Physics of quantum information — the study of the information-theoretic properties of quantum systems and the protocols that exploit them. Foundations: pure vs mixed states represented by density operators ρ (trace 1, positive…
quantum-information-physics
Density operator ρ: tr ρ = 1, ρ ≽ 0; pure ⇔ ρ² = ρ ⇔ tr ρ² = 1
Density operator ρ on a finite-dimensional Hilbert space H encodes the complete quantum-statistical state of the system: ρ is Hermitian (ρ†…
Bell states {|Φ±⟩, |Ψ±⟩}: maximally-entangled 2-qubit basis
The four Bell (Einstein-Podolsky-Rosen-Bohm) states are a maximally-entangled orthonormal basis for 2-qubit Hilbert space C² ⊗ C²: |Φ^±⟩ =…
CHSH operator C = AB + AB' + A'B − A'B': LHV ≤ 2, QM ≤ 2√2
Clauser-Horne-Shimony-Holt 1969 operator C = A⊗B + A⊗B' + A'⊗B − A'⊗B' on a bipartite quantum system, where A, A' are Alice-side dichotomic…
CHSH: LHV bound 2, Tsirelson 2√2, gap (2√2)²−2² = 4
Sympy-exact symbolic witness of the CHSH classical-vs-quantum bound gap. Step 1 — classical: on dichotomic ±1 observables the quantity C =…
No-cloning: ⟨φ|ψ⟩ = √2/2 forbids perfect copy; BH fidelity 5/6, shortfall 1/6
Sympy-exact symbolic witness of the Wootters-Zurek 1982 / Dieks 1982 no-cloning theorem + the Bužek-Hillery 1996 universal cloner bound. …
Bell singlet |Ψ⁻⟩: concurrence 1, ρ_A = 𝟙/2, S(ρ_A) = log 2
Sympy-exact matrix verification of the Bell singlet's maximal entanglement via the Wootters 1998 concurrence C = |⟨Ψ|Ψ̃⟩| with |Ψ̃⟩ = (σ_y…
Holevo bound chi = S(rho-bar) - sum p_i S(rho_i) >= I_acc; concavity on convex hull
Holevo-Schumacher-Westmoreland (HSW) accessible-information framework (Holevo 1973 Probl Inf Trans 9, 177; Schumacher-Westmoreland 1997 PR…
No-cloning theorem: U|psi>|0> = |psi>|psi> for all psi forces <phi|psi> in {0, 1}
No-cloning framework via inner-product preservation (Wootters-Zurek 1982 Nature 299, 802; Dieks 1982 Phys Lett A 92, 271). Setup: a…
Quantum Cramer-Rao: Var(theta-hat) >= 1/F_Q; F_Q = Tr(rho L_theta^2); SLD spectral resolution
Quantum Fisher Information / quantum Cramer-Rao framework (Helstrom 1976 Quantum Detection and Estimation Theory; Holevo 1982 Probabilistic…
Theorem: chi = log 2 for ensemble (1/2, |0><0|), (1/2, |1><1|) (orthogonal basis)
Theorem (Holevo-orthogonal-basis canonical): for the maximally distinguishable ensemble {(1/2, |0><0|), (1/2, |1><1|)}, rho-bar = (|0><0| +…
Theorem: x(1 - x) - (x - x^2) = 0 (cloning quadratic identity, only roots {0, 1})
Theorem (no-cloning quadratic-zero canonical): for inner product x = <phi|psi> in [0, 1], the cloning condition x = x^2 has factorisation…
Theorem: F_Q = 1 for pure-state family |psi> = cos(theta/2)|0> + sin(theta/2)|1>
Theorem (QFI-qubit-unity canonical): for the qubit-rotation family |psi(theta)> = cos(theta/2)|0> + sin(theta/2)|1>, the QFI is F_Q =…
No-cloning (1982)
W Wootters-W Zurek 1982 + Dieks 1982 no-cloning theorem; basis of QKD security; consequence of linearity of quantum mechanics.
Teleportation (Bennett 1993)
Bennett-Brassard-Crepeau-Jozsa-Peres-Wootters 1993 quantum-teleportation via 2-bit classical + EPR-pair; modern Northup 2024 ion +…
Entanglement (Werner 1989)
R Werner 1989 entangled vs separable states; modern Horodecki 2009 review; entanglement of formation + concurrence + monotones.
Bell (1964)
J Bell 1964 inequality; CHSH 1969 experimental form; Aspect 1982 + 2015 loophole-free + Nobel 2022 Aspect-Clauser-Zeilinger validation.
BB84 (Bennett-Brassard 1984)
Bennett-Brassard 1984 quantum key distribution; modern decoy-state + measurement-device-independent; commercial Toshiba + ID Quantique…
CQIT (Nielsen-Chuang 2000)
Nielsen-Chuang 2000 'Quantum Computation and Quantum Information'; canonical reference; modern Watrous 2018 + Wilde 2017…
Shor's algorithm (1994)
P Shor 1994 polynomial-time integer-factoring quantum; foundation post-quantum-cryptography + 2024 NIST PQC-standards Kyber.
Grover's algorithm (1996)
L Grover 1996 √N quantum-search; modern amplitude-amplification + 2024 fault-tolerant-Grover circuit-cost analysis.
BB84 (Bennett-Brassard 1984)
C Bennett-G Brassard 1984 QKD; modern fiber + satellite-Micius 1200 km + decoy-state + measurement-device-independent.
Quantum supremacy (Arute 2019)
F Arute Sycamore 2019 53-qubit; modern Jiuzhang 2020-2021 boson-sampling + Zuchongzhi 2021 + 2024 100-q hybrid-sampling.
Teleportation (Bennett 1993)
C Bennett 1993 4-message teleportation protocol; modern continuous-variable Furusawa 1998 + ground-satellite Micius 2017.
Distillation (Bennett 1996)
C Bennett 1996 entanglement-distillation; modern long-distance quantum-repeater + satellite + matrix-product state-distillation.