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Qubits, time and the equations of physics

Salomon S. Mizrahi

Departamento de Física, CCET, Universidade Federal de São Carlos

Time and MatterOctober 04 – 08, 2010, Budva -

Montenegro

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V FEYNFEST-2011FEYNMAN FESTIVAL – BRAZIL

MAY 02-06, 2011WWW.FEYNFEST2011.UFSCAR.BR

XII ICSSUR-2011INTERNATIONAL CONFERENCE ON SQUEEZED

STATES AND UNCERTAINTY RELATIONSMAY 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

Quantum computation, quantum criptography, search algorithms,

plus

Information and Communication

Theories

Quantum Informationtheory

P. Benioff,R.

Feynman,D.

Deutsch,P. SchorGrover

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 actionsI 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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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-Schrödinger6. The Dirac equation is represented by two qubits, 7. Summary and conclusions

Dynamical equations for qubitsand their carrier

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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 andon 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, = nn

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 evolutionof a qubit

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The generator of the motion can be generalized

An arbitrary initial state is

A qubit needs a carrier

We recognize as a mean value

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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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If one sets T(p) = μ, a constant,the variable q becomes irrelevant

Where, the carrier position becomes correlated with qubit state. The probabilities are

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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 soundgenerator

All the 4X4 matrices involved in Dirac´s theory of the electron can be written as two-qubit operators

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the matrices have structure of qubits

Dirac hamiltonian is

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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 U2. 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 numbers6. 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 equation8. 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 we write the kinetic energy of the free particle