is also interesting to show the LT and Lee inequalities for the standard Clauser-Horne-Shimony-Holt (CHSH) inequality as a concrete realization in the finite-dimensional space. These developments will be presented in subsequent publications.", 'Acknowledgement': 'N.O. is indebted to Sho Machiyama for assistance in understanding the LT formalism and to Shogo Tanimura for fostering a welcoming research environment in his group. We are grateful to Jaeha Lee for providing the talk slides, to Kenji Nishiwaki for helpful conversation, and to Juntaro Wada for useful comments. The work of K.O. is in part supported by the JSPS KAKENHI Grant Nos. JP19H01899 and JP21H01107.', 'References': '- [1] W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Z. Phys. 43 (1927) 172.\n- [2] E. H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen , Z. Phys. 44 (1927) 326.\n- [3] H. P. Robertson, The Uncertainty Principle , Phys. Rev. 34 (1929) 163.\n- [4] E. Schrodinger, About Heisenberg uncertainty relation , Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 19 (1930) 296 [ quant-ph/9903100 ].\n- [5] E. Arthurs and J. L. Kelly, B.S.T.J. briefs: On the simultaneous measurement of a pair of conjugate observables , Bell Syst. Tech. J. 44 (1965) 725.\n- [6] E. Arthurs and M. S. Goodman, Quantum correlations: A generalized Heisenberg uncertainty relation , Phys. Rev. Lett. 60 (1988) 2447.\n- [7] M. Ozawa, Uncertainty relations for joint measurements of noncommuting observables , Physics Letters A 320 (2004) 367.\n- [8] M. Ozawa, Universally valid reformulation of the heisenberg uncertainty principle on noise and disturbance in measurement , Phys. Rev. A 67 (2003) 042105.\n- [9] H. Yuen and M. Lax, Multiple-parameter quantum estimation and measurement of nonselfadjoint observables , IEEE Trans. Info. Theor. 19 (1973) 740.\n- [10] Y. Watanabe, T. Sagawa and M. Ueda, Uncertainty relation revisited from quantum estimation theory , Phys. Rev. A 84 (2011) 042121.\n- [11] Y. Watanabe and M. Ueda, Quantum estimation theory of error and disturbance in quantum measurement , 1106.2526 .\n- [12] J. Lee and I. Tsutsui, Geometric formulation of universally valid uncertainty relation for error , 2002.04008 .\n- [13] J. Lee and I. Tsutsui, Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems , Entropy 22 (2020) 1222.\n- [14] J. Lee, A Universal Formulation of Uncertainty Relation for Errors under Local Representability , 2203.08197 .'}