Geometric Quantum State Estimation

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Title Geometric Quantum State Estimation

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Authors Anza, Fabio Crutchfield, James P

Publication Date 2020-08-19

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California arXiv:2008.XXXXX

Geometric Quantum State Estimation

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) Density matrices capture all of a quantum system’s statistics accessible through projective and positive operator-valued measurements. They do not completely determine its state, however, as they neglect the physical realization of ensembles. Fortunately, the concept of geometric quantum state does properly describe physical ensembles. Here, given knowledge of a density matrix, possibly arising from a tomography protocol, we show how to estimate the geometric quantum state using a maximum entropy principle based on a geometrically-appropriate entropy.

PACS numbers: 05.45.-a 89.75.Kd 89.70.+c 05.45.Tp Keywords: Quantum mechanics, geometric quantum mechanics, maximum entropy estimation, density matrix

I. INTRODUCTION P(H). This notion of state, which we here dub geometric quantum state, straightforwardly generalizes the density Quantum mechanics defines a system’s state |ψi as an matrix formalism, to which it reduces in appropriate cases. element of a Hilbert space H. These are the pure states. The primary benefit is an informationally-richer concept To account for uncertainties in a system’s actual state |ψi of quantum state than the density matrix. one extends the definition to density operators ρ that act Geometric quantum states have been sporadically consid- on H. These operators are linear, positive semidefinite ered in the literature [9, 19–21], though mostly in special ρ ≥ 0, self-adjoint ρ = ρ†, and normalized Tr ρ = 1. ρ cases. Companion works, Refs. [22, 23], give a formal def- then is a pure state when it is also a projector: ρ2 = ρ. inition, invariant under changes of coordinates, in terms The spectral theorem guarantees that one can always de- of the infinite limit of finite convex combinations, together P with an extended geometric formalism that addresses the compose a density operator as ρ = i λi |λii hλi|, where λi ∈ [0, 1] are its eigenvalues and |λii its eigenvectors. En- notion of partial trace in a more flexible way. While ef- semble theory gives the decomposition’s statistical mean- forts have been invested to develop the tools of GQM, ing [1, 2]: λi is the probability that the system is in the to the best of our knowledge the notion of continuous pure state |λii. Thus, if we concede that an operational mixed state has remained (almost) exclusively a math- procedure aimed at preparing a system in a pure state ematical tool. With this in mind, the following lays a |ψi will not always be perfectly executed, we must admit bridge between density matrices and geometric quantum uncertainties of this kind. Moreover, if we admit limited states. The goal is to introduce a protocol for estimating knowledge of the interactions a system has with its envi- a geometric quantum state, starting from experimental ronment, uncertainties of this kind will be dynamically data. generated. Our development proceeds as follows. We briefly intro- P Most generally, density operators ρ = k pk |ψki hψk| are duce geometric quantum mechanics, highlighting its rela- convex combinations of pure states. Given a preparation tion to the more familiar density-matrix formalism. Then, scheme, they say that a system is in pure state |ψki with we show how to set up a maximum entropy estimate probability pk. In this sense, they are probability mea- for geometric quantum states, starting from knowledge sures whose sample space is H. This is formalized using of the density matrix. Eventually, we draw out several geometric quantum mechanics (GQM)—an equivalent for- consequences and discuss future challenges. mulation that exploits tools from differential geometry arXiv:2008.08679v1 [quant-ph] 19 Aug 2020 to describe the states and dynamics of quantum systems [3–18]. In GQM the states of a quantum system with II. GEOMETRIC QUANTUM MECHANICS Hilbert space H of dimension D are points in the projective manifold P(H) = CP D−1. Adopting GQM’s perspective, one realizes that pure states The following briefly covers geometric quantum mechanics are Dirac-delta measures on the pure-state manifold P(H), and its tools. More complete discussions are found in the while mixed states are discrete convex combinations of relevant literature [3–18] and in two companion works Direct-delta measures. This suggests considering con- [22, 23]. tinuous mixed states as generic probability measures on Manifold of quantum states. Given a system with a D- dimensional Hilbert space H, in GQM its pure states are points in the complex projective manifold P (H) = ∗ [email protected] CP D−1. To give coordinates we fix an arbitrary ba- † [email protected] D−1 sis {|eαi}α=0 of H. In this way, a pure state |ψi is 2 parametrized by D complex and homogeneous coordi- associated with the functional P : α nates Z = {Z }, up to normalization and an overall Z phase: Pq[I] = q(Z)dVFS = 1 . P(H) D−1 X α |ψi = Z |eαi , In this setting, since pure states |ψ0i are points in P(H), α=0 they are associated to functionals with Dirac-delta distri- ˜ butions p0(Z) = δ [Z − Z0]: where Z ∈ CD, Z ∼ λZ, and λ ∈ C/ {0}. In the case of a single qubit, for example, we have Z = Z √ √ qubit ˜ iν0 iν1 P0[O] = δ(Z − Z0)O(Z)dVFS , ( p0e , p1e ). P(H) Measurements. Quantum measurements on H are ˜ modeled using positive operator-valued measurements where δ(Z − Z0) is shorthand for a coordinate-covariant (POVMs) as follows. {E }n are collections of Hermi- Dirac-delta in arbitrary coordinates. In homogeneous j j=1 coordinates this reads: tian, nonnegative operators Ej ≥ 0, called effects. They n P D−1 are decompositions of the identity j=1 Ej = I. In GQM ˜ 1 Y they are described by finite collections of n real and non- δ(Z − Z0) := √ δ(X − X0)δ(Y − Y0) , det gFS negative functions Ej(Z) ≥ 0 on P(H): α=0

X α β where Z = X + iY . In (pα, να) coordinates this becomes: Ej(Z) = (Ej)α,β Z Z , α,β D−1 ˜ Y 0 0 ∗ δ(Z − Z0) = 2δ(pα − pα)δ(να − να) , where Eα,β = Eβ,α and whose sum is always equal to Pn α=1 the constant function 1: j=1 Ej(Z) = 1. For POVMs, observables O are also real quadratic functions: where the coordinate-invariant nature of the functionals P Pp[O] is manifest. Mixed states ρ = j λj |λji hλj| are D X α β thus convex combinations of these Dirac-delta function- O(Z) = Oα,βZ Z , P ˜ als: pmix(Z) = j λjδ(Z − Zj), where Zj ↔ |λji. The α,β=1 distribution q(Z) is called a geometric quantum state. ∗ where Oα,β = Oα,β. Geometric quantum states. Complex projective spaces, such as as P(H), have a preferred metric gFS and a related volume element dVFS, called the Fubini-Study volume element. The details surrounding these go beyond our present purposes. Fortunately, it is sufficient to give dVFS’s explicit form in the coordinate system we use for concrete calculations. The latter is the “probability + Figure 1. If the barycenter of two density matrices is the √ same, as for ρ and ρ , the standard formalism of quantum phase” coordinate system: Zα = p eiνα : A B α mechanics regards them as identical, while they are not. The D−1 difference is seen using the geometric formalism. PA and PB Y α are two Dirac-delta distributions peaked on different antipodal dV = pdet g dZαdZ FS FS points of the Bloch sphere. α=0 D−1 Y dpαdνα = . Density matrix formalism. Geometric quantum states 2 are logically prior to density matrices, essentially for two α=1 reasons. First, the latter can be computed from the former, as follows: In GQM, quantum states convey information about the ways real values are associated to observables and POVMs. q β ρ = P [ZαZ ] (1) Thus, it is quite natural to describe the latter as real- αβ q Z valued functionals P [O] that, via a probability measure α β = dVFS q(Z) Z Z . (2) on P(H), associate a number to an element O of the P(H) algebra A of observables: Second, the correspondence is many-to-one: many differ- Z ent geometric quantum states can have the same density Pq[O] = q(Z)O(Z)dVFS , P(H) matrix. Indeed, while the geometric quantum state q(Z) addresses the system’s state, the density matrix addresses where O ∈ A and q(Z) ≥ 0 is the normalized distribution the statistics of POVM outcomes, provided that the state 3

q is q(Z). Going from q(Z) to ραβ actually erases most The goal in maximum entropy estimation is to find HG’s information about the detailed statistics of the quantum maximum given the constraints CN = 0 (normalization) state and preserves only its POVM statistics. α β and Cαβ = 0 (a fixed density matrix Pq[Z Z ] = ραβ): Figure1 illustrates this. Prepare an ensemble with equally-likely copies of states |0i and |1i or of states Z CN := dVFS q(Z) − 1 , (4a) |+i and |−i. While the density matrix is the same for P(H) both, the geometric quantum state q(Z) of the system is Z α β not. Due to this, the density matrix contains no informa- Cα,β := dVFS q(Z)Z Z − ραβ , (4b) tion about the physical realization of the ensemble of a P(H) quantum system. For a more thorough discussion on this for all α and β. point see Ref. [22]. Applying Lagrange multipliers for the maximization, we This observation is particularly relevant in information define: processing, as found in quantum computing. There, one X is not only interested in measurement outcomes, but must Λ[~λ, γ, q(Z)] = H [q(Z)] − λ C − γC . (5) understand, predict, and control how a quantum system G αβ αβ N α,β changes its behavior due to external signals. By focusing only on the density matrix we lack an appropriate charac- Variation with respect to the Lagrange’s multipliers γ terization of the ensemble producing the pure states and, and {λαβ} impose the constraints Cαβ = 0 for all α, β therefore, miss a wealth of information that can help in † and CN = 0. Note that since ρ = ρ not all λs are linearly manipulating quantum states. ∗ independent as λαβ = λβα. The functional derivative With this set up and the key issues of state represen- with respect to the probability distribution q(Z) gives: tation laid out, we are now ready to imagine a state X β preparation protocol and a tomographically-complete [24] log q(Z) = −(1 + γ) − ZαZ λ . set of POVMs that allows to recover the density matrix αβ α,β from measurements. As we argued, this information is insufficient to recover the geometric quantum state. To This eventually turns into the following maximum entropy fill the gap we invoke the maximum entropy paradigm, estimate: which provides a principled and well-understood proce- PD α β − λαβ Z Z dure with which to infer a probability distribution from e α,β=1 partial knowledge. qME(Z) = (6a) Z(λαβ) Z PD α β − λαβ Z Z Z(λαβ) := dVFS e α,β=1 . (6b) III. MAXIMUM GEOMETRIC QUANTUM P(H) ENTROPY In this, we used Z = eγ+1. This fixes the probability Maximum entropy estimation [25–27] is applied ubiqui- distribution’s functional form to be exponential. tously in science and engineering. Its predictive power To guarantee that qME satisfies Eq. (4)’s constraints, the has enjoyed marked empirical successes, especially when Lagrange multipliers λj must satisfy the following nonlin- addressed to systems consisting of large numbers of de- ear equations: grees of freedom or when confronted with much missing ∂ log Z(λαβ) microscopic information. Here, we adapt this well-known − = ραβ , (7) technique to geometric quantum mechanics. ∂λαβ The relevant entropy is the equivalent of Shannon’s dif- for all α, β = 1,...,D. Equivalently, defining F (λ ) = ferential entropy [27] on a manifold. We call it geometric αβ − log Z(λ ) and using the Legendre transform of H : quantum entropy. Given a continuous and smooth proba- αβ G bility distribution q(Z) it is defined: D X Z Γ(λαβ) = λαβραβ − F (λαβ) , HG = − dVFS q(Z) log q(Z) . (3) α,β=1 P(H) one can straightforwardly show that Eq. (7) is equivalent This was first considered, to the best of our knowledge, to: by Ref. [28]. Its role in quantum thermodynamics and statistical mechanics was recently explored by the authors ∂Γ = 0 . (8) in Ref. [23]. Here, we do not dwell on the conceptual ∂λαβ implications of using HG in place of the von Neumann quantum entropy S. We simply note that HG is a gener- for all α, β = 1,...,D. alized version of the latter, to which it reduces when q(Z) Moreover, one can also show that the functional Γ(λαβ) is a convex sum over Dirac-delta distributions. is everywhere convex, thus proving that if a stationary 4 point—Eq. (8)—exists, it must be an absolute minimum. Given knowledge of only the density matrix, we have no This statement then transfers to the entropy maximum input about ~µ, which results in ~µ = 0 and Σ = ρ. ~ since, by solution Eq. (6), we have Γ(λαβ) = −HG[q(λ)].

Convexity of the functional Γ(λαβ) is established simply by showing that its Hessian Γ(2) matrix is positive αβ;˜αβ˜ semideﬁnite. This is done by noticing that: IV. CONCLUSIONS

2 (2) ∂ Γ Γαβ;µν = (9) ∂λαβ∂λµν The geometric quantum state q(Z) is a probability dis- h β ν i h βi h ν i = ZαZ ZµZ − ZαZ ZµZ tribution on a system’s pure-state manifold P(H). It E E E determines a system’s state at a deeper and more re- h α β µ ν i fined level than its density matrix ρ. The latter only = E Z Z Z Z − ραβρµν . contains information about POVM statistics, which are As with any covariance matrix, this is positive semidefi- essentially pairwise correlations. And so, to every geo- nite. Thanks to this reformulation and despite the fact metric quantum state q(Z) there corresponds a unique that a closed solution to Eq. (7) is not known, good itera- density matrix ρq that describes q(Z)’s POVM statistics; tive procedures that provide sufficiently-accurate solutions see Eq. (2). However, the correspondence is many-to- can be employed [29]. For example, gradient-descent opti- one: Given a density matrix ρ, there are various q(Z) mization can be directly leveraged to find the minimum of compatible with ρ’s POVM statistics. One consequence Γ(λ ). (For more in-depth discussion see Refs. [30–32].) stems from results presented in the two companion works, αβ Refs. [22, 23], where it is argued in fuller detail that the Taking advantage of its particularly simple form, we can geometric quantum state is a richer characterization of a extract an analytical form for Z(λαβ). AppendixA pro- quantum system’s physical state—one that goes beyond vides the details; here, we simply quote the result. First, its POVMs’ statistics. Here, we answered a complemen- since λαβ is a Hermitian and positive-definite matrix we tary, operational question, What is the best guess of q(Z), know we can always diagonalize it by means of a unitary assuming knowledge of the density matrix ρ? † ~ matrix U: UλU = λD, where λD = (λ0, . . . , λD−1). By reversing this relation, the unitary matrix diagonalizing λαβ provides a set of D functions such that λj = fj(λαβ). With these definitions in mind:

D−1 X e−fj (λαβ ) Z (λαβ) = Q . [fk(λαβ) − fj(λαβ)] k=0 j6=k By simple algebraic manipulations, the maximum-entropy estimation qME(Z) can be written as a multivariate Gaus- α ~ β ~ ∗ sian. Using vector notation Z = Z and Z = Z and Figure 2. Maximum-entropy geometric quantum state in β calling ~µ = hZαi and ~µ∗ = hZ i, we find: probability + phase coordinates (p, φ) determined by functional Eq. (11). Here, we started with density matrix ρ given by ∗ 1 − 1 (Z~ ∗−~µ∗)Σ−1(Z~−~µ) ρ00 = 1 − ρ11 = 0.45 and ρ01 = ρ10 = 0.2 − 0.3i. q (Z) = e 2 (10) ME Q Z − 1 (Z~ ∗−~µ∗)Σ−1(Z~−~µ) Q = dVFS e 2 , (11) This was answered using the constrained maximum en- P(H) tropy paradigm, an ubiquitous estimation technique in science and engineering. We showed that the entropy h α βi where Σ is the covariance matrix Σαβ = E Z Z − functional appropriate to quantum settings is the geomet- α β α α ric quantum entropy of Eq. (3), a transposition into the E [Z ] E[Z ], ~µ is the average vector µ = E [Z ], and quantum-state manifold of the classical notion of entropy they are linear combinations of the λαβ. Fixing Σ and of a phase space distribution. Due to Eq. (2)’s relation ~µ is equivalent to fixing all the λαβ and we can use this between the density matrix ρ and the geometric quantum parametrization to fix the Lagrange multipliers in a func- state q(Z), the resulting distribution q(Z) has Eq. (9)’s tionally simpler way: form of a multivariate Gaussian in which the covariance matrix Σ is directly related to the density matrix ρ. Given µα = [Zα] , E the Gaussian form, fixing its parameters amounts to an h α βi α β Σαβ = E Z Z − E [Z ] E[Z ] appropriate characterization of its average vector ~µ and covariance matrix Σ. Using this functional parametriza- = ραβ − µαµβ . tion we derived the maximum entropy estimate for a 5 geometric quantum state q(Z) with density matrix ρ: Gier, Samuel Loomis, Ariadna Venegas-Li for helpful discussions and the Telluride Science Research Center for its −1 − 1 Z~ ∗ρ−1Z~ qME(Z) = Q e 2 , with hospitality during visits. This material is based upon work Z supported by, or in part by, a Templeton World Char- − 1 Z~ ∗ρ−1Z~ Q = dVFS e 2 , ity Foundation Power of Information Fellowship, FQXi P(H) Grant FQXi-RFP-IPW-1902, and U.S. Army Research Laboratory and the U. S. Army Research Office under when det ρ 6= 0. contracts W911NF-13-1-0390 and W911NF-18-1-0028.

ACKNOWLEDGMENTS DATA AVAILABILITY STATEMENT F.A. thanks Marina Radulaski, Davide Pastorello, and Davide Girolami for discussions on the geometric for- The data that support the ﬁndings of this study are malism of quantum mechanics. F.A. and J.P.C. thank available from the corresponding author upon reasonable Dhruva Karkada for his help with the example, David request.

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Appendix A: Calculating the Partition Function

Recall:

Z P α β − λαβ Z Z Z (λαβ) = e α,β dVFS . P(H)

Since λαβ are the Lagrange multipliers of Cαβ we chose them to be Hermitian as they are not all independent. Thus, we can always diagonalize them with a unitary matrix:

X † UγαλαβUβ = lγ δγ . αβ

P α This allows us to deﬁne auxiliary integration variables Xγ = α UγαZ . Thanks to these, we express the quadratic † form in the exponent of the integrand using that (U U)αβ = δαβ:

X α β ZλZ = Z λαβZ αβ ˜ X X α β = Z δαα˜λα˜β˜δββ˜Z αβ α˜β˜ X X X β = ZαU † U λ U † U Z αa aα˜ α˜β˜ βb˜ bβ αβ α˜β˜ ab X 2 = |Xa| la . a Moreover, recalling that the Fubini-Study volume element is invariant under unitary transformations, we can simply iνa QD−1 dqadνa adapt our coordinate systems to Xa. And so, we have Xa = qae . This gives dVFS = k=1 2 . We get to the following simpler functional:

D−1 Z PD−1 − laqa Y dqadνa Z (λ ) = e a=0 αβ 2 P(H) k=1 D−1 Z P (2π) − laqa Y = e a dq . 2D−1 a ∆D−1 a Now, we are left with an integral of an exponential function over the D − 1-simplex. We can use Laplace transform trick to solve this kind of integral:

Z D−1 D−1 ! Y −lkqk X ID−1(r) := e δ qk − r dq1 . . . dqD−1 ∆D−1 k=0 k=0 Z ∞ −zr ⇒ I˜D−1(z) := e ID−1(r)dr , 0 n Y (−1)k Iñ(z) = (lk + z) k=0 n n(n+1) Y 1 = (−1) 2 , z − zk k=0 with zk = −lk ∈ R. The function Iñ(z) has n+1 real and distinct poles: z = zk = −lk. Hence, we exploit the partial fraction decomposition of Iñ(z), which is:

n n n(n+1) Y 1 n(n+1) X Rk (−1) 2 = (−1) 2 , z − zk z − zk k=0 k=0 where: Rk = (z − zk)I˜n(z) z=zk n n(n+1) Y (−1) 2 = . zk − zj j=0, j6=k

Exploiting linearity of the inverse Laplace transform plus the basic result:

1 L−1 (t) = e−atΘ(t) , s + a where: ( 1 t ≥ 0 Θ(t) = . 0 t < 0

We have for:

−1 In(r) = L [I˜n(z)](r) n X zkr = Θ(r) Rke . k=0 And so:

Z = ID−1(1) D−1 X e−lk = . QD−1 k=0 j=0, j6=k(lk − lj)

Now, consider that la are linear functions of the true matrix elements:

la = fa(λαβ) X † = UaαλαβUβa . αβ

We arrive at:

e−fk(λαβ ) Z(λ ) = . αβ QD−1 j=0, j6=k(lk(λαβ) − lj(λαβ))