Here is an extended answer that concludes

**Summary** On entropic grounds, gravitational radiative decoherence is similarly irreversible to all other forms of radiative decoherence, and in consequence, Nature's quantum state-spaces are effectively low-dimension and non-flat.

**Update B** For further discussion and references, see this answer to the *CSTheory.StackExchange* question "Physical realization of nonlinear operators for quantum computers."

**Update A** This augmented survey/answer provides an entropically naturalized and geometrically universalized survey of the physical ideas that are discussed by Jan Dereziski, Wojciech De Roeck, and Christian Maes in their article *Fluctuations of quantum currents and unravelings of master equations* (arXiv:cond-mat/0703594v2). Especially commended is their article's "Section 4: Quantum Trajectories" and the extensive bibliography they provide.

By deliberate intent, this survey/answer relates also to the lively (and ongoing) public debate that is hosted on *Gödel's Lost Letter and P=NP*, between Aram Harrow and Gil Kalai, regarding the feasiblity (or not) of scalable quantum computing.

**Naturalized survey of thermodynamics**

We begin with a review, encompassing both classical and quantum thermodynamical principles, following the exposition of Zia, Redish, and McKay's highly recommended *Making sense of the Legendre transform* (*AJP*, 2009). The fundamental thermodynamical relations are specified as

$$
\Omega(E)=e^{\mathcal{S}(E)}\,,
\quad\qquad
Z(\beta)=e^{-\mathcal{A}(\beta)}\,,\\[2ex]
\frac{\partial\,\mathcal{S}(E)}{\partial\,E} = \beta\,,
\quad\qquad
\frac{\partial\,\mathcal{A}(\beta)}{\partial\,\beta}= E\,,\\[3ex]
\mathcal{S}(E) + \mathcal{A}(\beta) = \beta E\,.
$$

In these relations the two conjugate thermodynamic variables

$$
E := \text{total energy}\,, \quad\qquad \beta := \text{inverse temperature}\,,
$$

appear as arguments of four fundamental thermodynamic functions

$$
\mathcal{S} := \text{entropy function}\,,
\quad\qquad
\mathcal{A} := \text{free energy function}\,, \\
{Z} := \text{partition function}\,,
\quad\qquad
{\Omega} := \text{volume function}\,.
$$

Any one of the four thermodynamic potentials $(\mathcal{S},\mathcal{A},Z,\Omega)$ determines the other three via elementary logarithms, exponentials, Laplace Transforms, and Legendre transforms, and moreover, any of the four potentials can be regarded as a function of either of the two conjugate variables.

**Aside** The preceding relations assume that only one quantity is globally conserved and locally transported, namely the energy $E$. When more than one quantity is conserved and transported — charge, mass, chemical species, and magnetic moments are typical examples — then the above relations generalize naturally to a vector space of conserved quantities and a dual vector space of thermodynamically conjugate potentials. None of the following arguments are fundamentally altered by this multivariate thermodynamical extension.

**Naturalized survey of Hamiltonian dynamics**

To make progress toward computing concrete thermodynamic potential functions, we must specify a Hamiltonian dynamical system. In the notation of John Lee's *Introduction to Smooth Manifolds* we specify the Hamiltonian triad $(\mathcal{M},H,\omega)$ in which

$$
\begin{array}{rl}
\mathcal{M}\ \ :=&\text{state-space manifold}\,,\\
H\,\colon \mathcal{M}\to\mathbb{R}\ \ :=&\text{Hamiltonian function on $\mathcal{M}$}\,,\\
\omega\,\llcorner\,\colon T\mathcal{M}\to T^*\mathcal{M}\ \ :=& \text{symplectic structure on $\mathcal{M}$}\,.
\end{array}\hspace{1em}
$$

The dynamical flow generator $X\colon \mathcal{M}\to T\mathcal{M}$ is given by Hamilton's equation

$$\omega\,\llcorner\,X = dH\,.$$

From the standard (and geometrically natural) ergodic hypothesis — that thermodynamic ensembles of Hamiltonian trajectories fill state-spaces uniformly, and that time averages of individual trajectories equal ensemble averages at fixed times — we have ${\Omega}$ given naturally as a level set volume

$$
\text{(1a)}\qquad\qquad\quad\quad
\Omega(E) = \int_\mathcal{M} \star\,\delta\big(E-H(\mathcal{M})\big)\,,
\qquad\qquad\qquad\qquad\qquad
$$

where "$\star$" is the *Hodge star operator* that is associated to the natural volume form $V$ on $\mathcal{M}$ that is given as the maximal exterior power $V=\wedge^{(\text{dim}\,\mathcal{M})/2}(\omega)$. This expression for $\Omega(E)$ is the geometrically naturalized presentation of Zia, Redish, and McKay's equation (20).

Taking a Laplace transform of (1a) we obtain an equivalent (and classically familiar) expression for the partition function $Z(\beta)$

$$
\text{(1b)}\qquad\qquad\qquad
Z(\beta) = \int_\mathcal{M} \star\exp\big({-}\beta\,H(\mathcal{M})\big)\,,
\qquad\qquad\qquad\qquad
$$

The preceding applies to Hamiltonian systems in general and thus quantum dynamical systems in particular. Yet in quantum textbooks the volume/partition functions (1ab) do not commonly appear, for two reasons. The first reason is that John von Neumann derived in 1930 — before the ideas of geometric dynamics were broadly extant — a purely algebraic partition function that, on flat state-spaces, is easier to evaluate than the geometrically natural (1a) or (1b). Von Neumann's partition function is
$$
\text{(2)}\qquad
Z(\beta) = \text{trace}\,\exp{-\beta\,\mathsf{H_{op}}}
\quad\text{where}\quad
[\mathsf{H_{op}}]_{\alpha\gamma} =
\partial_{\,\bar\psi_\alpha}\partial_{\,\psi_\gamma} H(\mathcal{M})\,.
\qquad\qquad
$$
Here the $\boldsymbol{\psi}$ are the usual complete set of (complex) orthonormal coordinate functions on the (flat, Kählerian) Hilbert state-space $\mathcal{M}$. Here $H(\mathcal{M})$ is real and the functional form of $H(\mathcal{M})$ is restricted to be bilinear in $\boldsymbol{\bar\psi},\boldsymbol{\psi}$; therefore the matrix $[\mathsf{H_{op}}]$ is Hermitian and uniform on the state-space manifold $\mathcal{M}$. We appreciate that $Z(\beta)$ as defined locally in (2) is uniform globally iff $\mathcal{M}$ is geometrically flat; thus von Neumann's partition function does not naturally extend to non-flat complex dynamical manifolds.

We naively expect (or hope) that the geometrically natural thermodynamic volume/partition functions (1ab) are thermodynamically consistent with von Neumann's elegant algebraic partition function (2), yet — surprisingly and dismayingly — they are not. Surprisingly, because it is not immediately evident why the geometric particion function (1b) should differ from von Neumann's partition function (2). Dismayingly, because the volume/partition functions (1ab) pullback naturally to low-dimension non-flat state-spaces that are attractive venues for quantum systems engineering, and yet it is von Neuman's partition function (2) that accords with experiment.

We would like to enjoy the best of *both* worlds: the geometric naturality of the ergodic expressions (1ab) and the algebraic naturality of von Neumann's entropic expression (2). The objective of restoring and respecting the mutual consistency of (1ab) and (2) leads us to the main point of this answer, which we now present.

**The main points: sustaining thermodynamical consistency**

**Assertion I** For (linear) quantum dynamics on (flat) Hilbert spaces, the volume function $\Omega(E)$ and partition function $Z(\beta)$ from (1ab) are thermodynamically inconsistent with the partition function $Z(\beta)$ from (2).

Here by "inconsistent" is meant not "subtly inconsistent" but "grossly inconsistent". As a canonical example, the reader is encourage to compute the heat capacity of an ensemble of weakly interacting qubits by both methods, and to verify that the (1ab) predict a heat capacity for an $n$-qubit system that is superlinear in $n$. To say it another way, for strictly unitary dynamics (1ab) predict heat capacities that are non-intensive.

So the second — and most important — reason that the volume/partition functions (1ab) are not commonly given in quantum mechanical textbooks is that strictly unitary evolution on strictly flat quantum state-spaces yields non-intensive predictions for thermodynamic quantities that experimentally *are* intensive.

Fortunately, the remedy is simple, and indeed has long been known: retain the geometric thermodynamic functions (1ab) in their natural form, and instead alter the assumption of unitary evolution, in such a fashion as to naturally restore thermodynamic extensivity.

**Assertion II** Lindbladian noise of sufficient magnitude to spatially localize thermodynamic potentials, when unraveled as non-Hamiltonian (stochastic) quantum trajectories, restores the thermodynamical consistency of the volume/partition functions $(\Omega(E),Z(\beta))$ from (1ab) with the partition function $Z(\beta)$ from (2).

Verifying Assertion II is readily (but tediously) accomplished by the Onsager-type methods that are disclosed in two much-cited articles: Hendrik Casimir's *On Onsager's Principle of Microscopic Reversibility* (*RMP* 1945)
and Herbert Callen's *The Application of Onsager's Reciprocal Relations to Thermoelectric, Thermomagnetic, and Galvanomagnetic Effects* (*PR*, 1948). A readable textbook (among many) that covers this material is Charles Kittel's *Elementary Statistical Physics* (1958).

To help in translating Onsager theory into the natural language of geometric dynamics, a canonical textbook is John Lee's *Introduction to Smooth Manifolds* (2002), which provides the mathematical toolset to appreciate the research objectives articulated in (for example) Matthias Blau's on-line lecture notes *Symplectic Geometry and Geometric Quantization* (1992).

Unsurprisingly, in light of modern findings in quantum information theory, the sole modification that naturality and universality require of Onsager's theory is this: the fluctuations that are the basis of Onsager's relations must be derived naturally from unravelled Lindblad processes, by the natural association of each Lindbladian generator to an observation-and-control process.

We note that it is neither mathematically natural, nor computationally unambiguous, nor physically correct, to compute Onsager fluctuations by non-Lindbladian methods. For example, wrong answers are obtained when we specify Onsager fluctuations as operator expectation fluctuations, because this procedure does not account for the localizing effects of Lindbladian dynamics.

Concretely, the fluctuating quantities that enter in the Onsager formulation are given as the data-streams that are naturally associated to Lindbladian observation processes … observation processes that are properly accounted in the overall system dynamics, in accord with the teaching of quantum information theory. Thereby Onsager's classical thermodynamical theory of global conservation and local transport processes straightforwardly naturalizes and universalizes — via the mathematical tool-set that quantum information theory provides — as a dynamical theory of the observation of natural processes.

**Physical summary** Consistency of the geometrically natural thermodynamic functions (1ab) with the algebraically natural thermodynamic function (2) is restored because the non-unitary stochastic flow associated to unraveled Lindbladian noise reduces the effective dimensionality of the quantum state-space manifold, and also convolutes the quantum state-space geometry, in such a fashion that as to naturally reconcile geometric descriptions of thermodynamics (1ab) with von Neumann-style algebraic descriptions of thermodynamics (and information theory) on Hilbert state-spaces (2).

**Assertion III** The thermodynamic consistency requires that, first, quantum dynamical flows be non-unitary and that, second, the resulting trajectories be restricted to non-flat state-spaces of polynomial dimensionality.

We thus appreciate the broad principle that quantum physics can make sensible predictions regarding physical quantities that are globally conserved and locally transported only by specifying non-unitary dynamical flows on non-flat quantum quantum spaces.

**Duality of classical physics versus quantum physics** The above teaching regards "classical" and "quantum" as well-posed and mutually consistent limiting cases of a broad class of naturalized and universalized Hamiltonian/Kählerian/Lindbladian dynamical frameworks. For practical purposes the most interesting dynamical systems are intermediate between fully classical and fully quantum, and the thrust of the preceding analysis is that the thermodynamical properties of these systems are naturally and universally defined, calculable, and observable.

**Duality of fundamental physics versus applied physics** The fundamental physics challenge of constructing a thermodynamically and informatically consistent description of non-unitary quantum dynamics on non-flat complex state-spaces — a challenge that is widely appreciated as difficult and perhaps even impossible — is appreciated as dual to the practical engineering challenge of efficiently simulating noisy quantum system dynamics … a challenge that is widely appreciated as feasible.

**Remarks upon gravitational decoherence** The above analysis establishes that decoherence associated to gravitational coupling — and more broadly the ubiquity of the *superradiant* dynamics that is associated to every bosonic field of the vacuum — and further supposing this decoherence to be "irreversible" (in Scott's phrase), would have the following beneficent implications:

- the naturality and universality of thermodynamics is thereby preserved, and
- quantum trajectories are
*effectively* restricted to low-dimension non-flat state-spaces, and
- the efficient numerical simulation of generic quantum systems is thus permitted.

From a fundamental physics point-of-view, the converse hypothesis is attractive:

**Kählerian hypothesis** Nature's quantum state-spaces are generically low-dimension and non-flat in consequence of irreversible decoherence mechanisms that are generically associated to bosonic vacuum excitations.

**Conclusions**

As with the *ergodic hypothesis*, so with the Kählerian hypothesis, in the sense that regardless of whether the Kählerian hypothesis is fundamentally true or not — and regardless of whether gravitation radiation accounts for it or not — for practical quantum systems engineering purposes experience teaches us that the Kählerian hypothesis *is* true.

The teaching that the Kählerian hypothesis is *effectively* true is good news for a broad class of 21st century enterprises that seek to press against quantum limits to speed, sensitivity, power, computational efficiency, and channel capacity … and it is very good news especially for the young mathematicians, scientists, engineers, and entrepreneurs who hope to participate in creating these enterprises.

**Acknowledgements** This answer benefited greatly from enjoyable conversations with Rico Picone, Sol Davis, Doug and Chris Mounce, Joe Garbini, Steve Flammia, and especially Aram Harrow; any remaining errors and infelicities are mine alone. The answer is also very largely informed by the ongoing debate of Aram Harrow with Gil Kalai, regarding the feasibility (or not) of scalable quantum computing, that has been hosted on the web page *Gödel's Lost Letter and P=NP*, regarding which appreciation and thanks are extended.

This post imported from StackExchange Physics at 2014-07-24 15:46 (UCT), posted by SE-user John Sidles