How possible write Pauli matrices X, Y, Z, without imaginary unit i?
With imaginary unit pauli matrices looks like this:
X= [0 1]
.....[1 0]
Y=[0 -i]
....[i 0]
Z=[1 0]
....[0 -1]
Each of Pauli matrices X, Y, Z, possible write without i, like this:
[(cosx; sinx) (cosx; sinx)]
[(cosx; sinx) (cosx; sinx)]
Cosx is x axis, and sinx is y axis (x; y) in decard coordinates.
So then possible all pauli matrices to write in decard coordinates (x; y) without imaginary unit "i":
X=[0 (cos0; sin0)]
....[(cos0; sin0) 0]
Y=[0 (cos270; sin270)]
....[(cos90; sin90) 0]
Z=[(cos0; sin0) 0]
....[0 (cos180; sin180)]
So I all Pauli matrices wrote in 2D space coordinates (x; y) and don't use imaginary unit i. This (x; y) coordinates describing phase. X matrix flip value 180 degrees. Y matrix fliping (rotating) phase 90 degrees or 270 degrees depending on value. Z matrix don't rotate phase or rotate phase 180 degrees, deoending on value.
One bit can be writen as:
|0>=[1]
.......[0]
|1>=[0]
.......[1]
Pauli matrices to 0 or 1 doing this:
X|0>=|1>
X|1>=|0>
Y|0>=(cos90; sin90)|0>
Y|1>=(cos270; sin270)|1>
Z|0>=|0>
Z|1>=-|1>
So if |0> or |1> is spin "up" or spin "down" (or imaginary electron rotation in one or another direction around his axis) then X matrix flips electron spin (electron rotation direction), Y matrix will rotate spin "up" phase 90 degrees (electron would make quantum leap by adding 1/4 his rotation energy) and spin "down" would change phase 270 degrees and would make quantum leap 3/4 adding speed. Z matrix spin spin "up" don't changing and spin down changing phase 180 degrees by adding 1/2 quantum leap.
Strange is quantum mechanic...
How with digital spin possible operate in many intractables ways? 
Every N dimensional complex vector space is isomorphic to a 2N dimensional Real vector space.
You just write every entry i in a matrix as the 2x2 Real matrix
( 0 1 )
( 1 0 )
You just write every entry i in a matrix as the 2x2 Real matrix
( 0 1 )
( 1 0 )
I not very good in understanding isomorphism, perturbatism and over wonderness. I was thinking that if I understood what is "i" and complex numbers then I would understood what a hell means pauli matrices, but when I understood complex numbers it's still nothing give about understanding of pauli matrices and how to operate with them... So I thinking maybe it's possible simplificies all this? For example all this pauli matrices very easy to describe on Bloch sphere
cosx|0>+(cosy; siny)sinx|1>, where (cosx)^2 is probability to measure spin up and (sinx)^2 is probability to measure spin down, and (cosy; siny) is possible phase of spin down (if you measure spin down). For |0> don't need phase becouse it's phase is transformed to |1> phase, becouse it's don't matter.
Why pauli matrices are exactly like this and not some rotations? And I don't understand how they test what pauli matrces is exactly this ones?
http://www-inst.eecs.berkeley.edu/~cs191/s...s/lecture11.pdf
cosx|0>+(cosy; siny)sinx|1>, where (cosx)^2 is probability to measure spin up and (sinx)^2 is probability to measure spin down, and (cosy; siny) is possible phase of spin down (if you measure spin down). For |0> don't need phase becouse it's phase is transformed to |1> phase, becouse it's don't matter.
Why pauli matrices are exactly like this and not some rotations? And I don't understand how they test what pauli matrces is exactly this ones?
http://www-inst.eecs.berkeley.edu/~cs191/s...s/lecture11.pdf
Rotations in 2d commute. The set of 2d rotation matrices are the group SO(2). When you convert this into a complex representation, it becomes U(1). U(1) is just a complex number and so you end up finding that the structure of the group is uninteresting, since doing a rotation by a and then b is the same as by b and then a.
The Pauli matrices form the group SU(2). They do not commute, so applying one then another is not the same as doing it in the reverse order. The properties of quantum mechanics is such that this kind of structure is common.
They are essentially the rotations in 2 complex dimensions, not two Real dimensions. You can partly convert this into a Real repretsentation by realising that SO(4) (the group of rotations in 4 real dimensions) has a Lie algebra which is the sum of two Lie algebras of SU(2). That's probably too technical for what you're talking about.
I would suggest doing some reading on representations and various kinds of morphisms, since they are the language of what you're asking about.
The Pauli matrices form the group SU(2). They do not commute, so applying one then another is not the same as doing it in the reverse order. The properties of quantum mechanics is such that this kind of structure is common.
They are essentially the rotations in 2 complex dimensions, not two Real dimensions. You can partly convert this into a Real repretsentation by realising that SO(4) (the group of rotations in 4 real dimensions) has a Lie algebra which is the sum of two Lie algebras of SU(2). That's probably too technical for what you're talking about.
I would suggest doing some reading on representations and various kinds of morphisms, since they are the language of what you're asking about.
In bloch sphere representation I guess possible write in this phorm:
(cosy+isiny)cosx|0>+(cosy-isiny)sinx|1>
and this means that |0> phase is rotates normal and |1> phase is rotating counterclock. And then if you rotate 360 degrees then it's means that you rotate 180 degrees, becouse phase and state are rotating by half angle. And if you write in this phorm:
cosx|0>+sinx(cosy+isiny)|1> then probability is rotates half angle and phase to all angle, normaly. And this two equations are equal, becouse phase is rotated in diferent directions for |0> and |1>. For example we want rotate probability by 360 and phase by 360 degrees, then this looks like this:
(cos180+isin180)cosx|0>+(cos180-isin180)sinx|1>=
=(1+i0)1|0>+(1-i0)0|1>=|0>
or
cos180|0>+sin180(cos360+isin360)|1>=
1|0>+0(1+i0)|1>=|0>.
Another example, phase we rotate 270 degrees and probability 720 degrees:
(cos135+isin135)cos360|0>+(cos135-isin135)sin360|1>=
=(-0.707+i0.707)1|0>+(-0.707-i0.707)0|1>=|0>
or
cos360|0>+sin360(cos270+isin270)|1>=
1|0>+0(0-i1)|1>=|0>.
So in both case phase shift between |0> and |1> is 90 degrees. In second case perhaps just simpliefed writing.
I just remeber that probability (cosx and sinx) can be rotated more than 90 degrees or more than 180/2 degrees...).
In link is showing like electron with arbitrary spin flying through magnetic field and somehow his spin is becoming in superposition of cosx|0>+sinx|1>. I can't understand this, if spin generating magnetic field along spin rotation axis then electron spin must be alinged according to local magnetic field, becouse local field has two poles and spin has two poles then opposit poles must pull and electron spin must align according polarization of magnetic field of magnets. But his some strange going in superposition and doing hell know what
I am more interesting how to rotate spin with magnetic field, how make superposition of spin and how change phase of one or another spin state in superposition. What is all possible ways of manipulating with spin.
(cosy+isiny)cosx|0>+(cosy-isiny)sinx|1>
and this means that |0> phase is rotates normal and |1> phase is rotating counterclock. And then if you rotate 360 degrees then it's means that you rotate 180 degrees, becouse phase and state are rotating by half angle. And if you write in this phorm:
cosx|0>+sinx(cosy+isiny)|1> then probability is rotates half angle and phase to all angle, normaly. And this two equations are equal, becouse phase is rotated in diferent directions for |0> and |1>. For example we want rotate probability by 360 and phase by 360 degrees, then this looks like this:
(cos180+isin180)cosx|0>+(cos180-isin180)sinx|1>=
=(1+i0)1|0>+(1-i0)0|1>=|0>
or
cos180|0>+sin180(cos360+isin360)|1>=
1|0>+0(1+i0)|1>=|0>.
Another example, phase we rotate 270 degrees and probability 720 degrees:
(cos135+isin135)cos360|0>+(cos135-isin135)sin360|1>=
=(-0.707+i0.707)1|0>+(-0.707-i0.707)0|1>=|0>
or
cos360|0>+sin360(cos270+isin270)|1>=
1|0>+0(0-i1)|1>=|0>.
So in both case phase shift between |0> and |1> is 90 degrees. In second case perhaps just simpliefed writing.
I just remeber that probability (cosx and sinx) can be rotated more than 90 degrees or more than 180/2 degrees...).
In link is showing like electron with arbitrary spin flying through magnetic field and somehow his spin is becoming in superposition of cosx|0>+sinx|1>. I can't understand this, if spin generating magnetic field along spin rotation axis then electron spin must be alinged according to local magnetic field, becouse local field has two poles and spin has two poles then opposit poles must pull and electron spin must align according polarization of magnetic field of magnets. But his some strange going in superposition and doing hell know what
I am more interesting how to rotate spin with magnetic field, how make superposition of spin and how change phase of one or another spin state in superposition. What is all possible ways of manipulating with spin.
...one normal and one cklockwise...
I read that take nuclear of say CO2 or H2O and this nuclear or molecule was flying trough magnetic field and was excpected that nuclear will leave measurment random line, but they sow that there is not line, but digitized or quantized points. So this means that nuclear with arbitrary spin was flying and then he is atracting to magnetic field then he don't having his some concrete state but is in both states simultnously or with some probability to be spin up and with some spin down and since in nuclear is many protons then they probabilisticly sum up and magnetic field probabilistacly atracing them and they therefore in superposition with some probability can go in one of some concrete number of points, becouse one protons can be oposit and some overs protons can have another spin...
So if between magnetic field flying electron then it is in arbitrary state or maybe in superposition but he behave like he is in superposition so say electron is in superposition and flying to magnetic field and then one of his spin superpostion want flying to one direction (up or right) and another spin state want fly in another direction (down or left). But imposible to find electron in two place at same time so electron would be obtained only in one place (up or down; right or left) with some probability. So there is reason to think that electron always acting in superposition and all atoms and molecules in termodinamic acting like in superposition and entangled. But how to make to be electron in one state if it is needed?
My teory can explain superposition as electron flying with arbitrary spin and if there is magnetic field then electron spliting into two parts and each part is with opposit to another part spin and with spin which is along magnetic field axis or more precisly spin generated magnetic moment, which is along local magnetic field axis, but with oposit magnetinc polarization. So then those two parts reaching some obstacle then one of those parts coming to another part with infinity speed using minimum space rule taking structure and then we optaning discreat quantized spin with some probability. IF flying some atom then his protons, which is initialy in arbitrary magnetic moment position spliting into 2^n parts, where n is number of protons in atom. This parts are with some quantized numbers of spins and this number of spin is like was number of protons. And so then if some part is measured then all over parts with infinity speed going back to this part, which is with quantized spins (up or down).
It's stange to me how two electrons can be entangled and what distance must be between them that they become entangled? And does they can be entangled weaker or stronger?
But electrons don't have normal live, they can't alignth to weak magnetic field not 180 degrees with magnetic moment but say 120 or 90 degrees. Or maybe can? If magnetic field is very weak then maybe in superposition and with some degrees of aligneth to magnetic field? So it's looks like electrons are less 'free' in space, but can be in many place and this looks like transition from quantity to quality. IF this we looking from quantum computer perspective or brain perspective then brain need quality and not quantity. And if use quantum computer then maybe possible win in some problems? But even if it's possible, like going from electric tubes to transistors then it's still don't means solvability of NP problems, becouse NP problems is infinity and you can't with something finity to solve somthing infinity. Okey, maybe quantum computer would be 10^10 - 10^30 times faster than classical computer and so what this proof? But since qc is analog in his nature it's possible that he wouldn't be faster than transistors processors.
I read that take nuclear of say CO2 or H2O and this nuclear or molecule was flying trough magnetic field and was excpected that nuclear will leave measurment random line, but they sow that there is not line, but digitized or quantized points. So this means that nuclear with arbitrary spin was flying and then he is atracting to magnetic field then he don't having his some concrete state but is in both states simultnously or with some probability to be spin up and with some spin down and since in nuclear is many protons then they probabilisticly sum up and magnetic field probabilistacly atracing them and they therefore in superposition with some probability can go in one of some concrete number of points, becouse one protons can be oposit and some overs protons can have another spin...
So if between magnetic field flying electron then it is in arbitrary state or maybe in superposition but he behave like he is in superposition so say electron is in superposition and flying to magnetic field and then one of his spin superpostion want flying to one direction (up or right) and another spin state want fly in another direction (down or left). But imposible to find electron in two place at same time so electron would be obtained only in one place (up or down; right or left) with some probability. So there is reason to think that electron always acting in superposition and all atoms and molecules in termodinamic acting like in superposition and entangled. But how to make to be electron in one state if it is needed?
My teory can explain superposition as electron flying with arbitrary spin and if there is magnetic field then electron spliting into two parts and each part is with opposit to another part spin and with spin which is along magnetic field axis or more precisly spin generated magnetic moment, which is along local magnetic field axis, but with oposit magnetinc polarization. So then those two parts reaching some obstacle then one of those parts coming to another part with infinity speed using minimum space rule taking structure and then we optaning discreat quantized spin with some probability. IF flying some atom then his protons, which is initialy in arbitrary magnetic moment position spliting into 2^n parts, where n is number of protons in atom. This parts are with some quantized numbers of spins and this number of spin is like was number of protons. And so then if some part is measured then all over parts with infinity speed going back to this part, which is with quantized spins (up or down).
It's stange to me how two electrons can be entangled and what distance must be between them that they become entangled? And does they can be entangled weaker or stronger?
But electrons don't have normal live, they can't alignth to weak magnetic field not 180 degrees with magnetic moment but say 120 or 90 degrees. Or maybe can? If magnetic field is very weak then maybe in superposition and with some degrees of aligneth to magnetic field? So it's looks like electrons are less 'free' in space, but can be in many place and this looks like transition from quantity to quality. IF this we looking from quantum computer perspective or brain perspective then brain need quality and not quantity. And if use quantum computer then maybe possible win in some problems? But even if it's possible, like going from electric tubes to transistors then it's still don't means solvability of NP problems, becouse NP problems is infinity and you can't with something finity to solve somthing infinity. Okey, maybe quantum computer would be 10^10 - 10^30 times faster than classical computer and so what this proof? But since qc is analog in his nature it's possible that he wouldn't be faster than transistors processors.
Here I give you a little teory about quantum computers.
Quantum computer is device, which using quantum mechainc to perform faster algorithms and computations...
For now there is only 3 algorithms for quantum computer and all they realated with problems, which grow exponentionlay then number of inputs grows linarly.
This 3 algorithms is a) Grover's algorihtm; b) Shor algorithm; c) atoms and molecules emulation algorithm.
Grover's algorithm is very weak. If for some same size classical computer need 2^n time, then for Grover's algorithm on QC need only (2^n)^0.5 time. There is very sirious problem with Grover's algorithm. If n (number of inputs) is very big then even for grover's algorithm need very much time. And becouse grover's algorithm can't work very much time then QC performing grover's algorithm can't be faster than classical computer more than about 10^10 - 10^20 times. If quantum computer working on about 10^10 Hz. And classical computer with many transistors and processors can be faster than quantum computer. So about 10^15 times don't means that quantum computer can solve effiecently NP problems and is just in best case very sucessfull hypercomputer.
Shor algorithm is very fast and if for classical computer need about 2^{n^0.3} times then for shor's algorihm on QC need only n^2 time. So shor algorithm is very fast... But this algorithm don't geting to humankind any benefit. This algorithm just for factoring numbers and RSA code breaking... So I don't know maybe exist fast factoring classical algorithm or maybe not, but shor algorithm isn't needed to someone. So for Shor's algorithm, place is trash.
About Atomic and moleculs emulation is very hard to say what will be benefit from it... This algorithm is exponentionaly faster than classical algorithm (like in shor algorithm case), so for classical computer need about 2^n time and for quantum computer performing this algorithm need only n time. But I am doubting about this algorithm usefulness, becouse if say there is HIV virus and this virus RNA has 10000 bases and this is about 300000 atoms. And all virus having say 10 milion atoms then to emulate this virus need at least 10^7 qubits. Okey, we can emulate this virus on quantum computer. And say now we can emulate some random molecule structure of thousands of atoms, which should kill this virus. But we don't know what exactly possibles combinations of atoms would kill this virus. So combinations can be of HIV killer molecule more than 10^1000000000000000000. Becouse atoms number of molecule can be bigger, can be smaller and in many possible combinations. So still nobody wouldn't know what combinations need to kill HIV virus. All vacines is maded by dead or 'weak' virus, which is obtained by human cell and which is manufactured and more and more to prepeare and kill real/alive some virus. So that becouse some virus can be destroid and it's nothing to do with that human somthing very clever understood about viruses or guess some combination...
So to sumarize all, grover algorithm can be ouperformed by more processors and transistors in processors; Shor algorithm nobody needing; quantum simulation is like real world and nothing can do better than real world... For example somebody can say that with atomic emulation will be possible creat virtual human, but human consist of 10^30 atoms and then need 10^30 qubits. Such number of qubits would take all surface of Earth. So according to this, NP problems still imposible to solve with quantum computer. A little bit faster - maybe... But I am now was talking optimistacaly and if QC wouldn't be able to operate in analog way with enough precision then all things can be more pesimistic...
Note. Exponentional classical algorithm can be described. Faster than Grover algorithm can't be described and this is proved. Quantum simulation can give some answer with small precision and you would be hard to say, which answer is good, and this means that maybe is needed to repeat about 1000y times, where y=1/x and x is good answer giving probability, to compare all answer and choose one which apearing most frenquently.
Quantum computer is device, which using quantum mechainc to perform faster algorithms and computations...
For now there is only 3 algorithms for quantum computer and all they realated with problems, which grow exponentionlay then number of inputs grows linarly.
This 3 algorithms is a) Grover's algorihtm; b) Shor algorithm; c) atoms and molecules emulation algorithm.
Grover's algorithm is very weak. If for some same size classical computer need 2^n time, then for Grover's algorithm on QC need only (2^n)^0.5 time. There is very sirious problem with Grover's algorithm. If n (number of inputs) is very big then even for grover's algorithm need very much time. And becouse grover's algorithm can't work very much time then QC performing grover's algorithm can't be faster than classical computer more than about 10^10 - 10^20 times. If quantum computer working on about 10^10 Hz. And classical computer with many transistors and processors can be faster than quantum computer. So about 10^15 times don't means that quantum computer can solve effiecently NP problems and is just in best case very sucessfull hypercomputer.
Shor algorithm is very fast and if for classical computer need about 2^{n^0.3} times then for shor's algorihm on QC need only n^2 time. So shor algorithm is very fast... But this algorithm don't geting to humankind any benefit. This algorithm just for factoring numbers and RSA code breaking... So I don't know maybe exist fast factoring classical algorithm or maybe not, but shor algorithm isn't needed to someone. So for Shor's algorithm, place is trash.
About Atomic and moleculs emulation is very hard to say what will be benefit from it... This algorithm is exponentionaly faster than classical algorithm (like in shor algorithm case), so for classical computer need about 2^n time and for quantum computer performing this algorithm need only n time. But I am doubting about this algorithm usefulness, becouse if say there is HIV virus and this virus RNA has 10000 bases and this is about 300000 atoms. And all virus having say 10 milion atoms then to emulate this virus need at least 10^7 qubits. Okey, we can emulate this virus on quantum computer. And say now we can emulate some random molecule structure of thousands of atoms, which should kill this virus. But we don't know what exactly possibles combinations of atoms would kill this virus. So combinations can be of HIV killer molecule more than 10^1000000000000000000. Becouse atoms number of molecule can be bigger, can be smaller and in many possible combinations. So still nobody wouldn't know what combinations need to kill HIV virus. All vacines is maded by dead or 'weak' virus, which is obtained by human cell and which is manufactured and more and more to prepeare and kill real/alive some virus. So that becouse some virus can be destroid and it's nothing to do with that human somthing very clever understood about viruses or guess some combination...
So to sumarize all, grover algorithm can be ouperformed by more processors and transistors in processors; Shor algorithm nobody needing; quantum simulation is like real world and nothing can do better than real world... For example somebody can say that with atomic emulation will be possible creat virtual human, but human consist of 10^30 atoms and then need 10^30 qubits. Such number of qubits would take all surface of Earth. So according to this, NP problems still imposible to solve with quantum computer. A little bit faster - maybe... But I am now was talking optimistacaly and if QC wouldn't be able to operate in analog way with enough precision then all things can be more pesimistic...
Note. Exponentional classical algorithm can be described. Faster than Grover algorithm can't be described and this is proved. Quantum simulation can give some answer with small precision and you would be hard to say, which answer is good, and this means that maybe is needed to repeat about 1000y times, where y=1/x and x is good answer giving probability, to compare all answer and choose one which apearing most frenquently.