Universality in Uncertainty Relations for a Quantum Particle
UUniversality in Uncertainty Relationsfor a Quantum Particle
Spiros Kechrimparis ∗ and Stefan Weigert † Department of Mathematics, University of YorkYork, YO10 5DD, United Kingdom22 June 2016
Abstract
A general theory of preparational uncertainty relations for a quantum particle inone spatial dimension is developed. We derive conditions which determine whethera given smooth function of the particle’s variances and its covariance is boundedfrom below. Whenever a global minimum exists, an uncertainty relation has beenobtained. The squeezed number states of a harmonic oscillator are found to be univer-sal : no other pure or mixed states will saturate any such relation. Geometrically, weidentify a convex uncertainty region in the space of second moments which is boundedby the inequality derived by Robertson and Schr ¨odinger. Our approach provides aunified perspective on existing uncertainty relations for a single continuous variable,and it leads to new inequalities for second moments which can be checked experi-mentally.
Inspired by Heisenberg’s analysis [1] of Compton scattering, Kennard [2] proved thepreparational uncertainty relation ∆ p ∆ q ≥ ¯ h ∆ p and ∆ q of momentum and position of a quantum particlewith a single spatial degree of freedom. Experimentally, they are determined by mea-surements performed on an ensemble of systems prepared in a specific state | ψ (cid:105) . Theonly states which saturate the bound (1) are squeezed states with a real squeezing parame-ter [3, 4, 5] (we follow the review [6] regarding the naming of squeezed states). Squeezedstates are conceptually important since they achieve the best possible localization of aquantum particle in phase-space, and they are easily visualized by “ellipses of uncer-tainty”. Each squeezed state may be displaced rigidly in phase space without affectingthe value of the variances, resulting in a three-parameter family of states saturating thelower bound (1).Not many other uncertainty relations are known. The sum of the position and mo-mentum variances is bounded [7, 8] according to the relation ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] O c t p + ∆ q ≥ ¯ h , (2)which holds in a system of units where the physical dimensions of both position andmomentum equal √ ¯ h . Only the ground state of a harmonic oscillator with unit mass andfrequency saturates this inequality (ignoring rigid displacements in phase-space). TheRobertson-Schr ¨odinger (RS) inequality [9, 10], ∆ p ∆ q − C pq ≥ ¯ h C pq defined in Eq. (8).Eq. (3) is saturated by the two-parameter family of squeezed states with a complex squeez-ing parameter [6], again ignoring phase-space displacements. The additional free pa-rameter describes the phase-space orientation of the uncertainty ellipse which, in theprevious case, was aligned with the position and momentum axes.By introducing the observable ˆ r = − ˆ p − ˆ q , which satisfies the commutation relations [ ˆ q , ˆ r ] = [ ˆ r , ˆ p ] = ¯ h / i , one obtains a bound on the product of the variances of three pairwisecanonical observables, ∆ p ∆ q ∆ r ≥ (cid:18) τ ¯ h (cid:19) , τ = csc (cid:18) π (cid:19) ≡ (cid:114)
43 . (4)This triple product uncertainty relation has been found only recently [13]. Ignoring phase-space translations, only one state exists which achieves the minimum. Since the varianceof ˆ r is given by ∆ r = ∆ p + ∆ q + C pq , (5)the left-hand-side of (4) can also be considered as a function of the three second moments.The inequalities (1) to (4) and the search for their minima arise from one single math-ematical problem: Does a given smooth function of the second moments have a lower bound ? If so, which states will saturate the inequality ?In this paper, we answer these questions for a quantum particle with a single spatialdegree of freedom by presenting a systematic approach to studying uncertainty relationsderived from smooth functions f ( ∆ p , ∆ q , C pq ) . Proceeding in three steps we1. identify a universal set of states E which can possibly minimize a given functional f ( ∆ p , ∆ q , C pq ) ;2. spell out conditions which determine the extrema of the functional f as a subset ofthe universal set, E ( f ) ⊆ E ; if no admissible extrema exist, the functional has nolower bound;3. determine the set of states M ( f ) ⊆ E ( f ) which minimize the functional f , leadingto an uncertainty relation in terms of the second moments.The inequalities studied here will be preparational in spirit: they apply to scenarios inwhich the quantum state of the particle | ψ (cid:105) is fixed during the three separate runs of themeasurements required to determine the numerical values of the second moments. Theseinequalities do not describe the limitations of measuring non-commuting observables simultaneously . 2he paper is divided into two major sections. In Sec. 2 we introduce uncertainty func-tionals and explain how to determine their extrema and minima. The impatient readermay jump directly to Sec. 3 where we derive new families of uncertainty relations and de-termine the states minimizing them. We conclude the paper with a summary and discussfurther applications. To begin, we introduce the uncertainty functional [11] J [ ψ ] = f (cid:0) ∆ p , ∆ q , C pq (cid:1) − λ ( (cid:104) ψ | ψ (cid:105) − ) , (6)which sends each element | ψ (cid:105) of the one-particle Hilbert space H to a real number de-termined by the real differentiable function f ( x , x , x ) of three variables. The Lagrangemultiplier λ ensures the normalization of the states. The variances of position and mo-mentum are defined by ∆ p = (cid:104) ψ | ˆ p | ψ (cid:105) − (cid:104) ψ | ˆ p | ψ (cid:105) , (7)etc., and the covariance of position and momentum reads C pq = (cid:104) ψ | ( ˆ p ˆ q + ˆ q ˆ p ) | ψ (cid:105) − (cid:104) ψ | ˆ p | ψ (cid:105)(cid:104) ψ | ˆ q | ψ (cid:105) . (8)The second moments form the real, symmetric covariance matrix C = (cid:18) ∆ p C pq C pq ∆ q (cid:19) ≡ (cid:18) x ww y (cid:19) , (9)with state-dependent matrix elements x ≡ x ( ψ ) etc. The covariance may take any finitereal value, w ∈ R , while the variances of position and momentum take (finite) positive values only, x , y >
0. States of a quantum particle with vanishing position (or momentum)variance and diverging momentum (or position) variance are not taken into account sincethey only arise for non-normalizable states which cannot be prepared experimentally.Nevertheless, position (or momentum) eigenstates can be approximated arbitrarily wellby states within the set we consider.It will be convenient to work with states in which the expectation values of bothmomentum and position vanish, (cid:104) ψ | ˆ q | ψ (cid:105) = (cid:104) ψ | ˆ p | ψ (cid:105) =
0. This can be achieved by rigidlydisplacing the observables using the unitary operatorˆ T α = exp [ i ( p ˆ q − q ˆ p ) /¯ h ] , α = √ h ( q + ip ) , (10)where p = (cid:104) ψ | ˆ p | ψ (cid:105) , etc. This transformation leaves invariant the values of the secondmoments (7) and (8) and has thus no impact on the minimization of the functional J [ ψ ] .A lower bound of a functional J [ ψ ] of the form (6) will result in an uncertainty relationassociated with the function f ( x , x , x ) . To determine such a bound, we apply a methodused in [11, 12, 13] (see also [14, 15, 16]). First, we derive an eigenvalue equation for theextrema of the functional J [ ψ ] which also need to satisfy a set of consistency conditionsgiven in Sec. 2.3. Then, we introduce a “space of moments” to visualize these results(Sec. 2.4) and, finally, we determine the minimizing states whenever the functional isguaranteed to be bounded from below (see Sec. 2.5 and Sec. 3).3 .1 Extrema of uncertainty functionals When comparing the values of the functional J at the points | ψ (cid:105) and | ψ + ε (cid:105) ≡ | ψ (cid:105) + ε | e (cid:105) ,for any unit vector | e (cid:105) and a small parameter ε , we find to first order that J [ ψ + ε ] − J [ ψ ] = ε D ε J [ ψ ] + O (cid:0) ε (cid:1) , (11)where the expression D ε = (cid:104) e | δδ (cid:104) ψ | + δδ | ψ (cid:105) | e (cid:105) , (12)denotes a Gˆateaux derivative. If the functional J [ ψ ] does not change under this variation, D ε J [ ψ ] = (cid:104) e | (cid:18) δδ (cid:104) ψ | f ( x , y , w ) − λ | ψ (cid:105) (cid:19) + c.c. =
0, (13)it has an extremum at the state | ψ (cid:105) . More explicitly, this condition reads (cid:104) e | (cid:18) ∂ f ∂ x δ x δ (cid:104) ψ | + ∂ f ∂ y δ y δ (cid:104) ψ | + ∂ f ∂ w δ w δ (cid:104) ψ | − λ | ψ (cid:105) (cid:19) + c.c. = | e (cid:105) and its dual (cid:104) e | can bevaried independently (just consider their position representations e ∗ ( x ) and e ( x ) ), theexpression in round brackets must vanish identically which implies that the complexconjugate term will also vanish. Using δ x δ (cid:104) ψ | ≡ δ ∆ p δ (cid:104) ψ | = δ (cid:104) ψ | ˆ p | ψ (cid:105) δ (cid:104) ψ | = ˆ p | ψ (cid:105) , (15)a similar relation for δ y / δ (cid:104) ψ | , and the identity δ w δ (cid:104) ψ | ≡ ( ˆ q ˆ p + ˆ p ˆ q ) | ψ (cid:105) , (16)we arrive at an Euler-Lagrange -type equation, (cid:18) ∂ f ∂ x ˆ p + ∂ f ∂ y ˆ q + ∂ f ∂ w ( ˆ q ˆ p + ˆ p ˆ q ) − λ (cid:19) | ψ (cid:105) =
0. (17)The parameter λ can be eliminated by multiplying this equation with the bra (cid:104) ψ | fromthe left and solving for λ ; substituting the value obtained back into Eq. (17), one finds anonlinear eigenvector-eigenvalue equation, (cid:18) f x ˆ p + f y ˆ q + f w ( ˆ q ˆ p + ˆ p ˆ q ) (cid:19) | ψ (cid:105) = (cid:0) f x x + f y y + f w w (cid:1) | ψ (cid:105) , (18)using the standard shorthand for partial derivatives.Eq. (18) is our first result following from the approach conceived in [11]: the extremaof arbitrary smooth functions of the second moments are encoded in an eigenvalue equa-tion for a Hermitean operator quadratic in position and momentum. However, the equa-tion is not linear in the state | ψ (cid:105) because the quantities x , y , . . . , f w are functions of expec-tation values taken in the yet unknown state. Previously, similar results had only beenfound for specific uncertainty functionals such as the product of the standard deviationsor position and momentum. 4et us briefly illustrate the crucial features of Eq. (18) in a simple case before systemat-ically investigating its solutions. For a function linear in x , y , and w , the derivatives f x , f y ,and f w will be fixed constant numbers. In this case, the operator on the left-hand-sideof (18) represents a quadratic form in the position and momentum operators, falling intoone of three possible categories [17]. Up to a multiplicative constant, the operator will beunitarily equivalent to the Hamiltonian of (i) a harmonic oscillator with unit mass andfrequency, ˆ p + ˆ q , (ii) a free particle, ˆ p , or (iii) an inverted harmonic oscillator, ˆ p − ˆ q .In the first case, the spectrum of the operator will be discrete and bounded from below(or above); the spectra of the operators in the other two cases are continuous which istantamount to the absence of normalizable eigenstates. Thus, a linear function f ( x , y , w ) possesses a non-trivial bound only if it gives rise to an operator in (18) which is unitarilyequivalent to the Hamiltonian of a harmonic oscillator. Generally, our method will signalthe absence of lower bounds corresponding to the cases (ii) and (iii). To find a lower bound of the functional J [ ψ ] , we will determine all its extrema and thenpick those where J [ ψ ] assumes its smallest value. However, Eq. (18) is not a standardeigenvalue equation: even for a linear function f , the right-hand-side of (18) dependsnon-linearly on the as yet unknown state | ψ (cid:105) , and if the function f is non-linear, theoperators on the left-hand-side of the equation acquire state-dependent coefficients givenby its partial derivatives.Nevertheless, the eigenvalue problem can be solved systematically, in a self-consistent way. Initially, we treat the expectations x , y , w , and f x , f y , f w in (18) as parameters withgiven values, i.e. we ignore their dependence on the state | ψ (cid:105) . The solutions | ψ ( x , y , w ) (cid:105) will depend on these parameters which means that the solutions must be checked forconsistency since the relations (7) now require that x = (cid:104) ψ ( x , y , w ) | ˆ p | ψ ( x , y , w ) (cid:105) , etc. Itmay or may not be possible to satisfy these restrictions on the parameters.To begin, we write the operator on the left-hand side of (18) in matrix form, ( ˆ p , ˆ q ) (cid:18) f x f w /2 f w /2 f y (cid:19) (cid:18) ˆ p ˆ q (cid:19) ≡ ˆ z (cid:62) · F · ˆ z . (19)Williamson’s theorem [18] ensures that any positive or negative definite matrix can bemapped to a diagonal matrix by conjugation with a symplectic matrix Σ . We will assumefrom now on that the matrix F is positive definite. The negative definite case is easilydealt with by considering − f ( x , y , w ) instead of f ( x , y , w ) . Applied to the 2 × F ,Williamson’s result states that we can write Σ (cid:62) · F · Σ = c I , Σ ∈ Sp ( R ) , c > I is the identity matrix, whenever F > F ≡ f x f y − f w /4 > f x > f y > f ( x , y , w ) : since the operators ˆ p and ˆ p − ˆ q result in matrices F with zero or negative determinant, the left-hand-side of (18) cannotbe mapped to an oscillator Hamiltonian by means of a symplectic transformation.A direct calculation shows that the matrix F is diagonalized by the symplectic matrix Σ = ( S γ G b ) − , where 5 b = (cid:18) b (cid:19) , and S γ = (cid:18) e − γ e γ (cid:19) , (22)with real parameters b = f w f y ∈ R and γ =
12 ln (cid:18) f y √ det F (cid:19) ∈ R , (23)leading to c = √ det F in Eq. (20). The symplectic matrices S γ and G b give rise to theIwasawa (or K A N ) decomposition of the matrix Σ − ∈ Sp ( R ) (cf. [19], for example)if they are written in opposite order and the parameter b is replaced by be γ ; the thirdfactor happens to be the identity.Next, we observe that the linear action of the matrices G and S on the canonical pairof operators ( ˆ p , ˆ q ) (cid:62) can be implemented by conjugation with suitable unitary operators,known as metaplectic operators [19]. We have, for example, (cid:18) b (cid:19) (cid:18) ˆ p ˆ q (cid:19) = e ib ˆ p /2¯ h (cid:18) ˆ p ˆ q (cid:19) e − ib ˆ p /2¯ h , (24)or, in matrix notation, G b · ˆ z = ˆ G b ˆ z ˆ G † b (25)where the unitary operator ˆ G b = e ib ˆ p /2¯ h (26)describes a momentum gauge transformation . Similarly, the squeeze operatorˆ S γ = e i γ ( ˆ q ˆ p + ˆ p ˆ q ) /2¯ h , (27)symplectically scales position and momentum according to S γ · ˆ z = ˆ S γ ˆ z ˆ S † γ . (28)With Σ − = S γ G b in (20), we rewrite (19) asˆ z (cid:62) · F · ˆ z = √ det F ( S γ · G b · ˆ z ) (cid:62) · ( S γ · G b · ˆ z ) . (29)Finally, using the identities (25) and (28) and multiplying Eq. (18) with the unitary ˆ S † γ ˆ G † b from the left, the condition for the existence of extrema of the functional J [ ψ ] takes on thedesired form, 12 (cid:0) ˆ p + ˆ q (cid:1) | ψ ( b , γ ) (cid:105) = (cid:18) x f x + y f y + w f w √ det F (cid:19) | ψ ( b , γ ) (cid:105) . (30)Thus, the solutions | ψ ( b , γ ) (cid:105) ≡ ˆ S † γ ˆ G † b | ψ (cid:105) (31)must be proportional to the eigenstates | n (cid:105) , n ∈ N , of a unit oscillator , i.e. a quantummechanical oscillator with unit mass and unit frequency. Equivalently, the candidates forstates extremizing the functional J [ ψ ] are given by the family of states, | n ( b , γ ) (cid:105) = ˆ G b ˆ S γ | n (cid:105) , b , γ ∈ R , n ∈ N . (32)6pon rewriting the operator ˆ S † γ ˆ G † b these states are seen to coincide with the squeezed num-ber states [6]. As shown in in Appendix A, the product of a squeeze transformation ˆ S γ (with real parameter γ ) and a momentum gauge transformation ˆ G b equalsˆ G b ˆ S γ = ˆ S ( ξ ) ˆ R ( χ ) , (33)i.e. the product of a rotation in phase space,ˆ R ( χ ) = e i χ ˆ a † ˆ a χ ∈ [
0, 2 π ) , (34)and a squeeze transformation (with complex ξ ) along a line with inclination θ ,ˆ S ( ξ ) = e ( ξ ˆ a †2 − ξ ˆ a ) , ξ = re i θ ∈ C . (35)Summarizing our findings, we draw two conclusions:1. The complete set of solutions of Eq. (30) coincides with the squeezed number states , E = ∞ (cid:91) n = E n ≡ ∞ (cid:91) n = (cid:8) | n ( α , ξ ) (cid:105) = ˆ T α ˆ S ( ξ ) | n (cid:105) , α , ξ ∈ C (cid:9) , (36)where non-zero expectation values of position and momentum have been reintro-duced via the translation operator ˆ T α (see Eq. (10)) and irrelevant constant phaseshave been suppressed.2. The value of the right-hand-side of Eq. (30) can take only specific values, x f x + y f y + w f w √ det F = (cid:18) n + (cid:19) ¯ h , n ∈ N , (37)given by the eigenvalues of the unit oscillator. This relation constrains the state-dependent quantities of the left-hand-side which needs to be checked for consis-tency, just as Eq. (21) does.We have thus obtained our second main result. The extrema E of an arbitrary functional J [ ψ ] characterized by a function f ( x , y , w ) are necessarily squeezed number states , a setwhich is independent of the function at hand. In other words, the set E containing allthe states which may arise as minima of an uncertainty functional J [ ψ ] , is universal . Theminima of any functional must be a subset E ( f ) ⊆ E which will depend explicitly on thefunction f ( x , y , w ) , determined by the consistency conditions to be studied next. We now spell out the conditions which must be satisfied by the states | n ( b , γ ) (cid:105) in (32) –or, equivalently, the states | n ( α , ξ ) (cid:105) in (36) – to qualify as extrema for a specific functional J [ ψ ] :1. Recalling that x ≡ ∆ p , etc., the relations x = (cid:104) n ( b , γ ) | ˆ p | n ( b , γ ) (cid:105) , y = (cid:104) n ( b , γ ) | ˆ q | n ( b , γ ) (cid:105) , (38)and w = (cid:104) n ( b , γ ) | ( ˆ p ˆ q + ˆ q ˆ p ) | n ( b , γ ) (cid:105) , (39)represent three, generally nonlinear consistency equations between the second mo-ments since the parameters b and γ are functions of x , y and w (cf. Eq. (23)).7. The values of the moments x , y and w must satisfy Eq. (37).3. The matrix F of the first derivatives must be positive definite.Using (32), (25) and (28), the first consistency condition in (38) leads to x = (cid:104) n ( b , γ ) | ˆ p | n ( b , γ ) (cid:105) = e γ (cid:104) n | ˆ p | n (cid:105) = e γ (cid:18) n + (cid:19) ¯ h , n ∈ N , (40)or, recalling the definition of γ in (23), x √ det F = (cid:18) n + (cid:19) ¯ h f y , n ∈ N . (41)Similar calculations result in y √ det F = (cid:18) n + (cid:19) ¯ h f x , n ∈ N , (42)and − w √ det F = (cid:18) n + (cid:19) ¯ h f w , n ∈ N , (43)respectively. These conditions may be expressed in matrix form, F · C √ det F = (cid:18) n + (cid:19) ¯ h I , n ∈ N , (44)involving both the covariance matrix C and F .Taking the trace of the last relation shows that Eq. (37) is satisfied automatically. With-out specifying a function f ( x , y , w ) , no conclusions can be drawn about the validity ofEqs. (41-43) or the positive definiteness of the matrix F . We visualize the interplay of the consistency conditions by expressing them in the form x f x = y f y , x f w = − w f y , (45)and xy − w = (cid:18) n + (cid:19) ¯ h , n ∈ N , (46)following easily from either (41-43) or (44). The third constraint is universal since it doesnot depend on the function f ( x , y , w ) . Using the variables u = ( x + y ) > v = ( x − y ) ∈ R ,we define the three-dimensional space of ( second ) moments, with coordinates ( u , v , w ) . Foreach non-negative integer, the third condition u − v − w = e n , e n = (cid:18) n + (cid:19) ¯ h , n ∈ N , (47)determines one sheet of a two-sheeted hyperboloid, located in the “upper” half of thespace of moments, i.e. u > v , w ∈ R (cf. Fig. 1). The points on the n -th sheet, which8 n E u wv Figure 1: Space of (second) moments, with points ( u , v , w ) : the extremal states of smoothfunctionals J [ ψ ] are located on a discrete set of nested hyperboloids E = (cid:83) ∞ n = E n the firstthree of which are shown, using light ( n = n =
1) and dark shading ( n = ) ,respectively. The accessible uncertainty region for a quantum particle is given by the pointson and inside of the convex surface E : u − v − w = ¯ h /4 which coincides with theminima M ( f RS ) of the RS inequality, i.e. squeezed states with minimal uncertainty.intersects the u -axis at u = + e n , are in one-to-one correspondence with the squeezedstates originating from the number state | n (cid:105) , forming the set E n in (36).The consistency conditions (45) clearly depend on the function f ( x , y , w ) at hand. Theconstraints will only be satisfied for specific subsets E n ( f ) of points on the hyperboloids E n , resulting in the f -dependent set of states E ( f ) = ∞ (cid:91) n = E n ( f ) which contains all the candidates possibly minimizing the functional J [ ψ ] . The candidatesets E ( f ) may depend on one or two parameters, or contain isolated points only. If theconsistency conditions cannot be satisfied, then the functional J [ ψ ] has no lower bound.Furthermore, if the matrix F is not positive definite for any of the states in E ( f ) , themethod makes no predictions about the minima of the functional J [ ψ ] . We have, however,not found any non-trivial cases of this behaviour.Finally, we need to evaluate the functional J [ ψ ] for all candidate states E ( f ) and pickthe smallest possible value. The states achieving this minimum value constitute the so-lutions M ( f ) ⊆ E ( f ) of the minimization problem. In their entirety, the minima M ( f ) may consist of isolated states or of sets depending on one or two parameters. Usually,the states saturating the bound are located on the sheet E .Let us briefly introduce the concept of a space of moments , defined by the triples ofnumbers ( u , v , w ) (cid:62) ∈ R . For n =
0, Eq. (47) is equivalent to (3) which implies that not allpoints of this set can arise as moment triples. The accessible part of the space, boundedby the extremal hyperboloid E defined in Eq. (47), is called the uncertainty region (cf.9ig. 1). The boundary of an analogously defined uncertainty region for a quantum spin s [20] is not convex. The relation between moment triples and the underlying pure ormixed states of quantum particles is discussed elsewhere [21]. To illustrate our approach we re-derive three of the four bounds mentioned in the intro-duction: the uncertainty relations by
Robertson-Schr¨odinger , by
Heisenberg-Kennard , andthe triple-product inequality.
Robertson-Schr¨odinger uncertainty relation . Defining f RS ( x , y , w ) = xy − w , (48)the matrix of first-order derivatives associated with the quadratic form (19) is given by F = (cid:18) y − w − w x (cid:19) , (49)and, interestingly, its determinantdet F = xy − w ≡ f RS ( x , y , w ) (50)coincides with the original functional. At this point of the derivation, it is not yet knownwhether the matrix F is strictly positive.The relations Eq. (45) do not constraint the parameters x , y , and w , since they are satis-fied automatically, leaving Eq. (46) as the only restriction. Since the left-hand-side of (46)coincides with the function f RS ( x , y , w ) , all squeezed states are candidates to minimizethe RS functional, E ( f RS ) = E . (51)Therefore, the function f RS comes with the largest possible set of candidates to minimizeit, given by the union of the sets E n in Fig. 1. The lower bound on f RS now follows directlyfrom combining (46) with (48), f RS ( x , y , w ) = (cid:18) n + (cid:19) ¯ h ≥ ¯ h F ispositive everywhere in the uncertainty region.The hyperboloid closest to the origin of the ( u , v , w ) -space provides the states mini-mizing the function f RS , M ( f RS ) = E , (53)i.e. the set of squeezed states based on the ground state | (cid:105) of a unit oscillator. This is, ofcourse, a two-parameter family since the relations (23) take the form b = − zx , and γ =
12 ln ( x ) , (54)meaning that, with x > z ∈ R , both b and γ take indeed arbitrary real values.Thus, each squeezed state can be reached and, when adding phase-space translations,we obtain the four-parameter family of all squeezed states as minima of f RS : M α ( f RS ) = ˆ T α M ( f RS ) . (55)10his property singles out the RS functional among all uncertainty functionals.If an uncertainty functional associated to a function f , is different from the RS func-tional, the first two consistency relations will, in general, not be satisfied automaticallybut impose non-trivial constraints on the second moments. Therefore, the extrema of thefunctional must be a proper subset of those of the RS functional, i.e. E ( f ) ⊂ E , as thefollowing example shows. Heisenberg’s uncertainty relation . Let us determine the minimum of the product of thestandard deviations ∆ p and ∆ q by considering the function f H ( x , y , w ) = √ xy . (56)Its partial derivatives satisfy2 f Hx = (cid:114) yx , 2 f Hy = (cid:114) xy , f Hw = F , namely, F = (cid:18)(cid:112) y / x (cid:112) x / y (cid:19) , det F = > √ xy = (cid:18) n + (cid:19) ¯ h ≡ e n , n ∈ N , (59)which determine the value of the product of the standard deviations at the extrema of f H ( x , y , w ) , labeled by the positive integers. In the ( u , v , w ) -space, the intersections ofthe surfaces defined by (59) and the hyperboloids (47) consist of hyperbolas in the ( u , v ) -plane containing the points ( e n , 0 ) , n ∈ N . The union of these hyperbolas define the set E ( f H ) , corresponding to the potential minima of the function f H ( x , y ) (cf. Fig. 2). Thethird condition, Eq. (43), implies that w =
0. Combining (59) with (56), we obtain thebound f H ( x , y , w ) = (cid:18) n + (cid:19) ¯ h ≥ ¯ h n = γ in (23), leading to f y = x /¯ h .Since the consistency conditions do not impose any other condition on the variance x , itmay take any positive value implying that γ ∈ R . Since f w = b =
0, the set ofstates minimizing the Heisenberg’s uncertainty relation is given by squeezed states withreal squeezing parameter, M α ( f H ) = ˆ T α E ( f H ) ≡ (cid:8) ˆ T α ˆ S γ | (cid:105) , α ∈ C , γ ∈ R (cid:9) , (61)where we have re-introduced arbitrary phase-space displacements. Triple product inequality.
Using Eq. (5), we see that we need to find the minimum ofthe expression f T ( x , y , w ) = xy ( x + y + w ) (62)11 u E (cid:0) f H (cid:1) E (cid:0) f H (cid:1) E (cid:0) f H (cid:1) M (cid:0) f H (cid:1) = E (cid:0) f H (cid:1) Figure 2: Hyperbolas in the ( u , v ) -plane through the points ( e n , 0 ) , n ∈ N , stemmingfrom intersections of the hyperboloids (47) and the surfaces defined by (59). The unionof the hyperbolas defines the set E ( f H ) which represents the location of all possible min-ima of the function f H ( x , y ) ; the points on the “lowest” (darkest) hyperbola E ( f H ) cor-respond to the set of states M ( f H ) which saturate Heisenberg’s uncertainty relation.in order to reproduce the triple product uncertainty relation (4). The first consistencycondition in Eq. (45) implies that x = y ; using this identity in the second condition, onefinds x ( w + x )( w + x ) = x >
0, the corre-lation w must equal either − x or − x /2. According to (5), the first case would imply ∆ r ≡
0, which is impossible since the operator ˆ r has no normalizable eigenstates. There-fore, using the solution w = − x /2 of (63) and x = y in the third consistency condition,one finds that x = (cid:18) n + (cid:19) ¯ h , n ∈ N , (64)must hold. It is now straightforward to evaluate f T ( x , y , w ) at its extrema to find itsglobal minimum, f T ( x , y , w ) = x = (cid:32)(cid:114) (cid:18) n + (cid:19) ¯ h (cid:33) ≥ (cid:18) τ ¯ h (cid:19) ,which reproduces (4). It is easy to confirm that the matrix F is positive definite withdeterminant det F = ¯ h /3. Since the minimum occurs for n =
0, the values of the secondmoments are given by x = y = − w = ¯ h √ ≡ τ ¯ h b =
12 and γ =
14 ln τ . (66)12 T E T E T M T = E T u w Figure 3: Candidate states E ( f T ) ≡ E T possibly minimizing the product of three vari-ances f T ( u , v , w ) , represented by dots located on the intersections of the hyperboloids(47) and the planes defined by the consistency conditions (69); the point closest to theorigin, M ( f T ) ≡ M T , represents the state | Ξ (cid:105) achieving the minimum of the tripleproduct uncertainty relation (4) (and of any other S -invariant inequality associated witha functional f ( ) N in (91)).Using (32) or (36) we obtain one single state which saturates the triple uncertainty, namely | Ξ (cid:105) ≡ ˆ G ˆ S ln τ | (cid:105) = ˆ S i ln 3 | (cid:105) . (67)If one includes rigid phase-space translations, the set of states minimizing the triple un-certainty is finally given by the two-parameter family M α ( f T ) = (cid:8) ˆ T α | Ξ (cid:105) , α ∈ C (cid:9) , (68)in agreement with [13].Geometrically, the state | Ξ (cid:105) arises from the intersection of the sequence of hyper-boloids with the surfaces defined by u = τ e n , v = w = − τ e n , n ∈ N . (69)The planes defined by constant values of u have concentric circles in common with thehyperboloids, and the vertical uw -plane (given by v =
0) intersects with each of thecircles in two points only. Finally, the condition on the variable w selects a single one ofthe points with the same value of u . According to (69), the candidate states in ( u , v , w ) -space are located on a straight line, E ( f T ) = τ e n − , n ∈ N , (70)and the state | Ξ (cid:105) corresponds to the point closest to the origin (see Fig. (3)).13 New uncertainty relations
The linear combination of second moments f L ( x , y , w ) = µ x + ν y + λ w , µ , ν , λ , ∈ R , (71)leads to the uncertainty relation µ ∆ p + ν ∆ q + λ C pq ≥ ¯ h (cid:113) µν − λ , µ , ν > µν > λ . (72)The constraints on the parameters follow from the matrix F in (19) being strictly positivedefinite. The consistency conditions (45) associated with f L relate both y and w to x according to y = µν x , w = − λν x . (73)Then, Eq. (46) simplifies to (cid:0) µν − λ (cid:1) ν x = (cid:18) n + (cid:19) ¯ h = e n ¯ h , (74)which is consistent due to det F = µν − λ >
0. Expressing the functional f L ( x , y , w ) interms of x only, we obtain the bound given in (72), f L ( x , y , w ) = (cid:0) µν − λ (cid:1) ν x = e n ¯ h (cid:113) µν − λ ≥ ¯ h (cid:113) µν − λ . (75)Up to phase-space translations ˆ T α , a single squeezed state saturates the bound, namely M ( f L ) = (cid:40) | µ , ν , λ (cid:105) = ˆ G λν ˆ S ln (cid:18) ν √ µν − λ (cid:19) | (cid:105) (cid:41) . (76)When expressing the correlation term C pq in terms of the variance ∆ r according toEq. (5), we obtain, for µ = ν = λ = triple sum uncertainty relation ∆ p + ∆ q + ∆ r ≥ √ h , (77)derived in [13], and the minimum is achieved for the state |
1, 1, 1/2 (cid:105) ≡ | Ξ (cid:105) which alsominimizes the triple product uncertainty (cf. Eq. (67) and Fig. 3). Sums of powers of position and momentum variances are bounded from below accord-ing to the inequality µ (cid:0) ∆ p (cid:1) m + ν (cid:0) ∆ q (cid:1) m (cid:48) ≥ (cid:18) ¯ h (cid:19) mm (cid:48) m + m (cid:48) (cid:32) µ (cid:18) νµ m (cid:48) m (cid:19) mm + m (cid:48) + ν (cid:16) µν mm (cid:48) (cid:17) m (cid:48) m + m (cid:48) (cid:33) , m , m (cid:48) ∈ N ,(78)reducing to the pair sum uncertainty relation (2) in the simplest case ( µ = ν = m = m (cid:48) = generalized RS-uncertainty functional (48), f RSm , m (cid:48) ( x , y , w ) = ( xy ) m − µ w m (cid:48) , µ > m , m (cid:48) ∈ N . (79)14or arbitrary integers m and m (cid:48) , the consistency conditions cannot be solved in closedform. Setting m (cid:48) = m and assuming that both m > µ > (cid:0) ∆ p · ∆ q (cid:1) m − µ (cid:0) C qp (cid:1) m ≥ (cid:18) ¯ h (cid:19) m µ (cid:16) µ m − − (cid:17) m . (80)An interesting special case of f RSm ,2 m occurs for m = / and 0 < µ < ∆ p ∆ q − µ (cid:12)(cid:12) C pq (cid:12)(cid:12) ≥ ¯ h (cid:113) − µ , (81)which can be treated in spite of the presence of the non-differentiable term. The extremalstates depend on one free parameter, E α ( f RS / ) = (cid:40) | α , n (cid:105) = ˆ T α ˆ G ± µ en ¯ h x √ − µ ˆ S ln ( xen ¯ h ) | n (cid:105) (cid:41) , x > µ =
0, they reduce to the squeezed number stateswith a real parameter known to extremize Heisenberg’s inequality.Next, we present an example of an uncertainty relation which seems to be entirely outof reach of traditional derivations. Defining the functional f e ( x , y ) = x + µ e y / ν , µ , ν > ∆ p + µ e ∆ q / ν ≥ ( + W ( ¯ h /4 √ µν )) e W ( ¯ h /4 √ µν )) , (84)using the fact that Lambert’s W -function W ( s ) , defined as the inverse of s ( W ) = W exp W ,is a strictly increasing function. In the limit of µ → ∞ and assuming µ = ν , the left-hand-side of (84) turns into ( µ + ∆ p + ∆ q + O ( µ )) while the expansion of its right-hand-side produces the correct bound ( µ + ¯ h + O ( µ )) , since W ( s ) = s + O ( s ) .The position and momentum variances at the extremum with label n ∈ N are givenby x = µ W (cid:18) e n ¯ h √ µν (cid:19) e W (cid:16) en ¯ h √ µν (cid:17) (85)and y = ν W (cid:18) e n ¯ h √ µν (cid:19) , (86)respectively. Using Eqs. (23) with det F = ( µ / ν ) e y / ν , one finds b = γ =
14 ln (cid:16) µν (cid:17) + W (cid:18) e n ¯ h √ µν (cid:19) , (87)which means that only a single state (and its rigid displacements) will saturate the in-equality (84). If µ = ν , we recover x = y = ¯ h /4 as well as b = γ =
0, i.e. the groundstate of a unit oscillator since W ( ) =
0. 15et us point out that some general statements can be made about functionals of theform f = f ( xy , w ) and f = f ( µ x m + ν y m (cid:48) , w ) , i.e. generalizations of the expressions in(78) and (79), respectively. By examining the consistency conditions one can show thatthe extrema of the first expression come as a one-parameter set, while they are isolated ora one-parameter family in the second case. However, without knowing the explicit formof the functions no further conclusions can be drawn. Rational function s of the variances such us f r µ , ν ( x , y ) = x m y m µ x m (cid:48) + ν y m (cid:48) , µ , ν > m = m (cid:48) = / , for example, theextremal states of the functional are given by the set M α ( f r / , / ) = (cid:26) | α , n (cid:105) = ˆ T α ˆ S ln (cid:16) νµ (cid:17) | n (cid:105) (cid:27) . (89)However, none of these states minimizes the uncertainty functional. In any family ofsqueezed states with real squeezing parameter, the product of the position and momen-tum variances is constant while their sum increases without a bound. Consequently, thefunctional can be made arbitrarily small without ever reaching the value zero. The triple product uncertainty relation and the one derived by Heisenberg possess dis-crete symmetries. Here we investigate more general uncertainty functionals which areinvariant under the exchange of three and two variances. S -invariant functionals Consider a function of three variables which is invariant under the exchange of any pair, f ( ) ( x , y , z ) = f ( ) ( y , x , z ) = f ( ) ( x , z , y ) . (90)We now derive the lower bound of a large class of S -invariant uncertainty functionals J [ ψ ] and show that their minima coincide with the state | Ξ (cid:105) minimizing the triple prod-uct inequality. The variables x , y and z will denote the variances of the operators ˆ p , ˆ q andˆ r , respectively.More specifically, we study the minima of sums of completely homogeneous polyno-mials of degree n , with arbitrary non-negative coefficients, f ( ) N ( x , y , z ) = N ∑ n = ∑ j + k + (cid:96) = n a jk (cid:96) x j y k z (cid:96) , a jk (cid:96) ≥ a . The determinant of the associated F -matrix, det F ≡ f x f y + f y f z + f z f x , (92)is positive definite since x , y , z > S -permutations (90) implies that the coefficients must satisfythe conditions a jk (cid:96) = a kj (cid:96) = a j (cid:96) k , 0 ≤ j , k , (cid:96) ≤ n , (93)so that the first terms of the polynomials are given by f ( ) N ( x , y , z ) = a ( x + y + z )+ a (cid:0) x + y + z (cid:1) + a ( xy + yz + zx )+ a (cid:0) x + y + z (cid:1) + a (cid:0) x ( y + z ) + y ( z + x ) + z ( x + y ) (cid:1) + a xyz + . . . (94)If the only nonzero coefficients are a = a =
1, we recover the functionals associ-ated with the triple sum (77) or the triple product inequality (4), respectively. In general,a completely homogeneous S -symmetric polynomial in three variables of degree n ≥ κ n = (cid:106) ( n + ) + (cid:107) terms where the floor function (cid:98) s (cid:99) denotes the integer part ofthe number s : each term arises from one way to partition j + k + (cid:96) = n objects into threesets with j , k and (cid:96) elements, respectively [22]. Thus, a symmetric polynomial of degreeup to N depends on (cid:16) ∑ Nn = κ n (cid:17) independent coefficients if one ignores the constant term.The main result of this section follows from rewriting the consistency conditions (45)and (46) in terms of the variables x , y and z , x f x − y f y + ( x − y ) f z = z f z − x f x + ( z − x ) f y = ( xy + yz + zx ) − x − y − z = ( n + ) ¯ h , (97)where we have used the identity z = x + y + w given in (5). The conditions (95) and(96) imply that the extrema of any symmetric polynomial f ( ) N ( x , y , z ) occur whenever thethree variances take the same value, x = y = z . (98)To show that x = y holds we pick any nonzero term a jk (cid:96) x j y k z (cid:96) in the expansion (91)and assume that the powers of x and y are different, i.e. j (cid:54) = k ; the case j = k will beconsidered later. Due to the symmetry under the exchange x ↔ y , the sum also mustcontain the term a kj (cid:96) x k y j z (cid:96) , with a kj (cid:96) ≡ a jk (cid:96) . Defining t ( x , y , z ) = a jk (cid:96) (cid:0) x j y k + x k y j (cid:1) z (cid:96) , thefirst two terms of (95) take the form xt x − yt y = ( j − k ) a jkl (cid:16) x j y k − x k y j (cid:17) z (cid:96) . (99)Assuming that j = k + δ , with δ >
0, we find xt x − yt y = a k + δ kl δ ( x µ − y µ ) x k y k z (cid:96) (100) = ( x − y ) δ a k + δ k (cid:96) (cid:16) x δ − + x δ − y + . . . + xy δ − + y δ − (cid:17) z (cid:96) (101) ≡ ( x − y ) g + ( x , y , z ) , (102)where g + ( x , y , z ) >
0. Using this expression in (95), the consistency condition takes theform ( x − y ) ( g + ( x , y , z ) + t z ) = t z ( x , y , z ) . If δ < k = j − δ ≡ j + | δ | andeliminate k instead of j from (99), only to find that its left-hand-side again turns into ( x − y ) multiplied with a positive function. If the powers of x and y of the term a kj (cid:96) x k y j z (cid:96) are equal, j = k , one immediately finds that ( x ∂ x − y ∂ y ) a jkl x j y j z (cid:96) =
0, also reducingEq. (95) to ( x − y ) ∂ z a jkl x j y j z (cid:96) = all terms in the sum (91), and the positivity of thecoefficients a jk (cid:96) implies that the first consistency condition can only be satisfied for x = y .Using the symmetry of f ( ) N ( x , y , z ) under the exchange y ↔ z , an identical argumentleads to the identity y = z .Using (98) to evaluate the left-hand-side of Eq. (97) results in x = y = z = τ e n ¯ h , n ∈ N , (104)where τ = √ S -invariant function f ( ) N ( x , y , z ) ≥ f ( ) N ( x , y , z ) (cid:12)(cid:12)(cid:12) x = y = z = ¯ h / √ . (105)This result correctly reproduces the special cases of Eqs. (4) and (77), and there is onlyone state which saturates the inequality, namely | Ξ (cid:105) given in Eq. (67). Letting N → ∞ in Eq. (91), we conclude that the main result of this section, Eq. (105), also applies to any S -symmetric function f ( ) ∞ ( x , y , z ) with a Taylor expansion with positive coefficients andinfinite radius of convergence, as long as its first partial derivatives exist. S -invariant functionals Assume now that, in analogy to Eq. (90), we have a functional depending on just twovariances in a symmetric way, f ( ) ( x , y ) = f ( ) ( y , x ) . (106)An argument similar to the one given for the function f ( ) ( x , y , z ) results in the uncer-tainty relation f ( ) N ( x , y ) ≥ f ( ) N ( x , y ) (cid:12)(cid:12)(cid:12) x = y = ¯ h /2 , (107)which covers the cases of Heisenberg’s relation (1) and the pair sum inequality (2). Thus,the actual form of the function at hand determines whether the set of minima M ( f ( ) N ) will depend on a continuous parameter or not. If the functional is invariant under scal-ing transformation x → λ x , y → y / λ , in addition to the permutation symmetry, thereis a one-parameter family of solutions and the right-hand-side of Eq. (107) achieves itsminimum on the set of points with xy = ( ¯ h /2 ) , not just those with x = y = ¯ h /2.To derive (107), we suppose that the function f ( ) N ( x , y ) has an expansion in analogyto f ( ) N ( x , y , z ) in Eq. (91) but without the variable z . Adapting the reasoning applied to f ( ) N ( x , y , z ) , the consistency equations (45) are found to imply x = y and w =
0. Usingthis result in (46), the bound x ≥ ¯ h /4 follows immediately, so that the inequality (107)must hold for S -invariant functionals. 18 Summary and discussion
This paper responds to the fact that, without exception, the states minimizing the knownpreparational uncertainty relations for a quantum particle in one dimension are given by(sets of) squeezed states. To explain this fact we systematically study lower bounds forsmooth functions f ( ∆ p , ∆ q , C pq ) of second moments. The resulting theory explains theuniversal role of squeezed states for preparational uncertainty relations depending onsecond moments only, and it completely charts the landscape of inequalities of this type.The chain of inclusions E ⊇ E ( f ) ⊇ M ( f ) (108)concisely summarizes the general structure of our findings. First, we have shown thatonly squeezed number states of a quantum mechanical harmonic oscillator with unitfrequency and mass occur as extrema of an uncertainty functional J [ ψ ] depending onsecond moments. We denote this universal set of states by E . Second, the extrema of a specific functional J f [ ψ ] , associated with a function f ( ∆ p , ∆ q , C pq ) , form the subset E ( f ) of the universal set E . Third, the functional will assume its minimum for one or more ofthe extrema E ( f ) , a subset which we denote by M ( f ) . The set of minima may be empty, M ( f ) = Ø. If it is not empty, a lower bound on the functional J f [ ψ ] has been found, andit represents a preparational uncertainty relation in terms of the second moments.Strictly speaking, we obtained the relations E α ⊇ E α ( f ) ⊇ M α ( f ) instead of Eq. (108)as it is possible to move quantum states in phase space without affecting the values of thesecond moments. The four-parameter set E α ≡ ˆ T α E , for example, consists of the squeezedstates E plus those obtained from them by means of the translation operator ˆ T α definedin Eq. (10). Thus, each state saturating a specific inequality with vanishing expectationvalues gives rise to a two-parameter family of minima.Our results have a useful geometric representation in the real three-dimensional spaceof moments. The uncertainty region, consisting of all triples of moments which can arisefrom (pure or mixed) states of a quantum particle, turns out to be a convex set boundedby a one-sheeted hyperboloid. The boundary is invariant under the elliptic rotations,hyperbolic boosts and parabolic transformations which generate the group SO (
1, 2 ) . Thisobservation squares with the importance of the group SO (
1, 2 ) in quantum optics wherecoherent and squeezed states are ubiquitous.The invariance of the bounding hyperboloid can, in turn, also be understood as aninvariance of the functional J [ ψ ] defining the surface. Each point on this hyperboloid isassociated with a unique Gaussian state saturating the Robertson-Schr ¨odinger inequality.Changing from an active view of transformations (i.e. mapping one state with minimaluncertainty to another one) to a passive view, we see that the RS-functional (or a suitablesmooth function thereof) is invariant under the elements of the group Sp ( R ) appliedto the canonical pair ( ˆ p , ˆ q ) . The allowed symplectic transformations include rotations,scalings and linear gauge transformations, or shears.Repeatedly, we have encountered subsets of the maximal possible symmetry groupSp ( R ) (cf. [7]). The Heisenberg functional f H ( x , y , z ) in Eq. (56), for example, is in-variant under a scaling transformation ˆ p → λ ˆ p , ˆ q → ˆ q / λ , with λ >
0, resulting in a one-parameter set of states with minimal uncertainty depicted in Fig. 2. Similarly, uncertaintyfunctionals invariant under permutations of order two or three are minimized by stateswith corresponding symmetry properties. Examples are the triple product uncertaintyrelation (4) and, more generally, the functionals with discrete symmetries discussed inSec. 3.2.We also derived new and explicit uncertainty relations. Fig. 4 uses the ( b , γ ) -plane to19 b µ = / µ = / HeisenbergRobertson- S S Schr ¨odingerFigure 4: States on the boundary of the uncertainty region minimizing known and newuncertainty relations parameterized by the real numbers ( b , γ ) , with ¯ h =
1; each point ofthe plane corresponds to a squeezed state saturating the RS-inequality (3); points on thevertical dashed line represent minima of Heisenberg’s uncertainty relation (1); the twocurved dashed lines indicate the minima of the modified RS-inequality (81) with m = µ = µ = S -invariant functionals (107) such as the pair sum (2) and S -invariant functionals (105)such as the triple product (4).illustrate the sets of states which minimize (some of) the uncertainty relations discussedin this paper. The sets of minima M may depend on two parameters (all squeezed statesminimizing the RS-inequality), on one parameter (such as the real squeezed states satu-rating Heisenberg’s uncertainty relation) or consist of a single point only (associated with S -invariant inequalities such as the triple product inequality, for example). We have notbeen able to backward-engineer functionals which would be minimized by prescribedsubsets of the plane such as a circle or a disk. The minimizing states we found were allpure, located on the boundary of the uncertainty region. In principle, functionals couldalso take their minima inside this region although we have found only trivial exampleswith this property [21].We currently investigate three conceptually interesting generalizations of our ap-proach. First, there is no fundamental reason to restrict oneself to uncertainty functionalsdepending only on second moments in position and momentum [23]. On the contrary,higher order expectation values would enable us to move away from Gaussian quantummechanics which is largely reproducible in terms of a classical model “with an epistemicrestriction” of the allowed probability distributions [24]. Including fourth-order terms (cid:104) ψ | ˆ q | ψ (cid:105) , for example, will result in an eigenvalue equation (18) which is not related to aunit oscillator in a simple way. It is known that fourth-order moments for single-particleexpectations can give rise to inequalities which cannot be reproduced by models based onclassical probabilities [25, 26]. Thus, it might become possible to study truly non-classicalbehaviour in a systematic manner using suitable uncertainty functionals.Secondly, our approach can be generalized to the case of two or more continuousvariables. We expect that a systematic study of uncertainty functionals becomes possible,leading to criteria which would detect pure entangled states. For example, a generaliza-tion of the triple uncertainty relation (4) to bipartite systems has been shown to detectentangled states in a quantum optical setting [27]. Other scenarios are known which relyon intuitive choices of suitable bi-linear observables [28, 29].20inally, the comprehensive study [20] of uncertainty relations for a single spin s hasbeen limited to observables transforming covariantly under the group SU ( ) . The methodproposed here is easily adapted to investigate functionals depending on arbitrary func-tions of moments. Acknowledgments
The authors would like to thank Reinhard F. Werner for comments on an early draft ofthis paper and, more generally, for discussions of uncertainty relations. Tom Bullockkindly commented on a late version of our manuscript. S.K. has been supported viathe act “Scholarship Programme of S.S.F. by the procedure of individual assessment, of2011–12” by resources of the Operational Programme for Education and Lifelong Learn-ing of the ESF and of the NSF, 2007–2013.
A A Baker-Campbell-Hausdorff identity
The relation ˆ G b ˆ S γ = ˆ S ( ξ ) ˆ R ( χ ) in Eq. (33) can be shown by requiring that both productsmap the annihilation operator ˆ a = ( ˆ q + i ˆ p ) / √ h to the same operator. We obtainˆ G b ˆ S γ ˆ a ˆ S † γ ˆ G † b = ˆ a (cid:18) cosh γ − i b e γ (cid:19) + ˆ a † (cid:18) sinh γ + i b e γ (cid:19) (109)and ˆ S ( ξ ) ˆ R ( χ ) ˆ a ˆ R † ( χ ) ˆ S † ( ξ ) = ˆ ae − i χ cosh r − ˆ a † e − i χ e i θ sinh r , (110)respectively. Equating the coefficients of the operators ˆ a and ˆ a † leads to two equationscosh γ − i b e γ = e − i χ cosh r , (111)sinh γ + i b e γ = − e i ( θ − χ ) sinh r , (112)which we need to solve for the variables ξ ≡ re i θ and χ . Separating the real and imaginaryparts of the first equation, one finds that χ = arctan (cid:18) b + e − γ (cid:19) ∈ ( − π π ) . (113)In a similar way, the second equation allows one to solve for the function tan ( θ − χ ) which, upon using (113), leads to θ = arctan (cid:18) b − e − γ (cid:19) + arctan (cid:18) b + e − γ (cid:19) ∈ ( − π , π ) . (114)The case of γ = θ = ± π + arctan (cid:18) b (cid:19) ∈ ( − π , π ) . (115)Finally, the condition cosh r cos χ = cosh γ results in the expression r = arcosh (cid:18) cosh γ + b e γ (cid:19) ∈ [ ∞ ) , (116)21hich establishes the desired identity (33).The number states | n (cid:105) , n ∈ N , are eigenstates of phase-space rotations ˆ R ( χ ) . There-fore, the product ˆ G b ˆ S γ acts on those states according toˆ G b ˆ S γ | n (cid:105) ∼ = ˆ S ( ξ ) | n (cid:105) , (117)where an irrelevant phase has been suppressed. Thus, the operator ˆ S ( ξ ) generates allsqueezed states from | (cid:105) when the parameter ξ runs through the points of the complexplane. References [1] W. Heisenberg, Z. Phys. , 172 (1927)[2] E. H. Kennard, Z. Phys. , 326-52 (1927)[3] J. Pleb´anski, Phys. Rev. ,1076 (1967)[5] D. Stoler, Phys. Rev. D , 3217 (1970)[6] V. V. Dodonov, J. Opt. B , R1 (2002)[7] D. A. Trifonov, Schr¨odinger Uncertainty Relation and its Minimization States (preprint:arXiv:physics/0105035 [physics.atom-ph])[8] P. Busch, P. Lahti, R. F. Werner, Phys. Rev. A , 012129 (2014)[9] H. P. Robertson, Phys. Rev. , 163-164 (1929)[10] E. Schr ¨odinger, Sitzber. Preuss. Akad. Wiss. (Phys.-Math. Klasse) , 296 (1930)[11] R. Jackiw, J. Math. Phys. , 339 (1968)[12] S. Weigert, Phys. Rev. A , 2084-2088 (1996)[13] S. Kechrimparis and S. Weigert, Phys. Rev. A , 062118 (2014)[14] P. Busch, T. P. Sch ¨onbeck and F. Schroeck Jr., J. Math. Phys. , 2866 (1987)[15] I. Bialynicki-Birula and Z. Bialynicka-Birula: Phys. Rev. Lett. , 140401 (2012)[16] Ł. Rudnicki, Phys. Rev. A , 022112 (2012)[17] S. Weigert, J. Phys. A: Math. Gen. , 4169 (2002)[18] J. Williamson, Am. J. Math. , 141 (1936)[19] Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana , 471 (1995)[20] L. Dammeier, R. Schwonnek, and R. F. Werner, New J. Phys. , 093046 (2015)[21] S. Kechrimparis and S. Weigert, Mathematics , 49 (2016)[22] N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences , AcademicPress, 1995 2223] P. Busch, P. Lahti, R. F. Werner, J. Math. Phys. , 042111 (2014)[24] D. Bartlett, T. Rudolph and Robert W. Spekkens, Phys. Rev. A , 012103 (2012)[25] A. Bednorz, W. Belzig, Phys. Rev. A , 052113 (2011)[26] E. Kot, N. Grønbech-Jensen, B. M. Nielsen, J. S. Neergard-Nielsen, E. S. Polzik, A. S.Sørensen, Phys. Rev. Lett. , 233601 (2012)[27] E. C. Paul, D. S. Tasca, Ł. Rudnicki and S. P. Walborn, Phys. Rev. A , 012303 (2016)[28] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. , 2722 (2000)[29] A. Serafini, Phys. Rev. Lett.96