Quantum Theory
On the Formulations of Quantum Mechanics¶
Argument: Quantum Theory
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Why
-norm and why not -norm?Because if there are any linear transformations other than these trivial ones that preserve the
-norm, then either or .If
we get classical probability theory, while if we get quantum mechanics.Also all transformations must be norm preserving. So what kind of matrix or linear transformation preserves the
-norm? Unitary Matrices . -
Why complex numbers and not real numbers?
Axiom of continuity: There exists a continuous reversible transformation on a system between any two pure states of that system.
Thus, we need our field associated with vector spaces to be algebraically closed. Hence, we need complex numbers.
If you want every unitary operation to have a square root, then you have to go to the complex numbers.
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Why do transformations need to be linear?
If quantum mechanics were nonlinear, then one could build a computer to solve NP-complete problems in polynomial time. (Abrams and Lloyd).
History of Quantum Computing¶
Models of Classical Computation¶
The following serve as universal models of computation:
- Turing Machines
-calculus- Circuits
Super-Universal¶
Circuits can describe computations which are beyond what a Turing machine can do.
Apparently, we can solve the halting problem under some circuit model. But, what is that supposed to mean?
Quantum Computation¶
The idea of a QC model is to have information in superposition. If we want Turing Machines to be part of the quantum computational model then we will somehow have to make the read/write head to be in a superposition. But that is not possible (doubt). Hence we use a circuit where we keep the bits in a superposition and operate on them with gates.
History of Quantum Mechanics and Computation¶
Major questions in QM started arising with the EPR paradox in 1936 when faster than light travel was questioned.
Here's the paradox: Consider
In 1964 Bell gave his Theorem and then it was later verified too which proved that Quantum Mechanics is cool with EPR pair. But there was another problem that how is faster than light communication possible? It was later proved that faster than light communication is not possible indeed as you cannot actually send 'information' with the entanglement coordination. Basically you cannot manipulate the particles to actually transmit any useful information.
Non-cloning theorem: In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others.
Quantum Theory from 5 reasonable Axioms¶
State¶
Mathematical object that can be used to deter-mine the probability associated with the out-comes of any measurement that may be per-formed on a system prepared by the given preparation.
However, we do not need to measure all possible probability measurements to determine the state of a system.
- K (degrees of freedom): minimum number of probability measurements required to determine the state of a system, i.e., number of real parameters re-quired to specify the state.
- N (dimension): maximum number of states that can be reliably distinguished from one another in a single shot measurement.
Axioms¶
- Probabilities
- Simplicity:
- Subspaces: If state of a system
subspace - Composite systems:
- Continuity:
a continuous reversible transformation on a system between anytwo pure states of that system
Landauer's Principle¶
Landauer's Principle explains that when info is erased, it requires work.
Each bit erased
Hence to keep constant temp, that amount of work needs to be put in. This generalises to reversible processes requiring work.
Postulates of Quantum Mechanics¶
I shall state here the four major generic postulates of Quantum Mechanics stated in terms of both state-vector formalization and density-matrix formalization.
State is a vector¶
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Isolated physical system is given by its state vector operating on a certain Hilbert space.
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Evolution of a closed quantum system is given by a unitary transformation.
In its physical interpretation we have this postulate governed by the Schrodinger Equation, as stated.
The Hamiltonian is a hermitian operator and has a spectral decomposition,
. -
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
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Quantum measurements are described by a collection
of measurement operators acting on the state space of the system.Probability that upon measurement the outcome is
and the state of the system becomes as follows.Measurement operators also follow the completeness equation,
.
State is a density matrix¶
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Isolated physical system is given by its density matrix operating on a certain Hilbert space.
-
Evolution of a closed quantum system is given by a unitary transformation as
. -
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
-
Quantum measurements are described by a collection
of measurement operators acting on the state space of the system.Probability that upon measurement the outcome is
and the state of the system becomes as follows.Measurement operators also follow the completeness equation,
.
Measurement¶
Postulate¶
Quantum Measurements are descrivbed by a collection of measurement operators
If
Non-distinguishability of arbitrary states¶
We cannot distinguish any two arbitrary non-orthogonal quantum states.
Proof: Let
Then
Thus,
Hence, proved by contradiction.
Projective Measurements¶
We can use projective measurement formalism for any general measurement too.
In case of projective measurements,
Here,
POVM Measurements¶
POVM Measurements are a formalism where only measurement statistics matters.
Here, each of
Global Phase doesn't matter¶
We say
However, be aware that the global phase is quite different from the relative phase.
Density Matrices¶
We can represent a system as an ensemble of pure states
This formulation helps us a lot in dealing with quantum information, noisy systems and helps us represent measurements better, as well. Why? Because it provides a convenient means for describing quantum systems whose state is not completely known.
Postulates in Density Matrices formulation¶
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Isolated physical system is given by its density matrix operating on a certain Hilbert space.
-
Evolution of a closed quantum system is given by a unitary transformation as
. -
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
-
Quantum measurements are described by a collection
of measurement operators acting on the state space of the system.Probability that upon measurement the outcome is
and the state of the system becomes as follows.Measurement operators also follow the completeness equation,
.
Properties¶
Theorem 1: An operator
Converse is easy to prove. If an operator is both positive and has trace as one, then it shall have a spectral decomposition of the form
For the direct proof, let us consider
Theorem 2: The sets
where
As a consequence of the theorem, note that
Theorem 3: If
Theorem 4: Observable
Reduced Density Operator¶
This is the single-most important application of density operator formulation is the existence of reduced density operator. It is defined as follows.
This allows us to talk about sub-systems of a composite system.
Schmidt Decomposition¶
Suppose
where
Purification¶
Suppose we have a mixed state
Given
Bloch Sphere and Rotations¶
We can represent any
Single Qubit Operations¶
Thus,
In general, we have the above equation where
Some Algebra¶
Theorems¶
- Any arbitrary single qubit unitary operator can be written in the form
. - Suppose
is a unitary operation over a single qubit then such that . - There exists unitaries
for any given unitary such that and .
Proof that Bloch Sphere Unitaries as Rotations¶
A single qubit operator can be represented as
Also, such a unitary can also be represented this way,
and thus, we obtain the following equivalences.
Also from
where we define
Then define,
and further we get
Now, we define
By putting these in the other constraints we get,
CHSH Inequality¶
In a classical experiment, we have the following setup where Alice can choose to measure either Q or R and Bob chooses either S or T. The measurements are performed imultaneously and far off from each other.
and thereby we obtain the inequality
This is one of the set of Bell inequalities, the first of which was found by John Bell. This one in particular is named CHSH inequality.
Quantum Anomaly¶
In the quantum case, let us consider the measurements to be based on the following observables over the EPR pair
Then, we have the following result.
Thus, in other words, CHSH inequality doesn't hold.
Interpretation¶
The fact that CHSH doesn't hold in the quantum scenario implies that two of the major assumptions about nature is wrong in case of the classical experiment.
The assumptions are:
- Realism: Q, R, S, T are physical quantities which have defininte values irrespective of observation.
- Locality: Alice's measurement doesn't influence that of Bob's.
Thus, the result of CHSH being false when accounted for the quantum mechanical properties of nature (we can perform the experiment in a lab with particles) suggests that nature cannot be locally real and neither can any true mathematical representation of it be locally real.