ℏQuantum Impossibility Theorems

QUANTUM INFORMATION LIMITS QUANTUM IMPOSSIBILITY THEOREMS Fundamental Constraints on Quantum Information Processing An axiomatic flyer mapping the algebraic restrictions that protect and define quantum structures.

1. NO-BROADCASTING THEOREM (THE GENERALISED NO-CLONING) While the No-Cloning theorem forbids duplication of pure states, the No-Broadcasting Theorem extends this restric- tion to arbitrary mixed states ρ. It proves that a quantum channel cannot take a single mixed state and output two separate states that match the original marginals. The Mathematical Bound: Given an unknown density matrix ρ, it is impossible to find a universal valid quantum operation E capable of broadcasting it onto a target channel space: E(ρ ⊗θ) = eρ such that Tr1(eρ) = ρ and Tr2(eρ) = ρ The Constraint: Broadcasting is achievable if and only if the set of input states belongs to a mutually commuting family ([ρi, ρj] = 0).

2. NO-DELETION THEOREM (THE TIME-REVERSE COUNTERPART) Formulated by Pati and Braunstein in 2000, the No-Deletion Theorem serves as the precise time-reverse mirror of No-Cloning. It states that given two identical copies of an unknown quantum state, it is physically impossible to erase one copy using a linear quantum process. The Algebraic Mechanism: Assume a unitary process U attempts to overwrite one copy of |ψ⟩given a blank state |e⟩: U |ψ⟩|ψ⟩|A⟩= |ψ⟩|e⟩|Aψ⟩ Linearity forces the final state of the auxiliary apparatus |Aψ⟩to retain a perfect, permanent record of the state geometry. Thus, the information is never truly destroyed or deleted; it is merely shuffled into environmental degrees of freedom.

3. NO-HIDING THEOREM (CONSERVATION OF QUANTUM INFORMATION) The No-Hiding Theorem enforces that quantum information can never be truly hidden in the correlations between a system and its environment. If information disappears from a local subsystem via decoherence, it must completely leak into the environment. THE CONSERVATION PILE Let a state |ψ⟩undergo a process where its sub-space representation maps into a completely invariant state |0⟩, hiding it from local observation: U |ψ⟩S |0⟩E = |0⟩S |Φ(ψ)⟩E The Consequence: The transformation requires the map to the environment state |Φ(ψ)⟩E to be perfectly isometric. The original information remains completely intact, structured inside the environment, protecting information from absolute loss.

4. THE NO-TELEPORTATION LIMIT (WITHOUT CLASSICAL SIGNALLING) This limit theorem establishes that quantum states cannot be disassembled and reconstructed remotely without explicit coordinate adjustments over an accompanying classical channel. ✦The Bound: An entangled pair allows Alice to project a combined state, but Bob’s remote outcome remains a uniformly mixed random rotation: ρB = 12I. ✦Resolution: Bob requires exactly 2 classical bits from Alice to choose the proper corrective Pauli matrix (X, Y, Z, or I) to isolate the clean state vector.