arXiv:2008.XXXXX

Beyond Density Matrices: Geometric States

Fabio Anza∗ and James P. Crutchfield† Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616 (Dated: August 21, 2020) A quantum system’s state is identified with a density . Though their probabilistic interpre- tation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express an ensemble’s physical realization. Conveniently, when working only with the statistical outcomes of projective and positive -valued measurements this is not a hindrance. To track ensemble realizations and so remove the shortcoming, we explore geometric quantum states and explain their physical significance. We emphasize two main consequences: one in manipulation and one in .

PACS numbers: 05.45.-a 89.75.Kd 89.70.+c 05.45.Tp Keywords: Quantum , Geometric

Introduction. Dynamical systems theory describes long- However, this interpretation is problematic: It is not term recurrent behavior via a system’s attractors: sta- unique. One can write the same ρ using different decom- ble dynamically-invariant sets. Said simply there are positions, for example in terms of {|ψki}= 6 {|λii}: regions of —points, curves, smooth manifolds, X or fractals—the system repeatedly visits. These objects ρ = pk |ψki hψk| . are implicitly determined by the underlying equations k of and are modeled as probability distributions (measures) on the system’s state space. Given the interpretation, all the decompositions identify Building on this, the following introduces tools aimed the same quantum state ρ. While one often prefers Eq. at studying attractors for quantum systems. This re- (1)’s diagonal decomposition in terms of eigenvalues and quires developing a more fundamental concept of “state eigenvectors, it is not the only one possible. More tellingly, of a quantum system”, essentially moving beyond the in principle, there is no experimental reason to prefer it to standard notion of density matrices, though they can be others. In quantum mechanics, this fact is often addressed directly recovered. We call these objects the system’s ge- by declaring density matrices with the same barycenter ometric quantum states and, paralleling the Sinai-Bowen- equal. A familiar example of this degeneracy is that the Ruelle measures of dynamical systems theory [1], they are maximally mixed state (ρ ∝ I) has an infinite number specified by a on the manifold of of identical decompositions, each possibly representing a quantum states. physically-distinct ensemble. Quantum mechanics is firmly grounded in a vector formal- Moreover, it is rather straightforward to construct sys- ism in which states |ψi are elements of a complex Hilbert tems that, despite having the same , are space H. These are the system’s pure states, as opposed in different states. For example, consider two distinct to mixed states that account for incomplete knowledge state-preparation protocols. In one case, prepare states of a system’s actual state. To account for both, one em- {|0i , |1i} each with probability 1/2; while, in the other, ploys density matrices ρ. These are operators in H that always prepare states {|−i , |+i} each with probability are positive semi-definite ρ ≥ 0, self-adjoint ρ = ρ†, and 1/2. They are described by the same ρ. A complete and normalized Tr ρ = 1. unambiguous mathematical concept of state should not The interpretation of a density matrix as a system’s prob- conflate distinct physical configurations. Not only do such ambiguities lead to misapprehending fundamental mech- arXiv:2008.08682v1 [quant-ph] 19 Aug 2020 abilistic state is given by ensemble theory [2, 3]. Ac- cordingly, since a density matrix always decomposes into anisms, they also lead one to ascribe complexity where there is none. eigenvalues λi and eigenvectors |λii: Here we argue that an alternative—the geometric formal- X ρ = λ |λ i hλ | , (1) ism—together with an appropriately adapted measure i i i theory cleanly separates the primary concept of a system i state from the derived concept of a density matrix as the one interprets ρ as an ensemble of pure states—the set of all positive operator-valued measurement eigenvectors—in which λi is the probability of an observer generated by a system. interacting with state |λii. With this perspective in mind, we introduce a more inci- sive description of pure-state ensembles. The following ar- gues that geometric quantum mechanics (GQM), through ∗ [email protected] its notion that geometric quantum states are continuous † [email protected] mixed states, resolves the ambiguities. First, we introduce 2

GQM. Second, we discuss how it relates to the density Notice how p0 and ν0 are not involved. This is due to matrix formalism. Then we analyze two broad settings P(H)’s projective nature which guarantees that we can in which the geometric formalism arises quite naturally: PD−1 choose a coordinate patch in which p0 = 1 − α=1 pα quantum state manipulation [4] and quantum thermo- and ν0 = 0. dynamics [5]. After discussing the results, we draw out several consequences. Geometric quantum states. This framework makes it very Geometric quantum mechanics. References [6–21] give a natural to view a quantum state as a functional encoding comprehensive introduction to GQM. Here, we briefly that associates expectation values to , paral- summarize only the elements we need, working with leling the C∗-algebras formulation of quantum mechanics Hilbert spaces H of finite dimension D. [24]. Thus, states are described as functionals P [O] from Pure states are points in the complex projective manifold the algebra of observables A to the real line: P (H) = CPD−1. Therefore, given an arbitrary D−1 Z {|eαi}α=0 , a pure state ψ is parametrized by D complex α Pq[O] = q(Z)O(Z)dVFS , (4) homogeneous coordinates Z = {Z }, up to normalization P(H) and an overall phase: where O ∈ A, q(Z) ≥ 0 is the normalized distribution D−1 X associated with functional P : |ψi = Zα |e i , α Z α=0 Pq[I] = q(Z)dVFS = 1 , P(H) where Z ∈ CD, Z ∼ λZ, and λ ∈ C/ {0}. If the system consists of a single , for example, one can always use √ √ and Pq[O] ∈ R. amplitude-phase coordinates Z = ( 1 − p, peiν ). An is a quadratic real function O(Z) ∈ R that In this way, pure states |ψ i are functionals with a Dirac- associates to each point Z ∈ P(H) the expectation value 0 delta distribution p (Z) = δ [Z − Z ]: hψ| O |ψi of the corresponding operator O on state |ψi 0 e 0 with coordinates Z: Z P0[O] = δ(Z − Z0)O(Z)dVFS β e X α P(H) O(Z) = Oα,βZ Z , (2) α,β = O(Z0) = hψ0| O |ψ0i .

where Oαβ is Hermitian Oβ,α = Oα,β. δe(Z − Z0) is shorthand for a coordinate-covariant Dirac- Measurement outcome probabilities are determined by delta in arbitrary coordinates. In homogeneous coordi- n positive operator-valued measurements () {Ej}j=1 nates this reads: applied to a state [22, 23]. They are nonnegative opera- D−1 tors Ej ≥ 0, called effects, that sum up to the identity: 1 Y n δe(Z − Z0) := √ δ(X − X0)δ(Y − Y0) , P E = . In GQM they consist of nonnegative real det g j=1 j I FS α=0 functions Ej(Z) ≥ 0 on P(H) whose sum is always unity: where Z = X + iY . In (pα, να) coordinates this becomes X α β simply: Ej(Z) = (Ej)α,β Z Z , (3) α,β D−1 Y 0 0 Pn δe(Z − Z0) = 2δ(pα − pα)δ(να − να) , (5) where j=1 Ej(Z) = 1. α=1 Complex projective spaces, such as P(H), have a preferred metric gFS—the Fubini-Study metric [13]—and an associ- where the coordinate-invariant nature of the functionals ated volume element dVFS that is coordinate-independent Pq[O] is now apparent. and invariant under unitary transformations. The geomet- ric derivation of dVFS is beyond our immediate goals here. In this way, too, mixed states: That said, it is sufficient to give its explicit form in the √ α iνα X “probability + phase” Z = pαe ρ = λ |λ i hλ | that we use for explicit calculations: j j j j D−1 p Y α α are convex combinations of these Dirac-delta functionals: dVFS = det gFS dZ dZ α=0 X qmix(Z) = λjδe(Z − Zj) . D−1 Y dpαdνα j = . 2 α=1 Thus, expressed as functionals from observables to the 3 real line, mixed states are: POVM on the system that distinguishes between q1 and q . X 2 Pmix [O] = λj hλj| O |λji . (6) To emphasize, consider the example of two geometric j quantum states, q1 and q2, with very different character- istics: Equipped with this formalism, one identifies the distribu- tion q(Z) as a system’s geometric quantum state. This 1 − 1 Zρ−1Z q1(Z) = e 2 is the generalized notion of quantum state we develop in Q ˜ ˜ the following. q2(Z) = 0.864 δ(Z − Z+) + 0.136 δ(Z − Z−) , A simple example of an ensemble that is neither a pure 1 −1 R − 2 Zρ Z nor a mixed state is the geometric : where Q = 1 dVFSe , Z+ = (0.657, 0.418 + CP i0.627), and Z− = (0.754, −0.364 − i0.546). However, 1 q(Z) = e−βh(Z) , states q1 and q2 have same density matrix ρ (ρ00 = 0.45 = ∗ Qβ 1−ρ11 and ρ01 = 0.2−i0.3 = ρ10) and so the same POVM outcomes. From Fig.1 one appreciates the profound dif- where: ference between q1 and q2, despite the equality of their Z POVM statistics. −βh(Z) Qβ = dVFSe , h(Z) = hψ(Z)| H |ψ(Z)i , and H is the system’s Hamiltonian operator. This state was previously considered in Refs. [25, 26]. Reference [5] investigated its potential role in establishing a quantum foundation of thermodynamics that is an alternative to that based on Gibbs ensembles and von Neumann en- tropy. Moreover, it showed that the geometric ensemble genuinely differs from the Gibbs ensemble. This realiza- Figure 1. Geometric quantum states in (probability,phase) tion provides a concrete path to testing the experimental 1 consequences of geometric quantum states. coordinates (p, φ) of CP : (Left) Density matrix “state” q1 is the convex sum of two Dirac delta-functions, centered on Density matrix. The connection between geometric quan- the eigenvectors (p+, φ+) = (0.568, 0.983) and (p−, φ−) = tum states and density matrices is two-fold. On the one (0.432, 4.124) of density matrix ρ. (Right) Geometric quantum hand, when the distribution q(Z) falls into one of the two state q2 differs markedly: A smooth distribution across the 1 aforementioned cases—Dirac-deltas or finite convex com- entire pure-state manifold CP . However, q1 and q2 have the binations of them—the present formalism is equivalent to same density matrix ρq1 = ρq2 = ρ, where ρ00 = 1−ρ11 = 0.45, ∗ the standard one. However, not all functionals fall into ρ01 = ρ10 = 0.2 − 0.3i. ρ± are the eigenvalues of the density the Dirac-delta form. Given this, q(Z) is clearly a more matrix: ρ+ = 0.864 and ρ− = 0.136. Geometric quantum state q ’s structure is only sparsely reflected in the density-matrix general notion of a quantum system’s state. 2 “state” q1. On the other hand, given an arbitrary distribution q(Z), there is a unique density matrix ρq associated to q: This is particularly important for β processing where one encounters long-range and long- ρq = P [ZαZ ] αβ q lived correlational and mechanistic demands. Quantum Z α β computing immediately comes to mind. There, one is = dVFS q(Z) Z Z . (7) P(H) not only interested in measurement outcomes, but also in predicting and understanding how a quantum system Owing it to the fact that all POVMs are represented by evolves under repeated external manipulations imposed real and quadratic functions on P(H), recall Eq. (3), they by complex control protocols. q are sensitive to q(Z) via ρ . Therefore, if two distributions State manipulation. The following shows that the geo- q1 q2 q1 and q2 induce the same density matrix ρ = ρ , then metric formalism arises quite naturally when a discrete all POVMs produce the same outcomes. quantum system interacts and develops entanglement A well-known consequence of this fact is that two density with a continuous one. Imagine a protocol controlling a matrices with the same barycenter are considered equal, system’s continuous degrees of freedom to manipulate dis- even if they describe experiments with different physical crete ones that store a computation’s result. As a physical configurations. In these cases, the statistics of POVM reference, consider quantum with a given number outcomes are described by the same density matrix. Note of discrete degrees of freedom (e.g., ), confined to a that this statement does not mean that the two systems region R ⊆ R3. The results we derive do not depend on are in the same state. Rather, it means that there is no this choice, since the technical methods straightforwardly 4

d extend to other systems where continuous and discrete HM , we have: degrees of freedom are mixed. A helpful illustration is Z intra- entanglement [27], that couples position and hOi = Tr ρO = d~x |f(~x)|2 O(q(~x)) , spin degrees of freedom to create entangled states. In R this way, one manipulates the spin by only acting on the positional degrees of freedom, possibly via a potential. where O(q(~x)) = hq(~x)| O |q(~x)i. Comparing with Eq. (4) one realizes that the functions {ps(~x), φs(~x)} provide Consider a hybrid quantum system comprised of N con- N n an ~x-dependent embedding of R ⊆ R onto CP , with tinuous degrees of freedom and M qudits that are the n = dM − 1, or a submanifold, via: discrete ones. The entire system’s is:

c d Φ: ~x → Φ(~x) = Z(~x) , H = HN ⊗ HM , where: c where HN hosts the continuous degrees of freedom and has d 0 n infinite dimension, while HM hosts the discrete ones and Z = (Z ,...,Z ) , M c has dimension d . A basis for HN is provided by {|~xi}, M N d d −1 α p iφα(~x) ∗ where ~x ∈ R ⊆ R and a basis for HM is {|si}s=0 . with Z (~x) = pα(~x)e . Thus, letting R = Φ(R), Thus, a generic state is: we obtain: Z Z Z X d~x |f(~x)|2 O(q(~x)) = dV q(Z)O(Z) , |ψi = d~x ψs(~x) |~xi |si , FS R R∗ R s where: where ~x is a dimensionless counterpart of the physical continuous degrees of freedom, achieved by multiplying its |det DΦ(Z)| −1 2 value by appropriate physical quantities. So, the measure q(Z) = √ f(Φ (Z)) . det gFS d~x has no physical dimension. For an in a box, for example, this is achieved by renormalizing with the Here, DΦ denotes the Jacobian of the transformation Φ box’s total volume. and gFS is the Fubini-Study metric tensor and we assume the transformation is invertible. Generalizing to cases in The following theorem establishes that this can be done which Φ−1 is not invertible, due to the fact that different constructively. ~x might yield the same (ps(~x), φs(~x)), is left to future Theorem 1. Any state |ψi ∈ H can be written as: efforts. Let’s illustrate with a familiar system: an electron in a 2D Z rectangular box R = [x0, x1]×[y0, y1]. In this case, M = 1 |ψi = d~xf(~x) |xi |q(~x)i , (8) and d = 2 so that we have f(x, y), {p (x, y), φ (x, y)} . R s s s=0,1 This amounts to: where f(~x) is such that R d~x|f(~x)|2 = 1 and |q(~x)i is a R Z x1 Z y1 parametrized state of the discrete degrees of freedom: dx dy|f(x, y)|2O(q(x, y)) x0 y0 M d −1 1 Z 1 Z 2π X p iφs(~x) |q(~x)i = ps(~x)e |si , = dp dφ q(Z(p, φ)) O(Z(p, φ)) , 2 0 0 s=0

M where, for example, p0(x, y) = 1 − p1(x, y), p1(x, y) = where {ps(~x), φs(~x)}s is a set of 2(d − 1) real func- x−x0 y−y0 M , φ0(x, y) = 0, and φ1(x, y) = 2π . Pd −1 x1−x0 y1−y0 tions such that s=0 ps(~x) = 1, φs(~x) ∈ [0, 2π], and M In short, a generic quantum state |ψi of the whole system d −1 d {|si}s=0 is a basis on HM . uniquely defines a distribution q(Z) on the manifold of d dM −1 pure states P(HM ) = CP . The correspondence (The Supplementary Material gives the proof.) Equa- is not one-to-one as knowledge of q(Z) does not allow tion (8)’s state parametrization preserves key information recovering the entire state. The missing part is θ0(~x), the about the continuous degrees of freedom, namely |f(~x)|2, phase of f(~x). However, it does circumscribe the possible when working with the discrete degrees of freedom. In- states as it fixes the shape of the probability distribution deed, the partial over the continuous degrees of of the continuous variables |f(~x)|2. freedom yields: Note how the embedding functions ps(~x) and φs(~x) play Z a key role in determining whether we can cover the whole ρ = d~x |f(~x)|2 |q(~x)ihq(~x)| . CP n or just a submanifold. Consider the conditions R that guarantee the two extreme cases are covered: full covering of CP n and covering of tensor product states Continuing, given an observable O with support only on only CP d−1 ⊗ ... ⊗ CP d−1. In the first case, CP n is 5 a complex manifold that requires 2n independent real erases the functional information about the environment.  S S coordinates to be completely covered. For M qudits this Instead, the latter can be recovered from pα, χα as: means: q E S S S S Full ρ αβ = pαpβ χα χβ . M ≤ Mmax log (N/2 + 1) = . As dE grows, retaining this information as a set of proba- log d bilities and states quickly becomes unrealistic. However, the same information can be effectively encoded Instead, if we need to cover only the submanifold of tensor by switching to a geometric description. Indeed, at finite product states, the number of qudits we can control with d , ρS becomes: N continuous degrees of freedom is much larger: E d Prod E M ≤ Mmax S X E ˜ S  pd (Z) = pα δ Z − Z(χα) N E = . α=1 2(d − 1) and the thermodynamic limit is conveniently handled Most cases fall in between. And so, the number of qu- with: dits controllable with N continuous variables is M ∈ d  Full Prod E Mmax,Mmax . X E ˜ S  p∞(Z) = lim pα δ Z − Z(χα) . dE →∞ Thermodynamic framework. Another setting in which α=1 the geometric formalism arises naturally is quantum ther- modynamics. There, one is often interested in modeling Here the limit is performed, as usual, by keeping finite the the behavior of a small system in a thermal environment. average energy density limdE →∞ hHi /(NS + NE) = . For modest-sized environments one can naively treat the In this way, the geometric formalism emerges naturally system and environment as isolated and then simulate its in a quantum thermodynamics. In the limit of large evolution. As the environment’s size grows, this quickly environments, one simply cannot keep track of exactly becomes infeasible. Nonetheless, as we now show, the how an environment generates the ensemble of our system geometric formalism allows appropriately writing the sys- under study and so switch to a probabilistic description. tem’s reduced density matrix in a way that retains much Helpfully, the geometric formalism efficiently controls this. of the information about the environment. This can be See also Ref. [5] for an expanded exploration of the done due to Thm.1. geometric formalism in quantum thermodynamics. Consider a large quantum system consisting of M qudits Before proceeding, though, let’s highlight an interesting split in two asymmetric parts. Call the small part with NS discrepancy between the two applications presented. In qudits the “system” and let the rest be the “environment” the thermodynamic setting, knowledge of the ensemble  S S with NE = M − NS qudits. A generic state of the entire pα, χα allows fully recovering the global pure state PdS −1 PdE −1 |ψSEi. Indeed, it is easy to see that: system HS ⊗ HE is |ψSEi = k=0 α=0 ψkα |ski |eαi, where {|ski}k and {|eαi}k are bases for HS and HE, q S S respectively. ψkα = pα sk χα .

Given |ψSEi, it is not too hard to see that the system’s (reduced) state is: Substituting this into the pure state |ψSEi, we obtain a Schmidt-like decomposition in which the common label

dE runs over the dimension of the environment’s Hilbert S X S S S ρ = pα χα χα , (9) space: α=1 dE −1 q X S S where: |ψSEi = pα χα |eαi . α=0 dS −1 S X 2 pα = |ψkα| , The price paid for the decomposition is that the states S k=0 χα are not orthogonal. However, we gain a more detailed description of our system’s state. As we can see, here and the challenge of recovering |ψSEi disappears thanks to S dS −1 pα ∈ R. We comment on this discrepancy with the other S 1 X χ = ψkα |ski . case shortly. α p S pα k=0 Discussion. Standard quantum mechanics’ concept of state is the density matrix. However, while density ma- In numerical analysis one often retains only the dS × dS trices provide a complete account of POVM statistics, matrix elements of ρS in a certain basis. However, this they are not in one-to-one correspondence with the en- 6 sembles that generated them. This is a well-known fact one can always choose f(~x) ∈ R. We leave exploring the that underlies the freedom in writing a decomposition connection between recovering the global pure state from of the density matrix in terms of probabilities and pure a local geometric quantum state and a gauge principle for states. All such decompositions yield the same POVM a future investigation. statistics, but they are not physically equivalent since Conclusion. Geometric quantum mechanics is an alter- they are realized in physically different ways. native to the standard vector-based formalism. We in- From a purification perspective [28], the physical infor- troduced and then explored the concept of geometric mation about an ensemble’s realization can always be quantum state p(Z) as a probability distribution on the thought of as coming from a larger system that is in a manifold of pure states, inspired by the statistics of chaotic pure state. While the additional information about how attractors from the theory of dynamical systems or, more the ensemble is realized is not relevant for the measure- appropriately, its Sinai-Bowen-Ruelle measures [1]. This ment statistics on our system, it does provide a much characterization of a quantum state accounts for the fact richer description. It preserves part (if not all) of the that singling out the density matrix as the sole descrip- structural information about how the system’s POVM tor of a quantum system’s state entails ignoring how an statistics result from interactions with its surroundings. ensemble is physically realized. While this does not have Geometric quantum mechanics and its concept of geo- consequences for POVM statistics, in concrete situations metric quantum state provide a framework that allows the information about the ensemble realization can be retaining such information. This yields a richer picture of key to accurate modeling. Reference [4] gives an example. the system’s state which goes beyond the system’s POVM That said, density matrices can be readily computed as statistics, taking into account the physical way in which quadratic averages from p(Z) via Eq. (7). an ensemble has been realized. The geometric formalism’s We explored the physical relevance of geometric quantum benefits emerge in at least two important cases: (i) Hy- states via an in which a (finite) brid continuous-discrete systems, e.g., or other system under study is in contact with a larger environ- particles with spin or other discrete degrees of freedom, ment and the joint state is assumed to be pure. In this and (ii) the thermodynamic setting of a system in contact thermodynamic setting, portions of the structural infor- with a large environment. mation about the joint pure state is directly preserved in The geometric formalism directly handles the continuous the geometric quantum state of the smaller system under nature of hybrid systems and the large number of degrees study. The result is a markedly richer picture of the sys- of freedom in thermodynamics. And, it does so in a fairly tem’s state—a picture that goes substantially beyond the simple way. This allows working with the full geometric density matrix and its POVM statistics. quantum state, thus retaining the structural information about how the ensemble is generated. While the two applications considered are similar, a crucial difference ACKNOWLEDGMENTS does appear. If we assume a finite environment, knowl- edge of the geometric quantum state of our system is sufficient to recover the global pure state of system and F.A. thanks Marina Radulaski, Davide Pastorello, and environment. This does not occur for a hybrid discrete- Davide Girolami for discussions on the geometric for- continuous system, where knowledge of the geometric malism of quantum mechanics. F.A. and J.P.C. thank quantum state does not allow inferring the phase θ0(~x) Dhurva Karkada for his help with the example, David of f(~x). Notably, fully recovering the overall pure state, Gier, Samuel Loomis, and Ariadna Venegas-Li for helpful whose physical relevance can be argued on the ground discussions and the Telluride Science Research Center for of continuity with the finite-dimensional case, effectively its hospitality during visits. This material is based upon translates into a U(1) gauge principle on the overall sys- work supported by, or in part by, a Templeton World Char- tem. The requirement that states differing from a local ity Foundation Power of Information Fellowship, FQXi iϕ(~x) phase are physically equivalent—ψs(~x) ∼ e ψs(~x)— Grant FQXi-RFP-IPW-1902, and U.S. Army Research turns into a sufficient condition for recovering the global Laboratory and the U. S. Army Research Office under state from the geometric quantum state since, in this case, contracts W911NF-13-1-0390 and W911NF-18-1-0028.

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Fabio Anza and James P. Crutchfield

Appendix A: The Search for Quantum States

In those domains of the physical sciences that concern the organization and evolution of systems, a common first task is to determine a system’s distinct configurations or effective states. Ultimately, this turns on what questions there are to answer. One goal is prediction—of properties or behaviors. And, in this, quantum mechanics stands out as a particularly telling arena in which to define effective states. The very early history of its development can be construed partially as attempts to answer this question, from de Broglie’s phase-waves [29] and Schrodinger’s wave functions [30] to von Neumann’s statistical operators in Refs. [31] and [32, Chap. IV], later labeled density matrices by Dirac [33–35]. And, these were paralleled by Heisenberg’s “operational” that focused on experimentally accessible observables and so avoided imputing internal, hidden structure [36]. The abiding challenge is that effective states are almost always inferred indirectly and through much trial and error. Quantum mechanics heightens the challenge greatly due to its foundational axiom that the detailed, microscopic, and fundamental degrees of freedom cannot be directly and completely measured in principle. The main text revisits this perennial question, What is a quantum state?

Appendix B: Theorem1: Proof

In this Appendix we give the detailed proof of Theorem1 in the main text. Let’s first restate the setup of the theorem. Consider a hybrid quantum system comprised of N continuous degrees of freedom and M qudits that are the discrete ones. The entire system’s Hilbert space is:

c d H = HN ⊗ HM ,

c d where HN hosts the continuous degrees of freedom and has infinite dimension, while HM hosts the discrete ones and M M c N d d −1 has dimension d . A basis for HN is provided by {|~xi}, where ~x ∈ R ⊆ R and a basis for HM is {|si}s=0 . Thus, a generic state is:

Z X |ψi = d~x ψs(~x) |~xi |si , R s where ~x is a dimensionless counterpart of the physical continuous degrees of freedom, achieved by multiplying its value by appropriate physical quantities. So, the measure d~x has no physical dimension. Theorem1. Any state |ψi ∈ H can be written as: Z |ψi = d~xf(~x) |xi |q(~x)i , R

R 2 where f(~x) is such that R d~x|f(~x)| = 1 and |q(~x)i is a parametrized state of the discrete degrees of freedom:

dM −1 X p iφs(~x) |q(~x)i = ps(~x)e |si , s=0

M M M Pd −1 d −1 where {ps(~x), φs(~x)}s is a set of 2(d − 1) real functions such that s=0 ps(~x) = 1, φs(~x) ∈ [0, 2π], and {|si}s=0 d is a basis for HM .

Proof : The proof is constructive. Given an arbitrary {ψs(~x)}s, we can always find the set of functions f(~x), ps(~x), and φs(~x). The converse holds trivially: Given these functions one can always compute the {ψs(~x)}s. The set of transformations that maps one parametrization into the other is:

φs(~x) = θs(~x) − θ0(~x) , where:

ψs(~x) eiθs(~x) = . |ψs(~x)| Moreover: v udM −1 u X 2 iθ0(~x) f(~x) = t |ψs(~x)| e and s=0 |ψ (~x)|2 p (~x) := s , s PdM −1 2 l=0 |ψl(~x)|

2 It is easy to see how normalization of |f(~x)| and of ps(~x) emerges from the definitions:

Z Z dM −1 2 X 2 d~x |f(~x)| = d~x |ψs(~x)| R R s=0 = 1 ,

M M d −1 d −1 2 X X |ψs(~x)| p (~x) = s PdM −1 2 s=0 s=0 l=0 |ψl(~x)| = 1 , and

|ψs(~s)| |eiφs(~x)|2 = |ψs(~x)| = 1 .

The latter gives φs(~x) ∈ [0, 2π]. With these definitions we obtain:

iφs(~x) p p 2 iθs(~x) e f(~x) ps(~x) = |ψs(~x)| e

= ψs(~x) .

This in turn gives the desired result:

Z X |ψi = d~x ψs(~x) |~xi |si R s Z X iφs(~x)p = d~xf(~x) |~xi e ps(~x) |si R s Z = d~xf(~x) |xi |q(~x)i . R