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Introduction and Hilbert Space Formalism
In quantum mechanics, the state of a physical system is described by a state vector (also called a wavefunction or ket) in an abstract complex vector space known as a Hilbert space
web.eecs.utk.edu. This principle was formalized in the early 20th century by pioneers such as Paul Dirac and John von Neumann, who unified the matrix and wave formulations of quantum theory into a Hilbert space framework
reddit.com. In this formulation, each isolated system is associated with a Hilbert space H\mathcal{H}H, and the system’s state corresponds to a (normalized) vector ∣ψ⟩|\psi\rangle∣ψ⟩ in H\mathcal{H}H. Dirac introduced the convenient bra–ket notation to represent state vectors as kets (e.g. ∣ψ⟩|\psi\rangle∣ψ⟩) and their duals as bras (e.g. ⟨ψ∣\langle \psi|⟨ψ∣), greatly simplifying calculations
en.wikipedia.org. Von Neumann’s 1932 treatise, Mathematische Grundlagen der Quantenmechanik, rigorously cast quantum mechanics in terms of linear operators on Hilbert spaces
people.cs.rutgers.edu, introducing key notions like the density operator and laying the mathematical foundation still used today. In modern terms, Postulate 1 of quantum mechanics states that an isolated quantum system is completely described by a unit-length state vector in a Hilbert space
A Hilbert space H\mathcal{H}H is a complete vector space with an inner product, allowing one to define lengths and angles between vectors. For quantum states, the inner product induces the notion of probability amplitudes and normalization. If ∣ψ⟩,∣ϕ⟩∈H|\psi\rangle, |\phi\rangle \in \mathcal{H}∣ψ⟩,∣ϕ⟩∈H, their inner product ⟨ϕ∣ψ⟩\langle \phi | \psi \rangle⟨ϕ∣ψ⟩ is a complex number. The norm ⟨ψ∣ψ⟩\sqrt{\langle \psi|\psi\rangle}⟨ψ∣ψ⟩ gives the length of ∣ψ⟩|\psi\rangle∣ψ⟩, and physical state vectors are required to be normalized to unity (so that total probability is 1). Two vectors that differ only by a nonzero complex scalar correspond to the same physical state, since multiplying by a phase does not change any observable predictions
web.eecs.utk.edu. In other words, quantum states correspond to rays in projective Hilbert space (the vector ∣ψ⟩|\psi\rangle∣ψ⟩ and eiθ∣ψ⟩e^{i\theta}|\psi\rangleeiθ∣ψ⟩ represent the same state)
web.eecs.utk.edu. We therefore typically constrain state vectors to be normalized and ignore any overall phase. The use of bra–ket notation is ubiquitous: a ket ∣ψ⟩|\psi\rangle∣ψ⟩ represents a column vector in H\mathcal{H}H, while a bra ⟨ψ∣\langle \psi|⟨ψ∣ represents its Hermitian-conjugate row vector in the dual space
en.wikipedia.org. The inner product is written as ⟨ϕ∣ψ⟩\langle \phi | \psi \rangle⟨ϕ∣ψ⟩, and an outer product ∣ψ⟩⟨ϕ∣|\psi\rangle\langle \phi|∣ψ⟩⟨ϕ∣ represents a linear operator (matrix) that maps ∣ϕ⟩|\phi\rangle∣ϕ⟩ to ∣ψ⟩|\psi\rangle∣ψ⟩. An operator acting on state vectors (such as a Hamiltonian or measurement operator) can be expressed in this notation as well.
Every Hilbert space has an orthonormal basis {∣i⟩}\{ |i\rangle \}{∣i⟩} (where ⟨i∣j⟩=δij\langle i | j \rangle = \delta_{ij}⟨i∣j⟩=δij). Any state vector can be expanded in a basis: ∣ψ⟩=∑ici ∣i⟩|\psi\rangle = \sum_i c_i\,|i\rangle∣ψ⟩=∑ici∣i⟩. The basis could be finite or countably infinite dimensional depending on the system. For example, a qubit has a two-dimensional state space spanned by ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩, whereas a quantum harmonic oscillator has an infinite-dimensional Fock space spanned by number states ∣0⟩,∣1⟩,∣2⟩,…|0\rangle, |1\rangle, |2\rangle, \dots∣0⟩,∣1⟩,∣2⟩,…
web.eecs.utk.edu. In all cases, the vector components cic_ici are complex amplitudes. The superposition principle holds that any linear combination of valid state vectors (within H\mathcal{H}H) is also a valid state vector. This reflects the linear structure of the theory and is embodied in the use of vector space methods for quantum states.
Superposition and Basis States
The concept of superposition is fundamental: a quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ can be written as a superposition of basis states. If {∣i⟩}\{|i\rangle\}{∣i⟩} is an orthonormal basis of the Hilbert space, we can write:
∣ψ⟩ = ∑ici ∣i⟩,|\psi\rangle \;=\; \sum_i c_i\, |i\rangle,∣ψ⟩=∑ici∣i⟩,
where ci∈Cc_i \in \mathbb{C}ci∈C are complex coefficients called amplitudes. The state is normalized when ∑i∣ci∣2=1\sum_i |c_i|^2 = 1∑i∣ci∣2=1. Superposition means that the system is simultaneously in all the basis states in a weighted sense until a measurement is made. A simple example is a single qubit (two-level system): one can choose the basis {∣0⟩,∣1⟩}\{|0\rangle, |1\rangle\}{∣0⟩,∣1⟩} (often called the computational basis). An arbitrary qubit state is
∣ψ⟩=α ∣0⟩+β ∣1⟩,|\psi\rangle = \alpha\,|0\rangle + \beta\,|1\rangle,∣ψ⟩=α∣0⟩+β∣1⟩,
with complex amplitudes α,β\alpha, \betaα,β such that ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1∣α∣2+∣β∣2=1
michaelnielsen.org. Here ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩ might represent two energy levels of an atom, or spin-up vs. spin-down of an electron, etc. The coefficients α\alphaα and β\betaβ encode the probability distribution for the qubit’s value upon measurement (as discussed below). Superposition does not mean the system is literally in two classical states at once; rather, it resides in a definite state vector that is a linear combination of basis vectors. The relative magnitudes and phases of α\alphaα and β\betaβ have physical consequences (they determine interference effects and measurement outcomes probabilities), but as noted, any overall global phase is physically irrelevant. We emphasize that while a classical bit must be in either state 0 or 1, a qubit can exist in a continuum of superposed states of 0 and 1 until measured
The choice of basis {∣i⟩}\{|i\rangle\}{∣i⟩} is often dictated by the physical context or the measurement one plans to perform. Different complete bases are related by unitary transformations (change-of-basis rotations in Hilbert space). For instance, for a qubit one convenient alternative basis is ∣+⟩=12(∣0⟩+∣1⟩)|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)∣+⟩=21(∣0⟩+∣1⟩) and ∣−⟩=12(∣0⟩−∣1⟩)|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)∣−⟩=21(∣0⟩−∣1⟩), which are superpositions of the computational basis and correspond to eigenstates of the Pauli X operator. The ability to expand states in any basis and switch bases unitarily underlies quantum interference phenomena and quantum computing algorithms. What makes superposition truly distinct from classical mixtures is that the coefficients are complex amplitudes, allowing interference (positive or negative) between probability amplitudes. As a result, measuring a superposed state can yield different outcomes with probabilities given by the squared magnitudes of overlaps with eigenstates, rather than revealing a pre-existing definite classical state.
Global vs. relative phase: If ∣ψ⟩=∑ici∣i⟩|\psi\rangle = \sum_i c_i |i\rangle∣ψ⟩=∑ici∣i⟩, the state eiθ∣ψ⟩=∑i(eiθci)∣i⟩e^{i\theta}|\psi\rangle = \sum_i (e^{i\theta}c_i)|i\rangleeiθ∣ψ⟩=∑i(eiθci)∣i⟩ represents the same physical situation for any global phase θ\thetaθ. Only relative phases between components (phases of cic_ici relative to each other) have observable effects
web.eecs.utk.edu. This fact is often stated as: quantum states are rays in projective Hilbert space
web.eecs.utk.edu. The global phase symmetry is one reason probabilities involve absolute squares of amplitudes (which are phase-independent). For example, a qubit state ∣0⟩+∣1⟩|0\rangle + |1\rangle∣0⟩+∣1⟩ and ∣0⟩−∣1⟩|0\rangle – |1\rangle∣0⟩−∣1⟩ have a relative phase difference and will behave differently under certain interference experiments, whereas ∣0⟩+∣1⟩|0\rangle + |1\rangle∣0⟩+∣1⟩ and eiπ/4(∣0⟩+∣1⟩)e^{i\pi/4}(|0\rangle + |1\rangle)eiπ/4(∣0⟩+∣1⟩) are physically indistinguishable in all outcomes.
Measurement Postulates and the Born Rule
A defining feature of quantum theory is the role of measurement. Unlike a classical state, a quantum state vector generally cannot be directly observed; instead, experiments yield outcomes probabilistically according to the Born rule. The measurement postulates can be summarized as follows
- Observables as Hermitian operators: Every measurable quantity AAA is represented by a Hermitian (self-adjoint) operator on the Hilbert space. The possible measurement outcomes are the eigenvalues aia_iai of AAA, and the associated eigenvectors ∣ai⟩|a_i\rangle∣ai⟩ (assume for simplicity a non-degenerate spectrum) form an orthonormal basis of state space.
- Outcome probabilities (Born rule): If the system is in state ∣ψ⟩|\psi\rangle∣ψ⟩ (normalized) and we measure observable AAA, the probability of obtaining outcome aia_iai (i.e. the system collapsing to eigenstate ∣ai⟩|a_i\rangle∣ai⟩) is given by p(ai) = ∣⟨ai∣ψ⟩∣2,p(a_i) \;=\; |\langle a_i|\psi\rangle|^2,p(ai)=∣⟨ai∣ψ⟩∣2, which is the squared magnitude of the projection of ∣ψ⟩|\psi\rangle∣ψ⟩ onto the eigenstate ∣ai⟩|a_i\rangle∣ai⟩en.wikipedia.org. Equivalently, p(ai)=⟨ψ∣Pi∣ψ⟩p(a_i) = \langle \psi | P_i | \psi \ranglep(ai)=⟨ψ∣Pi∣ψ⟩, where Pi=∣ai⟩⟨ai∣P_i = |a_i\rangle\langle a_i|Pi=∣ai⟩⟨ai∣ is the projector onto the eigenspace corresponding to aia_iaien.wikipedia.org. These probabilities satisfy ∑ip(ai)=1\sum_i p(a_i) = 1∑ip(ai)=1 as a consequence of state normalization and completeness of the projectors (∑iPi=I\sum_i P_i = I∑iPi=I). For example, measuring a qubit state α∣0⟩+β∣1⟩\alpha|0\rangle + \beta|1\rangleα∣0⟩+β∣1⟩ in the {∣0⟩,∣1⟩}\{|0\rangle,|1\rangle\}{∣0⟩,∣1⟩} basis yields outcome 0 with probability ∣α∣2|\alpha|^2∣α∣2 and outcome 1 with probability ∣β∣2|\beta|^2∣β∣2michaelnielsen.org.
- Post-measurement state (collapse): Immediately after obtaining outcome aia_iai, the post-measurement state of the system is (for an ideal projective measurement) the corresponding eigenvector ∣ai⟩|a_i\rangle∣ai⟩ (or, in the degenerate case, the normalized projector Pi∣ψ⟩/⟨ψ∣Pi∣ψ⟩P_i|\psi\rangle/\sqrt{\langle \psi|P_i|\psi\rangle}Pi∣ψ⟩/⟨ψ∣Pi∣ψ⟩ in that eigenspace). This discontinuous, non-unitary change is often called the collapse of the wavefunction or projection postulate. It reflects the update of our knowledge/state due to the measurement. Notably, von Neumann formulated the measurement process as a two-step evolution: a deterministic unitary interaction between system and apparatus, followed by a non-unitary collapse associated with our observationpeople.cs.rutgers.edu.
Remark: The Born rule (named after Max Born, who introduced the probabilistic interpretation in 1926) is one of the fundamental postulates of quantum mechanics; it cannot be derived from the Schrödinger equation alone and must be assumed. It introduces inherent probabilism: even if a system is prepared in a known state vector, the result of a measurement may not be certain, only the probability distribution of outcomes is determined by the state. Measurement thus has a special status in quantum theory and gives rise to conceptual issues (the measurement problem, etc.), though these are beyond our scope here.
Measurements not only reveal information but generally disturb the state. If ∣ψ⟩|\psi\rangle∣ψ⟩ was not already an eigenstate of the observable measured, the act of measurement forces it into one of the eigenstates, destroying the original superposition (often phrased as “wavefunction collapse”). For example, if a qubit is in (∣0⟩+∣1⟩)/2(|0\rangle+|1\rangle)/\sqrt{2}(∣0⟩+∣1⟩)/2 and we measure in the {∣0⟩,∣1⟩}\{|0\rangle,|1\rangle\}{∣0⟩,∣1⟩} basis, we randomly get ∣0⟩|0\rangle∣0⟩ or ∣1⟩|1\rangle∣1⟩ (each with 50% probability), and the post-measurement state is correspondingly ∣0⟩|0\rangle∣0⟩ or ∣1⟩|1\rangle∣1⟩, eliminating the initial superposition. Repeating the same measurement immediately would then yield the same result with certainty (the state has collapsed into an eigenstate). This aspect has no analog in classical physics and is crucial to understanding quantum phenomena like uncertainty and the no-cloning theorem.
It is worth noting that the measurement postulates can be generalized beyond simple projective measurements. In the most general formulation, a measurement is described by a set of positive operators {Em}\{E_m\}{Em} (POVM elements) that sum to identity, such that p(m)=⟨ψ∣Em∣ψ⟩p(m) = \langle \psi|E_m|\psi\ranglep(m)=⟨ψ∣Em∣ψ⟩ is the probability of outcome mmm. Upon outcome mmm, the state updates as ∣ψ⟩→Mm∣ψ⟩p(m)|\psi\rangle \to \frac{\sqrt{M_m}|\psi\rangle}{\sqrt{p(m)}}∣ψ⟩→p(m)Mm∣ψ⟩ for some operator MmM_mMm with Mm†Mm=EmM_m^\dagger M_m = E_mMm†Mm=Em. However, for many theoretical discussions and for all practical purposes in this overview, we restrict to the simpler (yet very important) case of projective measurements onto eigenstates, which is the content of the standard postulates above.
Pure vs. Mixed States and the Density Matrix
Thus far we have discussed pure states, which are represented by a single state vector ∣ψ⟩|\psi\rangle∣ψ⟩ (up to phase). A pure state encapsulates maximal knowledge of a quantum system — it cannot be expressed as a non-trivial mixture of other states. In contrast, a mixed state represents a statistical ensemble of different possible pure states. Mixed states arise in situations where we have incomplete information about the system or when the system is entangled with an environment such that we consider only the subsystem. The formalism of density matrices (or density operators) is used to describe both pure and mixed states in a unified way
Mathematically, the density operator ρ\rhoρ for a quantum system is defined as:
- For a pure state ∣ψ⟩|\psi\rangle∣ψ⟩, ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle \psi|ρ=∣ψ⟩⟨ψ∣. This is a projector onto the state ∣ψ⟩|\psi\rangle∣ψ⟩. Such ρ\rhoρ satisfies ρ2=ρ\rho^2 = \rhoρ2=ρ and has trace Tr(ρ)=1\mathrm{Tr}(\rho) = 1Tr(ρ)=1 and Tr(ρ2)=1\mathrm{Tr}(\rho^2)=1Tr(ρ2)=1en.wikipedia.org. In an orthonormal basis, the density matrix elements are ρij=ψiψj∗\rho_{ij} = \psi_i \psi_j^*ρij=ψiψj∗.
- For a mixed state, ρ\rhoρ is a positive semi-definite, Hermitian operator with unit trace, which can be expressed as a convex combination of pure-state projectors: ρ=∑kpk ∣ψk⟩⟨ψk∣\rho = \sum_k p_k\, |\psi_k\rangle \langle \psi_k|ρ=∑kpk∣ψk⟩⟨ψk∣, where pkp_kpk are classical probabilities (non-negative, summing to 1) for the system being in the pure state ∣ψk⟩|\psi_k\rangle∣ψk⟩en.wikipedia.org. This representation is not unique – the same mixed state ρ\rhoρ can be composed of different sets of pure states. An extreme example of a mixed state is the maximally mixed state in a $d$-dimensional Hilbert space, ρmix=1dId\rho_{\text{mix}} = \frac{1}{d}I_dρmix=d1Id, which can be seen as an equal mixture of any orthonormal basis of pure states and satisfies Tr(ρmix2)=1/d\mathrm{Tr}(\rho_{\text{mix}}^2) = 1/dTr(ρmix2)=1/d. In general, one can quantify the “purity” of a state by Tr(ρ2)\mathrm{Tr}(\rho^2)Tr(ρ2), which equals 1 for pure states and is less than 1 for mixed states (reaching a minimum of 1/d1/d1/d for the maximally mixed state)en.wikipedia.org.
The density matrix provides a powerful way to compute expectation values: for any observable AAA, the expected value in state ρ\rhoρ is ⟨A⟩=Tr(ρ A)\langle A \rangle = \mathrm{Tr}(\rho\,A)⟨A⟩=Tr(ρA). This formula yields ⟨ψ∣A∣ψ⟩\langle \psi|A|\psi\rangle⟨ψ∣A∣ψ⟩ for a pure state ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle\psi|ρ=∣ψ⟩⟨ψ∣, and for a mixed state it gives the probabilistic average of the expectations in each pure component. The density operator formalism was introduced by von Neumann and others in the 1930s and is essential for treating open systems and thermal states. Von Neumann also defined the von Neumann entropy S(ρ)=−Tr(ρlogρ)S(\rho) = -\mathrm{Tr}(\rho \log \rho)S(ρ)=−Tr(ρlogρ) as a measure of uncertainty or mixedness of a quantum state
people.cs.rutgers.edu. For a pure state S=0S=0S=0, while for a maximally mixed state S=logdS = \log dS=logd. Thus, entropy provides a bridge between quantum state purity and classical uncertainty.
Physical interpretation: A mixed state ρ\rhoρ can be understood in two (equivalent) ways: (1) Classical ignorance: with probability pkp_kpk the system was prepared in ∣ψk⟩ |\psi_k\rangle∣ψk⟩. Here the mixture reflects our lack of knowledge about which pure state was actually prepared (as in a classical random draw of states). (2) Entanglement with an environment: the system is part of a larger pure state, but if we lack access to the environment, the best description of the subsystem is a density matrix. By tracing out (averaging over) the environment’s degrees of freedom, the subsystem is left in a mixed state. In fact, any mixed state can be “purified” by introducing an environmental system such that ρ=Trenv(∣Ψ⟩SE⟨Ψ∣)\rho = \mathrm{Tr}_{env}(|\Psi\rangle_{SE}\langle\Psi|)ρ=Trenv(∣Ψ⟩SE⟨Ψ∣) for some joint pure state $|\Psi\rangle_{SE}$ of system+environment. In this sense, pure vs mixed is relative to what information is accessible.
The collapse postulate can also be formulated in the density matrix language: after a measurement yielding outcome aia_iai, a pure state $\rho = |\psi\rangle\langle\psi|$ updates to $\rho’ = |a_i\rangle\langle a_i|$. More generally, a density matrix $\rho$ would update as $\rho’ = \frac{P_i \rho P_i}{\mathrm{Tr}(\rho P_i)}$ for a projective measurement outcome $i$. Between measurements, an isolated system’s $\rho$ evolves unitarily (as described next). The density operator formalism is especially convenient when dealing with mixed states, decoherence, and quantum statistical mechanics.
Time Evolution and Unitary Dynamics
Isolated quantum systems evolve in time according to the Schrödinger equation, which governs the time-dependence of the state vector. For a state $|\psi(t)\rangle$ in a system with Hamiltonian (energy operator) $H$, the Schrödinger equation is:
iℏddt∣ψ(t)⟩ = H ∣ψ(t)⟩.i\hbar \frac{d}{dt}|\psi(t)\rangle \;=\; H \,|\psi(t)\rangle.iℏdtd∣ψ(t)⟩=H∣ψ(t)⟩.
Here $\hbar$ is the reduced Planck constant. The Hamiltonian $H$ is a Hermitian operator on the Hilbert space, and its spectrum of eigenvalues corresponds to possible energy levels of the system. This equation is a linear differential equation in the state vector. Its formal solution can be written as
∣ψ(t)⟩ = U(t,t0) ∣ψ(t0)⟩,|\psi(t)\rangle \;=\; U(t,t_0)\,|\psi(t_0)\rangle,∣ψ(t)⟩=U(t,t0)∣ψ(t0)⟩,
where $U(t,t_0)$ is the time-evolution operator from time $t_0$ to $t$. For a time-independent $H$, $U(t,t_0) = \exp!\left[-\frac{i}{\hbar}H (t-t_0)\right]$
web.eecs.utk.edu. In general, $U(t,t_0)$ is unitary, and for time-dependent $H(t)$ one can use a time-ordered exponential. Stone’s theorem guarantees that a time-continuous unitary evolution is generated by some Hermitian $H$.
Postulate 2 (Unitary dynamics): The evolution of a closed quantum system is described by a unitary transformation on its state
web.eecs.utk.edu. Equivalently, the state vector $|\psi(t)\rangle$ at time $t$ is related to the state $|\psi(t_0)\rangle$ at an earlier time by a unitary operator $U(t,t_0)$ depending only on the Hamiltonian and the time interval: $|\psi(t)\rangle = U(t,t_0),|\psi(t_0)\rangle$
web.eecs.utk.edu. Unitarity means $U U^\dagger = U^\dagger U = I$, which ensures conservation of inner products: $\langle \psi(t)|\phi(t)\rangle = \langle \psi(t_0)|\phi(t_0)\rangle$. This guarantees that normalization and the total probability remain constant in time, as required for consistency. Intuitively, time evolution in a closed system is like a rotation in Hilbert space (sometimes called a “complex rotation”
web.eecs.utk.edu), mixing the amplitudes while preserving the overall length of the state vector.
For example, if a qubit has Hamiltonian $H = \frac{\hbar \omega}{2}\sigma_Z$ (where $\sigma_Z$ is the Pauli Z matrix and $\omega$ a frequency), then $U(t) = e^{-i \omega t \sigma_Z/2} = \begin{pmatrix} e^{-i\omega t/2} & 0 \ 0 & e^{i\omega t/2} \end{pmatrix}$, which imparts a relative phase rotation between the $|0\rangle$ and $|1\rangle$ components of the qubit state. More generally, any fixed Hamiltonian causes deterministic, reversible unitary evolution. In the context of quantum computing, logic gates applied to qubits are represented by unitary matrices $U$ acting on the state vectors, which is just the discrete-time version of this continuous unitary evolution postulate.
If we use the density matrix description, the Schrödinger equation yields the von Neumann equation for the density operator:
iℏdρdt=[H,ρ]=Hρ−ρH,i\hbar \frac{d\rho}{dt} = [H,\rho] = H\rho – \rho H,iℏdtdρ=[H,ρ]=Hρ−ρH,
whose solution is ρ(t)=U(t) ρ(0) U(t)†\rho(t) = U(t)\,\rho(0)\,U(t)^\daggerρ(t)=U(t)ρ(0)U(t)†. This shows explicitly that a pure state remains pure under unitary evolution (since $\rho(t) = |\psi(t)\rangle\langle\psi(t)|$ if $\rho(0)=|\psi(0)\rangle\langle\psi(0)|$), and the purity $\mathrm{Tr}(\rho^2)$ is conserved
It should be stressed that the above unitary evolution law applies only to closed systems (no interaction with an environment). If a system is open (interacting with or being measured by an external system), then its evolution need not be unitary when considered by itself – indeed, environmental interactions can lead to decoherence and effectively non-unitary evolution (e.g. described by Lindblad master equations or quantum channels). In principle, one can always include the environment in an enlarged Hilbert space and treat the combination as a closed system evolving unitarily, but the reduced dynamics of the subsystem alone is more complicated (often resulting in a mixed state even if the total state is pure). In summary, a quantum state vector follows the Schrödinger equation in between measurements, and changes via a projection (collapse) during a measurement – these two processes (continuous unitary evolution and discontinuous collapse) together make up the foundational dynamics in standard quantum theory
Composite Systems, Tensor Products, and Entanglement
For a composite physical system composed of two or more subsystems, the Hilbert space of the whole system is the tensor product of the component Hilbert spaces. If system AAA lives in HA\mathcal{H}_AHA and system BBB in HB\mathcal{H}_BHB, then the joint system A+BA+BA+B is described in HAB=HA⊗HB\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_BHAB=HA⊗HB. A basis of HAB\mathcal{H}_{AB}HAB is given by products of basis vectors of each part, ∣i⟩A⊗∣j⟩B|i\rangle_A \otimes |j\rangle_B∣i⟩A⊗∣j⟩B (often abbreviated as ∣i⟩∣j⟩|i\rangle|j\rangle∣i⟩∣j⟩ or ∣ij⟩|ij\rangle∣ij⟩). A general pure state of the combined system is
∣Ψ⟩AB = ∑i,jcij ∣i⟩A⊗∣j⟩B,|\Psi\rangle_{AB} \;=\; \sum_{i,j} c_{ij}\; |i\rangle_A \otimes |j\rangle_B,∣Ψ⟩AB=∑i,jcij∣i⟩A⊗∣j⟩B,
with normalization ∑i,j∣cij∣2=1\sum_{i,j} |c_{ij}|^2 = 1∑i,j∣cij∣2=1. Such a state is said to be separable (or a product state) if it can be factored as ∣Ψ⟩AB=∣ψ⟩A⊗∣ϕ⟩B|\Psi\rangle_{AB} = |\psi\rangle_A \otimes |\phi\rangle_B∣Ψ⟩AB=∣ψ⟩A⊗∣ϕ⟩B for some individual states of $A$ and $B$. In contrast, if $|\Psi\rangle_{AB}$ cannot be written as a single product, it is an entangled state. By definition, an entangled state cannot be expressed as a product of states of the subsystems
arxiv.org; it is a superposition of multiple product terms. Entanglement is a uniquely quantum phenomenon with no counterpart in classical physics
en.wikipedia.org. It implies that the subsystems do not have their own well-defined pure states — their state is only defined jointly. The outcomes of measurements on one part can be correlated with outcomes on the other part in ways impossible for classical separable systems.
A famous example of an entangled state for two qubits (A and B) is the Bell state ∣Φ+⟩=12(∣0⟩A⊗∣0⟩B+∣1⟩A⊗∣1⟩B)|\Phi^+\rangle = \frac{1}{\sqrt{2}}\big(|0\rangle_A\otimes|0\rangle_B + |1\rangle_A\otimes|1\rangle_B\big)∣Φ+⟩=21(∣0⟩A⊗∣0⟩B+∣1⟩A⊗∣1⟩B). This state cannot be factored into ∣ψ⟩A⊗∣ϕ⟩B|\psi\rangle_A \otimes |\phi\rangle_B∣ψ⟩A⊗∣ϕ⟩B; any measurement on qubit A will instantaneously project qubit B into a corresponding state, even if they are far apart (in the sense that their outcomes are perfectly correlated in the same basis). Entangled states like this violate the assumption of local realism and were central to the famous EPR paradox and Bell’s theorem discussions
en.wikipedia.org. Operationally, entanglement means the joint state contains nonclassical correlations: the state of each subsystem cannot be described independently of the other
Entanglement has profound implications. If two particles are entangled, the maximal information one can have about the joint system is described by a pure state in the larger Hilbert space, yet each individual particle is in a mixed state (obtained by tracing out the other). For instance, for $|\Phi^+\rangle$ above, the reduced state of qubit A alone is $\rho_A = \frac{1}{2}(|0\rangle\langle0| + |1\rangle\langle1|)$ – a maximally mixed state. This exemplifies how entanglement leads to mixed local states despite the total state being pure. In general, if $|\Psi\rangle_{AB}$ is entangled, $\rho_A = \mathrm{Tr}_B(|\Psi\rangle\langle\Psi|)$ will have entropy $S(\rho_A) > 0$, indicating uncertainty in A’s state when B is unobserved.
Mathematically, criteria for entanglement include that a pure state’s coefficients $c_{ij}$ cannot be factored as $c_{ij} = a_i b_j$, and for mixed states one uses separability criteria (like the Peres–Horodecki PPT criterion). Entanglement is the key resource that enables quantum information protocols such as quantum teleportation, superdense coding, and certain quantum cryptography schemes. It also plays a central role in quantum computing, where entangled states of many qubits enable computational speedups that are believed to be impossible classically.
Illustration of the Bloch sphere, which geometrically represents all pure states of a single qubit (two-level system) as points on the surface of a unit sphere. Antipodal points on the sphere correspond to orthogonal states (e.g., the north and south poles might represent $|0\rangle$ and $|1\rangle$). Any pure qubit state can be written as ∣ψ⟩=cos(θ/2)∣0⟩+eiφsin(θ/2)∣1⟩|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\varphi}\sin(\theta/2)|1\rangle∣ψ⟩=cos(θ/2)∣0⟩+eiφsin(θ/2)∣1⟩ and mapped to a point with spherical coordinates $(\theta,\varphi)$ on this sphere. The interior of the Bloch sphere corresponds to mixed states (convex combinations of the pure-state points on the surface)
en.wikipedia.org. This visualization highlights that the space of quantum state vectors (for a qubit) is a continuous manifold, much richer than the two discrete states of a classical bit.
Quantum State Tomography and Quantum Information Applications
Because a quantum state vector encodes probability amplitudes (which are not directly observable), determining an unknown quantum state experimentally requires quantum state tomography. Quantum state tomography is the process of reconstructing the state (density matrix) of a quantum system by making many measurements in different bases on identically prepared systems
en.wikipedia.org. By gathering statistics from a tomographically complete set of measurements (enough to span the operator space on $\mathcal{H}$), one can infer the state $\rho$ that is most consistent with the observed data. For a single qubit, this might mean measuring $\sigma_X$, $\sigma_Y$, and $\sigma_Z$ Pauli observables on many copies to determine the Bloch vector; for higher-dimensional systems, more settings are needed. State tomography is an important tool in quantum information science for verifying that a quantum device (like a qubit or register of qubits) has been prepared in the desired state. However, it can be resource-intensive: the number of measurements needed grows rapidly with the dimension of the state space (for $n$ qubits, the state lies in a $2^n$-dimensional space, so full tomography requires on the order of $4^n$ measurement settings in general). This highlights the fact that quantum state vectors carry an exponential amount of information (in the number of qubits) — a double-edged sword, as it powers quantum algorithms but also makes characterization complex
In quantum information theory, the abstract properties of state vectors and their entanglement underlie tasks like quantum teleportation (where an arbitrary state is transmitted using entanglement and classical communication), quantum error correction (which protects state vectors against decoherence by encoding them in larger Hilbert spaces), and quantum algorithms. The famous textbook by Nielsen and Chuang uses state vectors as the starting point to discuss qubits, entangled states, and quantum gates
web.eecs.utk.edu. Entanglement, in particular, is viewed as a resource that can be consumed or distilled, and its presence (or absence) distinguishes quantum channels and codes. The superposition principle is what gives quantum algorithms their peculiar flavor of parallelism (though, as some argue, it’s entanglement and interference, rather than a naive “many parallel universes” picture, that provide quantum advantage
arxiv.org). Meanwhile, the measurement postulate underpins quantum cryptography, ensuring that eavesdropping disturbances can be detected (since measuring a quantum state generally changes it).
Finally, it’s worth noting the distinction between the state vector and the physical state. The state vector is an element of an abstract mathematical space with no direct physical existence, but it is a catalogue of probabilities for outcomes of all possible measurements on the system
plato.stanford.edu. As such, the quantum state (whether pure or mixed) encapsulates everything one can possibly know about the system’s measurable properties. Einstein, Dirac, and others debated whether the quantum state is a complete description of reality or just knowledge (the famous EPR argument and subsequent developments). In the standard interpretation, the state vector is taken as the most fundamental description of an individual quantum system’s state, and there is no deeper level of specification (unless one considers hidden variable theories). Modern experiments (violations of Bell inequalities, etc.) strongly suggest that if a deeper “hidden” state exists, it must be extraordinarily non-classical. Thus, quantum state vectors and their associated formalisms have proven to be an indispensable theoretical tool, from the foundational axioms laid down by Dirac and von Neumann to contemporary quantum computing applications
References: Quantum state vectors and Hilbert space formalism were introduced by Dirac and von Neumann in the 1930s
people.cs.rutgers.edu. Modern expositions can be found in textbooks such as Quantum Computation and Quantum Information by Nielsen & Chuang
web.eecs.utk.edu, which also discuss qubits, entanglement, and quantum information processing. The concepts of superposition, measurement (Born rule) and unitary evolution are standard in quantum mechanics literature (see, e.g., Dirac’s Principles of Quantum Mechanics and von Neumann’s work). For density matrices and mixed states, see von Neumann’s introduction of the entropy concept
people.cs.rutgers.edu or standard quantum theory texts. Quantum entanglement and its implications are reviewed in many sources (e.g., the Einstein-Podolsky-Rosen paper, Bell’s theorem experiments
en.wikipedia.org). Quantum state tomography techniques are discussed in quantum information contexts and in experimental papers
en.wikipedia.org. The above overview has synthesized these sources to provide a cohesive theoretical summary of quantum state vectors in modern quantum theory.
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