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Quantum Mechanics Foundations

Dirac Notation and Hilbert Spaces

Quantum states live in a Hilbert space H -- a complete inner product space over the complex numbers. For an n-qubit system, H = C^{2^n}.

Dirac notation provides compact formalism:

  • Ket |psi> represents a column vector (state vector) in H
  • Bra <psi| represents the conjugate transpose (dual vector)
  • Bracket <phi|psi> is the inner product, yielding a complex scalar
  • Outer product |psi><phi| is a linear operator (matrix)

A single qubit state is |psi> = alpha|0> + beta|1> where alpha, beta in C and |alpha|^2 + |beta|^2 = 1. The computational basis {|0>, |1>} corresponds to the standard basis of C^2.

Tensor products combine systems: |psi> tensor |phi> in H_A tensor H_B. For two qubits, the computational basis is {|00>, |01>, |10>, |11>}, spanning C^4.

Superposition

Superposition is the principle that any linear combination of valid quantum states is itself a valid quantum state. Given orthonormal basis {|e_i>}, an arbitrary state is:

|psi> = sum_i c_i |e_i>, where sum_i |c_i|^2 = 1

This is not merely classical ignorance. Interference between amplitudes -- constructive or destructive -- produces measurable effects with no classical analogue. The double-slit experiment demonstrates this: single particles exhibit interference patterns inconsistent with any classical trajectory model.

Measurement

Born Rule

Measuring observable A = sum_a a |a><a| on state |psi> yields outcome a with probability:

P(a) = |<a|psi>|^2

Post-measurement, the state collapses to |a> (projective measurement).

Projective (von Neumann) Measurement

Defined by a set of orthogonal projectors {P_m} satisfying sum_m P_m = I and P_m P_n = delta_{mn} P_m. Probability of outcome m is P(m) = <psi|P_m|psi>. Post-measurement state: P_m|psi> / sqrt(P(m)).

POVM (Positive Operator-Valued Measure)

Generalization where measurement elements {E_m} satisfy E_m >= 0 and sum_m E_m = I, but need not be orthogonal projectors. POVMs describe the most general measurement statistics achievable. Probability: P(m) = <psi|E_m|psi>. POVMs can always be realized as projective measurements on a larger (dilated) system via Naimark's theorem.

Key distinction: POVMs can have more outcomes than the dimension of the Hilbert space, enabling optimal state discrimination strategies impossible with projective measurements.

Density Matrices

The density operator rho generalizes state vectors to handle:

  • Pure states: rho = |psi><psi|, with Tr(rho^2) = 1
  • Mixed states: rho = sum_i p_i |psi_i><psi_i|, with Tr(rho^2) < 1

Properties: rho >= 0 (positive semidefinite), Tr(rho) = 1, rho is Hermitian.

Mixed states arise from classical uncertainty, decoherence, or tracing out part of an entangled system. The partial trace produces the reduced density matrix: rho_A = Tr_B(rho_{AB}).

Expectation values: = Tr(rho A). Time evolution: rho(t) = U rho(0) U^dagger (closed system) or via the Lindblad master equation (open system).

Bloch sphere representation (single qubit): rho = (I + r . sigma) / 2, where r is the Bloch vector (|r| <= 1, equality iff pure) and sigma = (X, Y, Z) are the Pauli matrices.

Entanglement

A bipartite pure state |psi>{AB} is entangled if it cannot be written as |a> tensor |b>. Mixed state rho{AB} is entangled if it is not separable, i.e., cannot be expressed as rho_{AB} = sum_i p_i rho_A^i tensor rho_B^i.

Bell States

The four maximally entangled two-qubit states forming an orthonormal basis:

|Phi+> = (|00> + |11>) / sqrt(2) |Phi-> = (|00> - |11>) / sqrt(2) |Psi+> = (|01> + |10>) / sqrt(2) |Psi-> = (|01> - |10>) / sqrt(2)

Each has maximally mixed reduced density matrices: rho_A = rho_B = I/2.

EPR Paradox

Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics is incomplete. Measuring one particle of an entangled pair instantaneously determines the other's state, suggesting either (a) the outcomes were predetermined (hidden variables) or (b) there is nonlocal influence. They favored (a), proposing local hidden variable (LHV) theories.

Bell's Theorem and CHSH Inequality

Bell (1964) proved that no LHV theory can reproduce all quantum predictions. The CHSH inequality provides a testable bound:

|<A_1 B_1> + <A_1 B_2> + <A_2 B_1> - <A_2 B_2>| <= 2 (classical)

Quantum mechanics allows violation up to 2*sqrt(2) (Tsirelson's bound), achieved by appropriate measurements on |Phi+>. Experimental violations (Aspect 1982, loophole-free tests 2015) confirm quantum nonlocality.

No-Cloning Theorem

Theorem: There exists no unitary U such that U(|psi> tensor |0>) = |psi> tensor |psi> for all |psi>.

Proof sketch: Suppose U|a>|0> = |a>|a> and U|b>|0> = |b>|b> for distinct states |a>, |b>. By unitarity, <a|b> = <a|b>^2, so <a|b> in {0, 1}. Thus only orthogonal states can be cloned, not arbitrary superpositions.

Implications: Prevents superluminal signaling via entanglement, underpins security of quantum key distribution, requires quantum error correction to use redundancy differently than classical repetition.

The no-broadcasting theorem extends this to mixed states.

Quantum Teleportation

Protocol to transmit an unknown qubit state |psi> = alpha|0> + beta|1> using one shared Bell pair and two classical bits:

  1. Alice and Bob share |Phi+>_{23}. Alice holds qubit 1 in state |psi>_1.
  2. Alice performs a Bell measurement on qubits 1 and 2, obtaining one of four outcomes (2 classical bits).
  3. Alice sends her measurement result to Bob classically.
  4. Bob applies the corresponding Pauli correction (I, X, Z, or XZ) to qubit 3.
  5. Bob's qubit is now in state |psi>.

No FTL communication: Bob needs Alice's classical message. The qubit state is destroyed at Alice's end (consistent with no-cloning). Resource cost: 1 ebit + 2 cbits.

Superdense Coding

Dual protocol to teleportation: transmit 2 classical bits using 1 qubit and 1 shared ebit.

  1. Alice and Bob share |Phi+>. Alice wants to send two classical bits (b1, b2).
  2. Alice applies Z^{b1} X^{b2} to her qubit:
    • 00 -> I -> |Phi+>
    • 01 -> X -> |Psi+>
    • 10 -> Z -> |Phi->
    • 11 -> ZX -> |Psi->
  3. Alice sends her qubit to Bob.
  4. Bob performs a Bell measurement on both qubits, recovering (b1, b2).

This achieves the Holevo bound for one-qubit communication augmented by entanglement. Together, teleportation and superdense coding demonstrate that 1 ebit + 1 qubit of communication = 2 cbits, establishing fundamental resource equivalences in quantum information.

Key Mathematical Tools

Schmidt decomposition: Any bipartite pure state |psi>_{AB} can be written as sum_i lambda_i |i_A>|i_B> with lambda_i >= 0, sum lambda_i^2 = 1. The number of nonzero Schmidt coefficients (Schmidt rank) determines entanglement: rank 1 iff separable.

Purification: Every mixed state rho_A can be expressed as the partial trace of a pure state |psi>_{AB} on a larger system. This purification is unique up to isometries on B.

Kraus representation: General quantum operations (CPTP maps) act as E(rho) = sum_k A_k rho A_k^dagger, where sum_k A_k^dagger A_k = I. This encompasses unitary evolution, measurement, and decoherence within a unified framework.