Transcript The qubits and the equations of physics

Qubits, time and the equations of physics
Salomon S. Mizrahi
Departamento de Física, CCET,
Universidade Federal de São Carlos
Time and Matter
October 04 – 08, 2010, Budva - Montenegro
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V FEYNFEST-2011
FEYNMAN FESTIVAL – BRAZIL
MAY 02-06, 2011
WWW.FEYNFEST2011.UFSCAR.BR
XII ICSSUR-2011
INTERNATIONAL CONFERENCE ON SQUEEZED
STATES AND UNCERTAINTY RELATIONS
MAY 02-06, 2011
WWW,ICSSUR2011.UFSCAR.BR
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TIME: At
2060
The decay of
the earth
According to
Newton
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Quantum Mechanics
plus
Information and
Communication
Theories
Quantum Information
theory
P. Benioff,
R. Feynman,
D. Deutsch,
P. Schor
Grover
Quantum computation, quantum criptography,
search algorithms,
New vision QM!
Would it be a kind of
information theory?
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What simple concepts of information theory
can tell about the very nature of QM?
Essentially, what can we learn about the more
emblematic equations of QM: the Schrödinger
and Dirac equations?
Dirac Eq. is Lorentz covariant, spatial coordinate plus spin
(intrinsic dof), S-O interaction
Pauli-Schrödinger Eq.
spatial coordinate plus spin (intrinsic dof)
Schrödinger Eq.
spatial coordinate
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Digest
Using the concept of sequence of actions
I will try to show that this is a plausible perspective,
I will use very simple formal tools.
A single qubit is sufficient for nonrelativistic dynamics.
Relativistic dynamics needs two qubits.
The dynamics of the spatial degree of freedom is
enslaved by the dynamical evolution of the IDOF.
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Dynamical equations for qubits
and their carrier
1. Bits, qubits in Hilbert space,
2. Action and a discrete sequence of actions.
3. Uniformity of Time shows up
4. The reversible dynamical equation for a qubit.
5. Information is physical, introducing the qubit carrier, a
massive particle freely moving, the Pauli-Schrödinger
6. The Dirac equation is represented by two qubits,
7. Summary and conclusions
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Bits and maps
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Bits, Hilbert space, action, map
The formal tools to be used:
[I,X] = 0
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Single action operation, map
U() is unitary
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we construct an operator composed
by n sequential events
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THE LABEL OF THE KET IS THE LINEAR CLASSICAL MAP
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The sequence of events is reversible
and norm conserving
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Now we go from a single bit to a qubit
We assume the coefficients real and
on a circle of radius 1
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If one requires a sequence of actions
to be reversible
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one needs
Thus necessarily
Parametrizing as
So, the i enters the theory due to the requirement of
reversibility and normalization of the vectors
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The n sequential action operators become
And we call
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Now, applying the composition law

then necessarily,n = n 
uniformity and
linearity follow
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Computing the difference between consecutive actions,
we get the continuous limit
and
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As an evolved qubit state is given by
it obeys the first order diferential equation for the evolution
of a qubit
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The generator of the motion can be generalized
We recognize  as a mean value
An arbitrary initial state is
A qubit needs a carrier
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Instead of trying to guess what should be  we
write the kinetic energy of the free particle
The solution (in coordinate rep.) is the
superposition
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Where, the carrier position becomes correlated with
qubit state. The probabilities are
If one sets T(p) = μ, a constant,the variable q
becomes irrelevant
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The relation
Finaly leads to the Pauli-Schrödinger
equation
For an arbitrary generator for a particle
under a generic field
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In the absence of the field that probes the qubit, or spin,
We have a decoupling
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For the electron Dirac found the sound
generator
All the 4X4 matrices involved in Dirac´s
theory of the electron can be written as twoqubit operators
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Dirac hamiltonian is
the
  matrices have structure of qubits
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In the nonrelativistic case Z1 is
absent.
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Going back the the usual representation, one
gets the entangled state
The solution to Dirac equation is a superposition of the
nonrelativistic component plus a relativistic complement,
known as (for λ=1) large and small components. However,
they are entangled to an additional qubit that controls
the balance between both components
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Summary
1. One bit and an action U
2. An action depending on exclusive {0,1} parameters is reversible,
3. A sequence actions is reversible, keeping track of the history
of the qubit state.
4. An inverse action with real parameters on a circle of radius 1
does not conserve the norm neither the reversibility of the
vectors.
5. Reversibility is restored only if one extends the parameters to
the field of complex numbers
6. Time emerges as a uniform parameter that tracks the sequence
of actions and we derive a dynamical equation for the qubit.
7. A qubit needs a carrier, a particle of mass m, its presence in
the qubit dynamical equation enters with its kinetic energy, leading
to the Schrödinger equation
8. Dirac equation is properly characterized by two qubits.
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Conclusion
It seems plausible to see QM as a particular
Information Theory where the spin is the
fundamental qubit and the massive particle is its
carrier whose dynamical evolution is enslaved by
the spin dynamics.
Both degrees of freedom use the same clock (a
single parameter t describes their evolution)
Thank you!
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Irreversible open system
The master equation has the solution
Whose solution is
At
 
there is a fix
point
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The generator of the motion can be generalized
Whose eigenvalues and eigenvectors are
For a generic state
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choosing
The probabilities for each qubit component are
And

is the mean energy:
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For an initial superposition
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A qubit needs a carrier or, information is
physical
R. Landauer, Information is Physical, Physics Today, 44, 23-29 (1991).
Doing the generalization
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Instead of trying to guess what should be
kinetic energy of the free particle

we write the
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