The Principles of Quantum Mechanics: Pt. 1 PHYS 476Q - Southern Illinois University February 15, 2018 PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 1 / 11
The Partial Trace and the Schmidt Decomposition The partial trace of any rank-one positive operator ψ ψ can easily be calculated in terms of a Schmidt decomposition of ψ. A Schmidt decomposition of ψ has the form ψ = r σ i α i β i i=1 where the α i are orthonormal as well as the β i. Then Tr A ψ ψ = r σi 2 β i β i Tr B ψ ψ = i=1 r σi 2 α i α i. i=1 PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 2 / 11
The Stern-Gerlach Experiment PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 3 / 11
Axiom 1: The State Space Axiom Every quantum system is represented by a Hilbert space H called state space. Physical states of the system are in a one-to-one correspondence with rays in the Hilbert space. Rays in a vector space are simply one-dimensional subspaces: {α ψ : α C}, where ψ is some unit vector. The convention is to represent every physical state by a unit vector ψ. Multiplying ψ by an overall phase e iφ does not change any of the experimental outcomes predicted in quantum mechanics. For a state ψ decomposed in a linear combination ψ = α 0 + β 1, a relative phase is a factor e iφ that is multiplied to just one of the kets: ψ = α 0 + βe iφ 1. Kets ψ and ψ represent different physical states. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 4 / 11
Axiom 2: The Unitary Evolution Axiom Every closed quantum system evolves unitarily in time. The axiom says that there exists a unitary operator U(t 0, t 1 ) acting on H such that if the system is in state ψ at time t 0, it will be in state U(t 0, t 1 ) ψ at time t 1. Since the evolution is unitary, U(t 0, t 1 ) ψ is still a unit vector. Note the entire evolution can be reversed by applying the unitary U(t 0, t 1 ). PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 5 / 11
Axiom 3: The Measurement Axiom A measurement on system H is represented by a hermitian operator X L(H) called an observable, and conversely, every hermitian operator X L(H) corresponds to a physical measurement on H. For an observable X with spectral decomposition X = n k=1 λ kp λk, its eigenvalues λ k are the different values that can be measured when performing the measurement described by X. If ψ H is the pre-measurement state of the system, then value λ k will be measured with probability p(λ k ) = ψ P λk ψ. When λ k is measured, the post-measurement state of the system is 1 p(λk ) P λ k ψ. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 6 / 11
Quantum measurement is a stochastic process. We can only assign a probability distribution {p(λ k )} n k=1 to the n different outcomes of a quantum measurement. Conditioned on outcome λ k, the system undergoes the transformation Pre-measurement: ψ Post-measurement: 1 p(λk ) P λ k ψ. The state of the system is projected into the λ k eigenspace of observable X whenever λ k is measured. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 7 / 11
States represented by eigenvectors of an observable are called eigenstates. If λ k is any eigenvector in the eigenspace V λk of X, then P λl λ k = δ lk λ k. This says that if l k, there is zero probability of obtaining outcome λ l when the system is prepared in state λ k. The only possible outcome is λ k, and the state λ k remains unchanged in the measurement process. For this reason, eigenstates of an observable are referred to as stationary states. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 8 / 11
Stern-Gerlach Example PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 9 / 11
Axiom 4: The Composite System Axiom If A and B are two quantum systems with state spaces H A and H B respectively, the state space of the their combined physical system is the tensor product space H AB := H A H B. For more systems, H A 1, H A 2,, H An, the joint state space is the n-fold tensor product space H A 1 H A 2 H An. Two quantum systems are often called a bipartite system while more than two systems are typically referred to as a multipartite system. A bipartite state ψ AB is entangled if it is not a tensor product state; i.e. ψ AB α A β B. A non-entangled state is also called a product or separable state. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 10 / 11
How to decide if a state is entangled? Proposition: A bipartite state ψ is a product state iff its operator representation M ψ is rank one. Recall a nonzero matrix is rank 1 iff all of its 2 2-minors vanish. Example Decide if the following C 3 C 3 state is entangled: ψ AB = a 11 + ab 12 + c 13 + ab 21 + b 22 + c b/a 23 c 33. PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018 11 / 11