Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Eker
Mosher, Mike, Field Producer has reference to this Academic Journal, PHwiki organized this Journal Postulates of Quantum Mechanics SOURCES Angela Antoniu, David Fortin, Artur Ekert, Michael Frank, Kevin Irwig , Anuj Dawar , Michael Nielsen Jacob Biamonte in addition to students Gates on Multi-Qubit State, a reminder Example of Complex quantum system of 3 qubits other realization of Toffoli, composed of 2-qubit gates All gates are at most 2-qubit Only CNOT as 2-qubit gates It has 6 not 5 interaction gates
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Linear Operators V,W: Vector spaces. A linear operator A from V to W is a linear function A:VW. An operator on V is an operator from V to itself. Given bases as long as V in addition to W, we can represent linear operators as matrices. An operator A on V is Hermitian iff it is self-adjoint (A=A). Its diagonal elements are real. Short review Eigenvalues & Eigenvectors v is called an eigenvector of linear operator A iff A just multiplies v by a scalar x, i.e. Av=xv eigen (German) = characteristic. x, the eigenvalue corresponding to eigenvector v, is just the scalar that A multiplies v by. x is degenerate if it is shared by 2 eigenvectors that are not scalar multiples of each other. Any Hermitian operator has all real-valued eigenvectors, which are orthogonal ( as long as distinct eigenvalues). Exam Problems Find eigenvalues in addition to eigenvectors of operators. Calculate solutions as long as quantum arrays. Prove that rows in addition to columns are orthonormal. Prove probability preservation Prove unitarity of matrices. Postulates of Quantum Mechanics. Examples in addition to interpretations.
Unitary Trans as long as mations A matrix (or linear operator) U is unitary iff its inverse equals its adjoint: U1 = U Some properties of unitary trans as long as mations (UT): Invertible, bijective, one-to-one. The set of row vectors is orthonormal. The set of column vectors is orthonormal. Unitary trans as long as mation preserves vector length: U = There as long as e also preserves total probability over all states: UT corresponds to a change of basis, from one orthonormal basis to another. Or, a generalized rotation of in Hilbert space Who an when invented all this stuff A great breakthrough Postulates of Quantum Mechanics Lecture objectives Why are postulates important they provide the connections between the physical, real, world in addition to the quantum mechanics mathematics used to model these systems Lecture Objectives Description of connections Introduce the postulates Learn how to use them in addition to when to use them
Physical Systems – Quantum Mechanics Connections Tensor product of components Composite physical system Postulate 4 Measurement operators Measurements of a physical system Postulate 3 Unitary trans as long as mation Evolution of a physical system Postulate 2 Hilbert Space Isolated physical system Postulate 1 Postulate 1: State Space Systems in addition to Subsystems Intuitively speaking, a physical system consists of a region of spacetime & all the entities (e.g. particles & fields) contained within it. The universe (over all time) is a physical system Transistors, computers, people: also physical systems. One physical system A is a subsystem of another system B (write AB) iff A is completely contained within B. Later, we may try to make these definitions more as long as mal & precise. A B
Closed vs. Open Systems A subsystem is closed to the extent that no particles, in as long as mation, energy, or entropy enter or leave the system. The universe is (presumably) a closed system. Subsystems of the universe may be almost closed Often in physics we consider statements about closed systems. These statements may often be perfectly true only in a perfectly closed system. However, they will often also be approximately true in any nearly closed system (in a well-defined way) Concrete vs. Abstract Systems Usually, when reasoning about or interacting with a system, an entity (e.g. a physicist) has in mind a description of the system. A description that contains every property of the system is an exact or concrete description. That system (to the entity) is a concrete system. Other descriptions are abstract descriptions. The system (as considered by that entity) is an abstract system, to some degree. We nearly always deal with abstract systems! Based on the descriptions that are available to us. States & State Spaces A possible state S of an abstract system A (described by a description D) is any concrete system C that is consistent with D. I.e., it is possible that the system in question could be completely described by the description of C. The state space of A is the set of all possible states of A. Most of the class, the concepts weve discussed can be applied to either classical or quantum physics Now, lets get to the uniquely quantum stuff
An example of a state space Schroedingers Cat in addition to Explanation of Qubits Postulate 1 in a simple way: An isolated physical system is described by a unit vector (state vector) in a Hilbert space (state space) Cat is isolated in the box Distinguishability of States Classical in addition to quantum mechanics differ regarding the distinguishability of states. In classical mechanics, there is no issue: Any two states s, t are either the same (s = t), or different (s t), in addition to thats all there is to it. In quantum mechanics (i.e. in reality): There are pairs of states s t that are mathematically distinct, but not 100% physically distinguishable. Such states cannot be reliably distinguished by any number of measurements, no matter how precise. But you can know the real state (with high probability), if you prepared the system to be in a certain state.
Postulate 1: State Space Postulate 1 defines the setting in which Quantum Mechanics takes place. This setting is the Hilbert space. The Hilbert Space is an inner product space which satisfies the condition of completeness (recall math lecture few weeks ago). Postulate1: Any isolated physical space is associated with a complex vector space with inner product called the State Space of the system. The system is completely described by a state vector, a unit vector, pertaining to the state space. The state space describes all possible states the system can be in. Postulate 1 does NOT tell us either what the state space is or what the state vector is. Revised Postulate 1 Distinguishability of States, more precisely Two state vectors s in addition to t are (perfectly) distinguishable or orthogonal (write st) iff st = 0. (Their inner product is zero.) State vectors s in addition to t are perfectly indistinguishable or identical (write s=t) iff st = 1. (Their inner product is one.) Otherwise, s in addition to t are both non-orthogonal, in addition to non-identical. Not perfectly distinguishable. We say, the amplitude of state s, given state t, is st. Note: amplitudes are complex numbers.
State Vectors & Hilbert Space Let S be any maximal set of distinguishable possible states s, t, of an abstract system A. Identify the elements of S with unit-length, mutually-orthogonal (basis) vectors in an abstract complex vector space H. The Hilbert space Postulate 1: The possible states of A can be identified with the unit vectors of H. Postulate 2: Evolution Postulate 2: Evolution Evolution of an isolated system can be expressed as: where t1, t2 are moments in time in addition to U(t1, t2) is a unitary operator. U may vary with time. Hence, the corresponding segment of time is explicitly specified: U(t1, t2) the process is in a sense Markovian (history doesnt matter) in addition to reversible, since Unitary operations preserve inner product
Example of evolution Time Evolution Recall the Postulate: (Closed) systems evolve (change state) over time via unitary trans as long as mations. t2 = Ut1t2 t1 Note that since U is linear, a small-factor change in amplitude of a particular state at t1 leads to a correspondingly small change in the amplitude of the corresponding state at t2. Chaos (sensitivity to initial conditions) requires an ensemble of initial states that are different enough to be distinguishable (in the sense we defined) Indistinguishable initial states never beget distinguishable outcome Wavefunctions Given any set S of system states (mutually distinguishable, or not), A quantum state vector can also be translated to a wavefunction : S C, giving, as long as each state sS, the amplitude (s) of that state. When s is another state vector, in addition to the real state is t, then (s) is just st. is called a wavefunction because its time evolution obeys an equation (Schrödingers equation) which has the as long as m of a wave equation when S ranges over a space of positional states.
Schrödingers Wave Equation We have a system with states given by (x,t) where: t is a global time coordinate, in addition to x describes N/3 particles (p1, ,pN/3) with masses (m1, ,mN/3) in a 3-D Euclidean space, where each pi is located at coordinates (x3i, x3i+1, x3i+2), in addition to where particles interact with potential energy function V(x,t), the wavefunction (x,t) obeys the following (2nd-order, linear, partial) differential equation: Planck Constant Features of the wave equation Particles momentum state p is encoded implicitly by the particles wavelength : p=h/ The energy of any state is given by the frequency of rotation of the wavefunction in the complex plane: E=h. By simulating this simple equation, one can observe basic quantum phenomena such as: Interference fringes Tunneling of wave packets through potential barriers Heisenberg in addition to Schroedinger views of Postulate 2 in this class we are interested in Heisenbergs view This is Heisenberg picture This is Schroedinger picture
Uncertainty Principle Positive Operator-Valued Measurements (POVM)
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