Quantum States & Qubits Overview

• Quantum states and qubits are the mathematical and physical frameworks that describe systems in quantum mechanics, characterized by superposition and entanglement.
• They are represented in Hilbert space using basis states, density matrices, and alternative tools like Wigner functions and probability simplexes to capture their evolution and gate operations.
• Practical implementations include resonant Rabi oscillations, deterministic state transfer protocols, and hybrid encodings, which are essential for scalable and robust quantum experiments.

A quantum state is the complete mathematical description of a physical system in quantum mechanics, with a qubit representing the fundamental unit of quantum information. Qubits are physical systems capable of existing in either of two distinct quantum states or in any superposition thereof. Modern qubit realizations span isolated two-level systems, multiply-degenerate atomic manifolds, photonic encodings, and hybrid discrete-continuous degrees of freedom. Understanding the structure, manipulation, and resource properties of quantum states and qubits is core to quantum information theory, computation, and experiment.

1. Structure of Quantum States and Formal Definitions

A conventional qubit is a quantum system with Hilbert space dimension two, typically denoted by orthonormal basis states $\{|0\rangle,|1\rangle\}$ separated by an energy $E_1-E_0=\hbar\omega_0$. An arbitrary pure qubit state is a complex linear combination $|\psi\rangle = \alpha|0\rangle+\beta|1\rangle$, $|\alpha|^2+|\beta|^2=1$. For a mixed state, the density matrix $\rho$ acts on the same Hilbert space, admitting a representation as a convex combination of projectors onto pure states.

More generally, a "degenerate" qubit can be formed from two energy levels, each with $n$-fold degeneracy, with logical subspace spanned by pairs $\{|0,m_i\rangle,|1,m_i\rangle\}_{i=1}^n$. Transition selectivity is enforced using angular-momentum selection rules and appropriately polarized laser fields, ensuring coherence between matched sublevels without mixing or population leakage among different $m_i$ (Bao et al., 6 Oct 2025).

Representation of quantum states extends beyond Hilbert space vectors. The 8-dimensional probability simplex and generalized phase-space methods provide alternative frameworks for encoding qubit states, their dynamics, and gate operations (Yavuz et al., 2023, Raussendorf et al., 2019).

2. Dynamical Evolution and Quantum Gates

Single-qubit gate operations correspond to unitary transformations on the two-level system, implementable as resonant Rabi oscillations under suitably chosen classical driving—for example, a laser field with $\vec{E}$ polarized along the quantization axis. The time-dependent state coefficients evolve as

$$\begin{align*} \alpha(t) &= \alpha(0)\cos(\Omega t/2), \ \beta(t) &= -i \alpha(0)\sin(\Omega t/2), \end{align*}$$

where $\Omega$ is the Rabi frequency, characterizing the strength of the driving field (Bao et al., 6 Oct 2025). For degenerate manifolds under $\pi$-polarization and $J_0 = J_1 = 1/2$, all $m_i$ transitions acquire identical $\Omega$, yielding $n$ independent two-level subsystems.

Static fields (e.g., Zeeman effect) lift degeneracies, resulting in differential phase accumulation and leading to fidelity penalties in average gate performance scaling as $O[(\mu_B B/\hbar\Omega)^2]$. Gate errors can thus be mitigated by increasing driving strength or by deliberately lifting degeneracies to stabilize logical subspaces (Bao et al., 6 Oct 2025).

Two-qubit gates such as the controlled-Z (CZ) can be realized via exchange-type interactions that selectively couple single-excitation subspaces, e.g.,

$$H_\text{int} = \hbar g (|0\rangle_A|1\rangle_B\langle 1|_A\langle 0|_B + h.c.)$$

Evolution under this Hamiltonian, combined with local phase rotations, yields the full CZ operation.

3. Phase Space, Representations, and Simulability

Quantum states may equivalently be described via generalized Wigner functions—a type of quasiprobability distribution over finite phase space. Each $n$-qubit state can be expanded as

$$\rho = \sum_{(\Omega,\gamma)\in\mathcal{V}} W_\rho(\Omega,\gamma) A_\Omega^\gamma$$

where each $A_\Omega^\gamma$ is a "phase-point" operator constructed from Pauli labels, and $W_\rho$ is the generalized Wigner function (Raussendorf et al., 2019).

Key features include:

• Stabilizer mixtures: Every convex combination of $n$-qubit stabilizer states is positively represented ($W_\rho\geq 0$ for all points).
• Non-stabilizer, but still positive: For $n\geq2$, there exist states outside the stabilizer polytope with $W_\rho\geq 0$.
• Simulability: Clifford+Pauli circuits on states with $W_\rho\geq 0$ admit efficient classical sampling ("weak simulation"). Negativity in $W_\rho$ is a necessary resource for quantum computational advantage; sampling cost in classical simulation scales as the square of the phase-space robustness $\mathfrak{R}(\rho)$ of the state.

This generalizes the known "magic-state" resource theory to fully capture the cost of simulating quantum circuits via phase-space methods, delineating the boundary between classically tractable and intractable quantum evolutions (Raussendorf et al., 2019).

4. Hybrid Quantum Encodings and State Transfer

Qubit states can interconvert between discrete-variable (DV), continuous-variable (CV), and more exotic encodings in both theory and experiment. Protocols have been developed for deterministic, universal transfer of arbitrary quantum states from a CV mode (e.g., oscillator) into a small register of qubits using sequences of Rabi-type interactions (Hastrup et al., 2021). The error from truncating the infinite-dimensional Hilbert space to a finite $N$-qubit register decays exponentially with $N$, and the protocol is robust to dephasing and amplitude damping.

In circuit QED architectures, deterministic state transfer between single-photon-state (SPS) qubits and coherent-state (CS) qubits is achievable via sequences of collective dispersive interactions, swaps, and conditional rotations orchestrated through a superconducting flux qutrit. Operation times can be rendered independent of $n$, and fidelities $>90\%$ are within reach experimentally (Su et al., 2021).

Additionally, topological-to-conventional qubit state transfer has been demonstrated in theoretical schemes leveraging joint parity measurements via the Aharonov–Casher effect. This enables coherent teleportation and entanglement between nonlocal (Majorana wire) and conventional (semiconductor DQD) qubits, with fidelity limited chiefly by conventional qubit relaxation/readout errors rather than topological-state stability (Bonderson et al., 2010).

5. Quantum State Tomography and Measurement Frameworks

Qubit tomography conventionally requires measurement in three Pauli bases. However, alternative frameworks, such as the symmetric informationally-complete (SIC)-POVM encoding, allow the full state to be specified by a four-vector of probabilities within a tetrahedral simplex in $\mathbb{R}^3$ (Murzin et al., 2024). Notably, within the "quantum potato chip" region—the intersection of a two-dimensional separable surface with the insphere—it is possible to fully reconstruct the qubit state using only two binary projective measurements, as the SIC vector reduces to the tensor product of two independent binary distributions. Outside this region, at least three measurements are necessary to avoid ambiguities in state assignment. This offers an operational reduction of measurement overhead for certain one-parameter state families but does not generalize to the whole Bloch sphere.

6. Entanglement, Many-Qubit States, and Resource Counting

Multipartite qubit states enable the creation of highly entangled resource states, such as GHZ states,

$$|\mathrm{GHZ}_N\rangle = \frac{1}{\sqrt{2}}(|00\dots 0\rangle + |11\dots 1\rangle).$$

Protocols leveraging networked cavities, three-level qutrits, and only virtual excitation of auxiliary states can deterministically and efficiently generate GHZ states distributed across multiple registers, with operation times independent of the system size and robust suppression of decoherence due to virtual occupation of higher-energy levels (Liu et al., 2020). Fractional revivals and the generation of multiple-cat states (superpositions of spin-coherent packets) are observed in systems coupling a single qubit to an $N$-qubit spin ensemble, where effective Kerr-type Hamiltonians generate squeezing and manifest measurement signatures in the phase-space quasi-probability distribution (Dooley et al., 2014).

Crucially, the finite precision of quantum state specification sets physical limits on the number of bits needed to describe an $N$-qubit state. For generic pure states in $2^N$-dimensional Hilbert space, $2^{N+1}-2$ real parameters are required. A physical cutoff (the minimal phase-space cell $\Delta \Omega=\hbar^{3N}$) renders the set of physically distinguishable states finite, with the effective classical bit-count scaling only linearly in $N$ in thermal equilibrium—restoring an information-theoretic parallel with classical discrete systems (Drossel, 2016).

7. Alternative Classical Encodings and the Probability Simplex

Quantum states can be mapped onto points in higher-dimensional classical probability simplexes, e.g., an 8-dimensional simplex for a single qubit using three classical probabilistic bits. This approach restores all one-qubit unitary dynamics and gate transformations as nonlinear affine maps on the simplex, preserving full equivalence to the Hilbert-space picture. For $N$ qubits, the tensor product structure embeds quantum states as probability vectors in a $2^{3N}$-dimensional simplex, enabling certain quantum operations (e.g., CNOT) as explicit block-stochastic maps (Yavuz et al., 2023). While efficient only for few qubits, such mappings offer fresh perspectives on the classical-quantum boundary and simulateability.

• "Do Qubit States have to be non-degenerate two-level systems?" (Bao et al., 6 Oct 2025)
• "Phase space simulation method for quantum computation with magic states on qubits" (Raussendorf et al., 2019)
• "Universal unitary transfer of continuous-variable quantum states into a few qubits" (Hastrup et al., 2021)
• "Transferring quantum entangled states between multiple single-photon-state qubits and coherent-state qubits in circuit QED" (Su et al., 2021)
• "Even-Balanced States Excitation in Two-Qubits System" (Koral et al., 2024)
• "Fractional Revivals, Multiple-Cat States and Quantum Carpets in the Interaction of a Qubit with $N$ Qubits" (Dooley et al., 2014)
• "Mapping of Quantum Systems to the Probability Simplex" (Yavuz et al., 2023)
• "Generation of quantum entangled states of multiple groups of qubits distributed in multiple cavities" (Liu et al., 2020)
• "Topological quantum buses: coherent quantum information transfer between topological and conventional qubits" (Bonderson et al., 2010)
• "How many bits specify a quantum state?" (Drossel, 2016)
• "Quantum Potato Chips" (Murzin et al., 2024)