Quantum Probability Representation

• Quantum Probability Representation is a framework that maps quantum states onto normalized, classical-like probability distributions (tomograms) via integral transforms.
• It employs symplectic, center-of-mass, and spin tomographic constructions to fully reconstruct density operators across continuous and discrete systems.
• Dynamic symmetries and entropic uncertainty relations underpin the approach, linking quantum information theory with experimental quantum state reconstruction.

Quantum probability representation re-expresses the formalism of quantum mechanics in terms of families of classical-like, fair probability distributions—termed tomograms—that fully characterize quantum states and observables for both continuous and discrete systems. This approach replaces the conventional wave function or density operator picture with an invertible mapping onto probability distributions constructed via integral transforms incorporating dynamical symmetries and group representations. The framework enables the analysis of quantum dynamics, uncertainty relations, contextuality, and entanglement directly in the language of probabilities, facilitating connections with information theory and experimental quantum state reconstruction.

1. Fundamental Principles and Framework

The quantum probability (tomographic) representation maps a quantum state—typically described by a wave function $|\psi\rangle$ or density matrix $\rho$—onto a parameterized family of genuine probability distributions (tomograms). Each tomogram is a real, nonnegative, normalized function over outcomes of appropriately chosen observables, and the full set uniquely determines the quantum state.

• For continuous variables, the symplectic tomogram is defined as:

$$M(X, u, v) = \operatorname{Tr} \left[ \rho\, \delta(X - u\cdot\hat{q} - v\cdot\hat{p}) \right]$$

where $X = u\,q + v\,p$, with $(u, v)\in\mathbb{R}^2$ parameterizing rotations, scalings, and shifts in phase space [(Man'ko et al., 2011), Eq. (14)].

• The density operator can be reconstructed via an inverse Radon (quantized Fourier) transform:

$$\rho = \int M(X,u,v)\, e^{i(X - u\cdot\hat{q} - v\cdot\hat{p})} dX\,du\,dv$$

• For finite-dimensional (spin) systems (e.g., qubits and qudits), the spin tomogram is:

$$w(m, u) = \langle m | u\,\rho\,u^{\dagger} | m \rangle$$

with $u\in U(N)$ parameterizing unitary transformations of basis; the probability vector $w(m, u)$ contains complete information about $\rho$ [(Man'ko et al., 2011), Eq. (104)].

Each of these constructions acts as an invertible mapping between the density matrix formalism and a (possibly overcomplete) set of classical probability distributions.

2. Tomographic Constructions: Symplectic, Center-of-Mass, and Spin Representations

Symplectic Tomography

Symplectic tomography generalizes classical Radon transforms on phase-space densities to quantum mechanics by averaging over projections of $\rho$ along rotated and scaled quadratures:

$$M(X,u,v) = \langle \delta(X-u\hat{q}-v\hat{p}) \rangle$$

The dynamic symmetry underlying this approach is the (multimode) symplectic group $Sp(2n,\mathbb{R})$, which ensures that canonical transformations correspond to group actions on tomogram parameters.

Center-of-Mass Tomography

The center-of-mass tomogram for multimode systems condenses the description to collective variables:

$$M_{cm}(X, u, v) = \operatorname{Tr}[\rho\, \delta(X - \sum_k u_k\hat{q}_k + v_k\hat{p}_k)]$$

This is particularly useful for assessing properties of subsystems or collective excitations [(Man'ko et al., 2011), Eq. (54)].

Spin (Unitary) Tomography

For discrete systems, tomographic probability representation is constructed as:

$w(m, u) = \langle m | u \rho u^\dagger | m \rangle$

where $m$ indexes the spin projection and $u$ runs over $U(N)$. The full set $\{w(m,u)\}$ as $u$ varies suffices to reconstruct any quantum state of the system [(Man'ko et al., 2011), Eq. (104)].

The quantum Fourier transform is introduced:

$F_{jk} = \frac{1}{\sqrt{N}} \exp(2\pi i\, jk / N)$

permitting the definition of complementary tomograms and bases necessary for expressing generalized uncertainty relations [(Man'ko et al., 2011), Eq. (114)].

3. Dynamic Symmetries and Group-Theoretic Structure

The probability representation leverages group-theoretic symmetries:

• Continuous variables: The symplectic group $Sp(2,\mathbb{R})$ and its multimode extensions govern canonical transformations, with rotations and squeeze operations represented by unitary operators acting on $(\hat{q}, \hat{p})$.
• Discrete variables: The unitary group $U(N)$ emerges as the dynamic symmetry for systems such as spins, where state transformations correspond to varying the measurement (tomogram) basis [(Man'ko et al., 2011), Section 4].

This group-theoretic underpinning guarantees that the mapping from operator language to the language of tomograms respects the structural requirements of quantum theory, including covariance under transformations and consistency with the uncertainty principle.

4. Entropic and Information-Theoretic Inequalities

One of the central developments is the translation of the uncertainty principle into the tomographic framework using entropic measures.

• Continuous variables: For symplectic tomograms, the tomographic (Shannon) entropy is

$$S(\mu, \nu) = - \int M(X,\mu, \nu) \ln M(X, \mu, \nu)\, dX$$

and entropic uncertainty relations take the form

$S(0) + S(\pi/2) \geq \ln(\pi e)$

where $S(0)$ and $S(\pi/2)$ correspond to orthogonal quadrature measurements, directly reflecting experimental homodyne data [(Man'ko et al., 2011), Eq. (98)].

• Spin systems: The spin tomogram entropy is

$H(u) = -\sum_m w(m, u) \ln w(m, u)$

and the corresponding Fourier-defined complementary entropy leads to

$H(u) + H(Fu) \geq \ln N$

for a spin-$j$ system of Hilbert space dimension $N = 2j + 1$ [(Man'ko et al., 2011), Eq. (159)].

These entropic bounds directly generalize classic uncertainty relations and can be operationally tested using tomographic data.

The approach also admits the derivation of multipartite inequalities, strong subadditivity relations (e.g., [(Man'ko et al., 2011), Equations (137)-(145)]), and Rényi entropies for analyzing quantum correlations and entanglement.

5. Contextuality and Classical-Quantum Boundaries

The tomographic formalism naturally embeds the investigation of contextuality:

• Contextuality is reflected in the impossibility of constructing a single joint probability distribution whose marginals reproduce all quantum measurement statistics.
• Violation of contextuality inequalities (e.g., pentagram/Klyachko inequalities for qutrits) can be phrased directly in terms of tomographic probabilities (Strakhov et al., 2013).

By expressing both states and observables as tomograms (using, e.g., dequantizer and quantizer operators), one can test for contextuality via violations of correlation and entropic inequalities, mapping these violations directly onto classical probability simplexes.

6. Relationship to Classical Descriptions and Experimental Realization

The mapping from quantum to probability representations bridges the conceptual gap with classical statistical mechanics while maintaining quantum-specific features via additional constraints (such as nonnegativity of the inverse transform and tomogram normalization).

• Operational measurements: The probability representation is experimentally accessible, particularly for continuous variables in quantum optics where optical tomograms are measured directly via homodyne detection (Man'ko et al., 2014).
• State and observable reconstruction: The entire formalism allows reconstructing $\rho$ from experimentally-measured probability distributions via explicit integral transforms.

The classical limit of the quantum tomographic evolution equations reduces to the Liouville equation for phase-space densities, ensuring compatibility with traditional classical statistical mechanics (Korennoy et al., 2015).

7. Applications and Implications

The quantum probability representation supports multiple lines of investigation and application:

• Quantum information theory: Entropic and probability-based measures facilitate the paper of uncertainty, information transmission, and entanglement in a classical-like probabilistic framework.
• Experimental quantum tomography: State reconstruction via homodyne tomography is naturally interpreted in terms of symplectic tomograms, making the theory highly compatible with experimental protocols.
• Dynamical and control systems: The representation is robust under quantum evolution, open-system dynamics, and can encode changes due to quantum channels or decoherence in a way that is accessible to classical intuition.
• General frameworks: The approach generalizes naturally to multimode fields, higher spin systems, and composite quantum systems with arbitrary Hilbert space structure.

In summary, the probability representation of quantum mechanics encodes all quantum information in fair probability distributions (tomograms) constructed via transformations dictated by the relevant dynamical symmetries. This framework enables the derivation of novel entropic uncertainty relations, transparent analysis of contextuality, and provides practical tools for experimental quantum state analysis, while maintaining direct correspondence with both classical statistical mechanics and the conventional operator formalism (Man'ko et al., 2011, Man'ko et al., 2011, Man'ko et al., 2014, Strakhov et al., 2013).