Quantum Information Basics: The No-Cloning Theorem, Tensor Products, and Density Matrix Diagonal Elements

What Is the No-Cloning Theorem?

The no-cloning theorem is a fundamental consequence of quantum mechanics stating that "no operation exists that can copy an unknown quantum state without disturbing it." More precisely, for any unknown state |ψ⟩, it asserts that

U(|ψ⟩ ⊗ |0⟩) = |ψ⟩ ⊗ |ψ⟩

there exists no unitary operator U satisfying this equation.

The proof is simple. Suppose such a U works for two distinct states |ψ⟩ and |φ⟩ alike:

U(|ψ⟩⊗|0⟩) = |ψ⟩⊗|ψ⟩,  U(|φ⟩⊗|0⟩) = |φ⟩⊗|φ⟩

Since unitary operations preserve inner products, taking the inner product of both sides gives the relation

⟨ψ|φ⟩ = ⟨ψ|φ⟩²

The only values of ⟨ψ|φ⟩ satisfying this equation are 0 (orthogonal) or 1 (identical states). In other words, there is no universal U capable of cloning an arbitrary unknown state that is neither orthogonal to nor identical with another.

The no-cloning theorem is precisely what underpins the security of quantum key distribution via BB84. An eavesdropper cannot "secretly copy a photon's state and then measure only the copy." Because measurement always disturbs the original state, traces of eavesdropping are statistically detectable. It is also why quantum information cannot achieve redundancy the way conventional error-correcting codes do — by directly duplicating information — which is why quantum error correction has no choice but to rely on indirect encoding using entanglement.

The Relationship Between Tensor Products and Entanglement

The state of a single qubit belongs to a two-dimensional vector space, but the state space of a system made up of n qubits is constructed as the tensor product of each qubit's state space, giving it dimension 2ⁿ. For a 2-qubit system, the basis consists of the four states |00⟩, |01⟩, |10⟩, |11⟩.

A general 2-qubit state can be written as |ψ⟩ = Σ c_ij |i⟩⊗|j⟩, but when this

|ψ⟩ = (a|0⟩ + b|1⟩) ⊗ (c|0⟩ + d|1⟩)

can be factored into a tensor product of each qubit's individual state, as shown above, the state is called a "product state" or a "separable" state. Conversely, a state that cannot be factored into this form no matter what is called an entangled state.

Let's check this with the representative example, the Bell state (|00⟩ + |11⟩)/√2. Suppose it could be factored into the form (a|0⟩+b|1⟩)⊗(c|0⟩+d|1⟩) = ac|00⟩ + ad|01⟩ + bc|10⟩ + bd|11⟩. Comparing coefficients requires ad = 0 and bc = 0, while simultaneously requiring ac = bd = 1/√2 ≠ 0. ad=0 means either a=0 or d=0, but either way one of ac or bd would have to be 0 — a contradiction. In other words, the Bell state cannot be factored into a product state, which algebraically proves that it is entangled.

This algebraic condition — whether or not a state can be factored into tensor-product form — underlies quantum communication protocols such as entanglement swapping, as well as many quantum algorithms.

The Diagonal Elements Are the Real Protagonist of Quantum Computation

A quantum state can be described by a density matrix ρ, which can represent statistical mixtures as well as pure states. Writing the density matrix out in the computational basis gives

ρ = Σᵢⱼ ρᵢⱼ |i⟩⟨j|

this form, in which the diagonal elements ρᵢᵢ and the off-diagonal elements ρᵢⱼ (i≠j) each carry entirely different meanings.

This picture is also directly tied to understanding decoherence (the phenomenon by which quantum behavior is lost through interaction with the environment). Decoherence is precisely the process by which the off-diagonal elements of the density matrix decay to zero over time; once the off-diagonal elements vanish, the density matrix becomes indistinguishable from a classical probabilistic mixture in which only the diagonal elements remain. The reason a quantum computer's coherence time is treated as such an important performance metric is that how long these off-diagonal elements can be preserved is exactly how long the computation can retain its "quantumness."

Summary

• The no-cloning theorem means no operation can copy an unknown quantum state without disturbing it, which is the basis for the security of QKD
• Multi-qubit systems are described by tensor products, and a state that cannot be factored into a product state is an entangled state
• The diagonal elements of the density matrix represent measurement probabilities, and the off-diagonal elements represent coherence (the phase relationship that causes interference)
• The measurement result of a quantum computation always comes from the diagonal elements; manipulating the off-diagonal elements is the process that leads there
• Decoherence is the process by which the off-diagonal elements vanish, which is why a quantum computer's coherence time matters so much