In QM, every state requires basis vectors to represent. Mixed states (combined states) require special basis vectors. This post discusses some of these topics.

Combined State of a Quantum System

Quantum Combined State

In quantum mechanics, the combined state of a composite system, such as a system composed of subsystems A and B, is described by the tensor product of the individual states of the subsystems. If subsystem A is in state ∣ψA⟩|\psi_A\rangle∣ψA​⟩ and subsystem B is in state ∣ψB⟩|\psi_B\rangle∣ψB​⟩, the combined state of the system is given by:

∣Ψ⟩=∣ψA⟩⊗∣ψB⟩|\Psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle∣Ψ⟩=∣ψA​⟩⊗∣ψB​⟩

The basis vectors for the combined state are formed by the tensor products of the basis vectors of the subsystems. If {∣ai⟩}\{|a_i\rangle\}{∣ai​⟩} and {∣bj⟩}\{|b_j\rangle\}{∣bj​⟩} are basis vectors for subsystems A and B, respectively, the basis vectors for the combined system are {∣ai⟩⊗∣bj⟩}\{|a_i\rangle \otimes |b_j\rangle\}{∣ai​⟩⊗∣bj​⟩}.

For example, if subsystem A has basis vectors ∣a1⟩|a_1\rangle∣a1​⟩ and ∣a2⟩|a_2\rangle∣a2​⟩, and subsystem B has basis vectors ∣b1⟩|b_1\rangle∣b1​⟩, ∣b2⟩|b_2\rangle∣b2​⟩, and ∣b3⟩|b_3\rangle∣b3​⟩, the combined system will have the following basis vectors:

∣a1⟩⊗∣b1⟩, ∣a1⟩⊗∣b2⟩, ∣a1⟩⊗∣b3⟩, ∣a2⟩⊗∣b1⟩, ∣a2⟩⊗∣b2⟩, ∣a2⟩⊗∣b3⟩|a_1\rangle \otimes |b_1\rangle, \ |a_1\rangle \otimes |b_2\rangle, \ |a_1\rangle \otimes |b_3\rangle, \ |a_2\rangle \otimes |b_1\rangle, \ |a_2\rangle \otimes |b_2\rangle, \ |a_2\rangle \otimes |b_3\rangle∣a1​⟩⊗∣b1​⟩, ∣a1​⟩⊗∣b2​⟩, ∣a1​⟩⊗∣b3​⟩, ∣a2​⟩⊗∣b1​⟩, ∣a2​⟩⊗∣b2​⟩, ∣a2​⟩⊗∣b3​⟩

In general, if subsystem A has $n$ basis states and subsystem B has $m$ basis states, the combined system will have $n\times m$ basis states.

Classical Combined State

In classical mechanics, the combined state of a system is described by the Cartesian product of the states of the subsystems. If subsystem A is described by coordinates (x1,p1)(x_1, p_1)(x1​,p1​) and subsystem B by coordinates (x2,p2)(x_2, p_2)(x2​,p2​), the state of the combined system is described by the tuple (x1,p1,x2,p2)(x_1, p_1, x_2, p_2)(x1​,p1​,x2​,p2​).

A classical pure state is a point in the phase space, and a classical mixed state is described by a probability density function ρ(x1,p1,x2,p2)\rho(x_1, p_1, x_2, p_2)ρ(x1​,p1​,x2​,p2​) over the phase space.

Comparison

• Quantum Basis Vectors: In quantum mechanics, the basis vectors for the combined state are formed by the tensor product of the basis vectors of the subsystems. These basis vectors can represent entangled states where the subsystems are not independent.
• Classical Basis Vectors: In classical mechanics, the state of the combined system is described by the Cartesian product of the states of the subsystems. The concept of basis vectors is not typically used in the same way as in quantum mechanics. Instead, the state of the system is described by a point or a probability density function in phase space.
• Independence of Subsystems: In a classical combined state, knowledge of the state of the entire system implies complete knowledge of each subsystem. However, in a quantum combined state, especially when entanglement is involved, the state of each subsystem can be mixed even if the state of the combined system is pure

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