Beyond Density Matrices: Geometric Quantum States
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arXiv:2008.XXXXX Beyond Density Matrices: Geometric Quantum 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 matrix. 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 operator-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 quantum state manipulation and one in quantum thermodynamics. PACS numbers: 05.45.-a 89.75.Kd 89.70.+c 05.45.Tp Keywords: Quantum Mechanics, Geometric Quantum Mechanics 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 state space—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 motion 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 probability distribution 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 density matrix, 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 statistics 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 observables, 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 basis 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 qubit, for example, one can always use √ √ and Pq[O] ∈ R. amplitude-phase coordinates Z = ( 1 − p, peiν ). An observable 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 (POVMs) {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” coordinate system 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 canonical ensemble: 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.