Since this work is aimed primarily at programmers, we will use the C/C++ notations for standard logical operations. However, since C/C++ usage of ^ and ! conflict with standard mathematical usage for these symbols we will use this font to distinquish logical operators, Viz:
C/C++ | name | Definition |
! | NOT | ! 0 = 1 ; ! 1 = 0 |
| | OR | (0 | 1) = (1 | 0) = (1 | 1) = 1 ; (0 | 0) = 0 |
& | AND | (0 & 0) = (0 & 1) = (1 & 0) = 0 ; 1 & 1 = 1 |
^ | XOR | (0 ^ 0) = (1 ^ 1) = 0 ; (0 ^ 1) = (1 ^ 0) = 1 |
Let us step back and ask what exactly we mean when we say that
an "outcome space" {↑,↓} is a "binary" state or property?
We usually mean that one (and only one) of the properties must "hold" at any one "instant" in "time".
That at any given time t we must have
EITHER (↑ & ( ! ↓)) OR (↓ & ( ! ↑))
which we can write as
( (↑ & ( ! ↓)) ^ (↓ & ( ! ↑)) ) = 1
.
We cannot have both at once,
and we cannot have neither. We must have one, but only one.
One "boolean variable" thus suffices to hold "both"
↑-ness and ↓-ness "properties" because ↑ = ! ↓ and the
two theoretical outcomes (↑ & ↓) and ( ! ↑ & ! ↓) cannot occurr.
It is by relaxing this last assumption that we enter the bizarre realm of quantum logic .
A "binary" property now has four possible discrete "states" or "outcomes", two of which
[ ↑ & ( ! ↓) and ( ! ↑) & ↓ ]
make sense classically
and two "phantom" states
[ ↑ & ↓ and ( ! ↑) & ( ! ↓) ]
which don't.
It is important here to recognise that ↑ and ↓ have not suddenly become "independant" variables.
We retain ↑ = ! ↓ in the sense that if we observe the ↑ and ↓
properties we insist on a classical outcome. It is the values of these states in the "spooky twilight zone"
between observations
that we "allow" to be non-classical.
Since ! ↑ is now a distinct concept from ↓ we represent ↓ by e_{2} orthogonal to e_{1} rather than by -e_{1}, which we also consider to represent ↑. We can then form weighted linear combinations of the "two" classically opposed states and the "weights" conventionally used in QM are complex numbers x + i y where i is an "imaginary" Ö-1 that commutes with everything.
A quantum bit or qubit is the superposition of the alternate
states of a classical binary state { ↑,↓}
y(t) = z_{1}(t) ï↑ñ + z_{2}(t) ï↓ñ
= z_{1}(t) e_{1} + z_{2}(t) e_{2}
where complex " probability amplitudes" z_{1},z_{2} are complex numbers
(ie. Â_{0,1} = Â_{2 +} multivectors) satisfying
|z_{1}|_{+}^{2} + |z_{2}|_{+}^{2} = 1 .
[ where |z|_{+} = (zz^{^})^{½} = (x^{2}+y^{2})^{½} is the traditional
complex modulus
]
Mathematicians refer to such "numbers" as a 2D Hilbert space denoted H_{2}.
The Rules of the Quantum Bit Game are that when you "observe" the qubit you get ↑ with probability
|z_{1}|_{+}^{2}
and y(t) "resets" to ï↑ñ. Otherwise you get ↓ and y(t) resets to ï↓ñ.
y(t) is normally considered to vary smoothly over time t between observations, "evolving" due to some physical law
(or, from a programmers perspecetive, "recurrance relation" or "recursive definition")
U(y(t),t) = 0
where U is a multi-dimensional-valued function of some kind, probably involving derivatives .
The act of reading the bit at t=t_{0} introduces a discontinuity in y(t) at t=t_{0} , "collapsing"
it either to ï↑ñ or to ï↓ñ.
Note that a qubit contains massively more "information" or "ambiguity" than a classical bit. We
superficially require two complex (four real) numbers rather than an integer from Z_{2}
to specify their state . [ One of the four reals being fixed by normalisation but with arbitarity of sign]
However, there is further redundancy here because any complex multiple of a qubit represents the same qubit.
Essentially, it is the complex number ratio z=z_{2}z_{1}^{-1} that is fundamental, although an explicit proviso
z=¥ for z_{1}=0 is required. We can thus specify a general qubit with two reals, plus a special case.
Quantum Register
A N-quantum register is a "combination" or entanglement of N qubits.
With each of the 2^{N} possible values of the classical N-register we associate
a pure state (eg. ï↑^{[1]}↓^{[2]}↓^{[3]}....↑^{[N]}ñ ) and a complex
probability amplitude.
y(t) = z_{1}(t)ï↑^{[1]}↑^{[2]}....↑^{[N]}ñ
+ z_{2}(t)ï↓^{[1]}↑^{[2]}....↑^{[N]}ñ
+ z_{3}(t)ï↑^{[1]}↓^{[2]}....↑^{[N]}ñ
+...+ z_{2N}(t)ï↓^{[1]}↓^{[2]}....↓^{[N]}ñ .
We will henceforth frequently ommitt the (t) from our z_{j} for brevity.
We say an N-register is normalised if
|z_{1}|_{+}^{2}
+ ...+ |z_{2N}|_{+}^{2} =
r_{1}^{2}
+ ...+ r_{2N}^{2} = 1 .
We will denote a normalised quantum register by y^{~}(t) .
We can represent a normalised N-quantum register at a given time t by a 2^{N}-dimensional complex 1-vector, the j^{th}
"coordinate" of which we write as z_{j} = r_{j}e^{fji} .
Real magnitude r_{j}^{2} ³ 0 corresponds to the classical probability
of the j^{th} classical bit combination;
real phase f_{j} Î [0,2p] is a "hidden variable" .
A quantum register is in pure state j if r_{j} is nonzero for one and only one
j Î { 1,2,...,2^{N} }.
A 1-quantum register is thus a qubit. We will refer here to an 8-quantum register as a qubyte.
We assume that we can initialise or reset a quantum register into the "first" (j=1) pure state ï↑^{[1]}↑^{[2]}...↑^{[N]}ñ at t=0 corresponding to all classical bits being zero. We further assume zero phases in the initial state so that z_{1}(0)=1, z_{k}(0)=0 " 1<k£2^{N} .
We can observe the value of a particular qubit, say the first in our suffix sequence,
and this will have the effect of "collapsing" y^{~}(t) into
y^{~}(t_{+}) =
(z_{1}ï↑^{[1]}↑^{[2]}....↑^{[N]}ñ
+ z_{3}ï↑^{[1]}↓^{[2]}....↑^{[N]}ñ
+...+ z_{2N-1}ï↑^{[1]}↓^{[2]}....↓^{[N]}ñ)^{~}
with probability
|z_{1}|_{+}^{2}+
|z_{3}|_{+}^{2}
+...+ |z_{2N-1}|_{+}^{2} or to
y^{~}(t_{+}) = (z_{2}ï↓^{[1]}↑^{[2]}....↑^{[N]}ñ
+ z_{4}ï↓^{[1]}↓^{[2]}....↑^{[N]}ñ
+...+ z_{2N}ï↓^{[1]}↓^{[2]}....↓^{[N]}ñ)^{~}
with probability
|z_{2}|_{+}^{2}+
|z_{4}|_{+}^{2}
+...+ |z_{2N}|_{+}^{2} ;
there being 2^{N-1} contributors to each summation.
The result of observing succesive qubits may thus depend on the
order in which they are observed.
We can evolve a quantum register by transforming the
complex amplitudes 1-vector y
in a specific way.
We can multiply it by any unitary complex 2^{N}×2^{N} matrix H,
by which we mean
a matrix satisfying H(H^{T}^{^}) = 1 .
Note that such a multiplication preserves normalisation and corresponds to a reversible
or nonsingular transformation.
Evolution preserves the "information" or "ambiguity" or "entropy"
of a quantum-register.
Observation, on the other hand, does not. Observation irreversibly
"resolves" some of the ambiguity. Having observed a qunatum resgister,
we can not restore the state y(t_{-}) from y(t_{+}) alone.
Multivector Representation of a Quantum Register
We shall see later that qubits have a natural geometric interpretation as multivectors in Â_{3,1+} ,
associating the m^{th} normalised qubit with a multivector
(1+r^{[m]}^{2})^{-½}(1+r^{[m]}e^{f[m]i[m]}e_{14}^{[m]})
½(1+e_{34}^{[m]})
fully specified by just two real scalar Riemanian parameters
r^{[m]}³0 and f^{[m]} for each qubit provided we also explicitly allow r^{[m]}=¥.
ï↑^{[i]}ñ = ½(1+e_{34}^{[i]}) with r^{[i]}=0
and f^{[i]} irrelevant ;
whereas ï↓^{[i]}ñ =
½e_{14}^{[i]}(1+e_{34}^{[i]}) =
½e_{13}^{[i]}(1+e_{34}^{[i]}) requires r^{[i]}=¥ .
We will also see that qubits are naturally measurable with regard to three alternate spacial "axis"
traditionally denoted ï↑ñ/ï↓ñ associated with direction e_{3};
ï←ñ/ï→ñ associated with e_{1}; and
ï´ñ/ï·ñ associated with e_{2}. The "definite" state ï↑ñ must then also be
regarded as a maximally ambiguous superposition of ï←ñ and ï→ñ , and also as
one of ï´ñ and ï·ñ.
On the face of it, this appears a complication of the simpler complex vector representation but we have just two parameters r and f rather than three for the normalised qubit.
N=2
For N=2, the product of two 4-D multivector spaces exists in the 16-D geometric algebra
generated by the six bivectors s_{i}^{[1]}=e_{i4}^{[1]} , s_{j}^{[2]}=e_{j4}^{[2]} .
This is twice the dimension of the
complex 4D 1-vector space traditionally used and we halve the effective dimension by forcing an equivalence between
i^{[1]}=e_{1234}^{[1]} and i^{[2]}=e_{1234}^{[2]} by means
of a further idempotent " correlator" geometric multiplier
½(1-i^{[1]}i^{[2]})=½(1-e_{1234}^{[1]}e_{1234}^{[2]}) so that our
2-quregister is represented by
y^{~} = (1+r^{[1]}^{2}))^{-½}
(1+r^{[2]}^{2}))^{-½}
(1+r^{[1]}e^{f[1]i[1]} s_{1}^{[1]} )
(1+r^{[2]}e^{f[2]i[2]} s_{1}^{[2]} )
½(1+ s_{3}^{[1]})
½(1+ s_{3}^{[2]})
½(1-i^{[1]}i^{[2]})
= (1+r^{[1]}^{2}))^{-½}
(1+r^{[2]}^{2}))^{-½}
(1+r^{[1]}e^{f[1]i[1]}e_{14}^{[1]} )
(1+r^{[2]}e^{f[2]i[2]}e_{14}^{[2]} )
b
where idempotent
b º
½(1+ s_{3}^{[1]})½(1+ s_{3}^{[2]})½(1-i^{[1]}i^{[2]})
= ½(1+e_{34}^{[1]})½(1+e_{34}^{[2]})½(1-e_{1234}^{[1]}e_{1234}^{[2]})
= 1/8(1+e_{34}^{[1]})(1+e_{34}^{[2]})(1-e_{12}^{[1]}e_{12}^{[2]})
has b^{2} = b^{†} = b^{#} = b ; bb^{§}=0
and can be regarded as a source of e_{34}^{[1]} and e_{34}^{[2]}, as a converter of e_{1234}^{[2]} s
into e_{1234}^{[1]} s (or e_{12}^{[1]} s), and also as
"condensing" the two e_{12}^{[i]} into an equivalent multiplier.
We thus regard
ï↑^{[1]}↑^{[2]}ñ = b
;
ï↓^{[1]}↑^{[2]}ñ
= e_{14}^{[1]}b
= e_{13}^{[1]}b
;
ï↑^{[1]}↓^{[2]}ñ
= e_{14}^{[2]}b
= e_{13}^{[2]}b
;
ï↓^{[1]}↓^{[2]}ñ
= e_{14}^{[1]}e_{14}^{[2]}b
= e_{13}^{[1]}e_{13}^{[2]}b .
The entangled "fermionic singlet" state
ï↑^{[1]}↓^{[2]}ñ
-ï↓^{[1]}↑^{[2]}ñ
is represented by (e_{14}^{[2]}-e_{14}^{[1]})b
General N
Consider the geometric space generated by the
3N 2-vectors
s_{i}^{[m]} .
Because any two s_{i}^{[m]} associated with the same qubit multiply to give
either a i^{[m]} s_{j}^{[m]} for the same qubit or a scalar, and commute with every s_{i}^{[m]}
associated with a different qubit this space has
has dimension 2^{3N} . However, the N 4-vectors
e_{1234}^{[m]} all commute with everything and square to -1 and we can "condense" them into a single
i using correlator
b º ( P_{i=1}^{k} ½(1+e_{34}^{[i]})) ( P_{j=1}^{N} ½(1-e_{1234}^{[1]}e_{1234}^{[j]}))
= ( P_{i=1}^{N} ½(1+e_{34}^{[i]})) ( P_{j=1}^{N} ½(1-e_{12}^{[1]}e_{12}^{[j]}))
= ( P_{i=1}^{N} ½e_{3}^{[i]}(e_{3}^{[i]}+e_{4}^{[i]})) ( P_{j=1}^{N} ½(1-e_{12}^{[1]}e_{12}^{[j]}))
which
represents the base state ï↑^{[1]}....↑^{[k]}ñ
and geometrically acts as a source of e_{34}^{[i]} for any i and converts any e_{1234}^{[i]} into
e_{1234}^{[1]} (or to e_{12}^{[1]} if preferrable, although this is less commutative ).
The multivector representation for the N-quregsiter
having qubits i_{1},i_{2},..,i_{j} set to ↓ and all the remainder set to ↑
is thus represented by the product
e_{13}^{[i1]}e_{13}^{[i2]}...e_{13}^{[ij]} b .
More generally, we have the representation
P_{m=1}^{N} (1+r^{[m]}^{2})^{-½}(1+r^{[m]}e^{f[m]e1234[m]}e_{14}^{[m]}) b
where r^{[m]} and f^{[m]} are the Riemanian paremtrs for the m^{th} qubit, but
things will be simplified when we move from "ket" representations like ï↑ñ to pensity representation
traditionally denoted ï↑ñá↑ï .
Quantum Cryptograhy
In Quantum Cryptograhy we allow Alice and Bob to receive a large number (comfortably in excess of 6N say) of "partnered" qubits. If either Alice or Bob
observe a qubit, the waveforms of that qubit and that of its distant partner qubit simultanmeously collapse to related states.
We will have more to say on such bizarre instantaneous-distant-effects later but for now the reader
should merely accept it as a given.
If Alice and Bob observe their qubits randomly with regard to three pre arranged quaxies , then they will get related Y_{es}/N_{o} answers
on those qubits which they observed using the same quaxis, approximatedly 2N of the original 6N.
Alice and Bob can establish which measurements had a common quaxis by open discussion on an insecure channnel.
By agreeing a random sample, say half, of these 2N readings and explicitly comparing the measured values, again on an insecure channel,
Alice and Bob can confirm that the recieved qubits are related as expected, establishing that
the qubits are unlikely to have been compromised by prior observation by Eve.
The remaining N readings then provide a random N-bit boolean key that can be exclusive ored with
an N-bit payload message prior to its transmission over an insecure channel.
Welsh-Hadamard Operator
Consider the real symmetric 2^{N}×2^{N} matrix having ±
2^{-½N}
in the j^{th} entry of the i^{th} row, the specific sign being + only if the parity
of the bitwise and (i-1) & (j-1) is even. It can be shown that W^{2}=1 so W is unitary.
Note that the -1 arise because we are here numbering our 2^{N} pure states from 1 rather than from 0.
W = ¼ | æ | +1 | +1 | +1 | +1 | ö |
ç | +1 | -1 | +1 | -1 | ÷ | |
ç | +1 | +1 | -1 | -1 | ÷ | |
è | +1 | -1 | -1 | +1 | ø |
Consider the unitary 2^{N}×2^{N} complex matrix S_{j}(q) having e^{qi} as the element of the j^{th} element of the j^{th} row ; 1 elsewhere on the lead diagonal ; and 0 everywhere else. Multiplication by S_{j}(q) has the effect of adding q to the phase of the j^{th} coordinate of y while preserving all the other coordinates.
Suppose normalised y^{~} 's j^{th} coordinate has real value
z+a for some real 0£a£z(2^{N}-1) and
every other coordinate has common positive real
value (z - a(2^{N}-1)^{-1}), where z is the averaged
coordinate 2^{-N}å_{k=1}^{2N} z_{k} .
The normalisation condition implies
(2^{N}-1)(z - a(2^{N}-1)^{-1})^{2}
+ (z + a)^{2} = 1
Þ
2^{N}z^{2} + a^{2}(1+(2^{N}-1)^{-1}) = 1
Þ
z = 2^{-½N}(1-a^{2}(1+(2^{N}-1)^{-1}))^{½} .
S_{j}(q) prserves all but the j^{th} coordinate which becomes
-(2^{-N} + a)^{½}
Suppose 2^{N}-D complex 1-vector y(t) = 2^{-N} (1,1,..,1)^{T}
Multiplication of a 2^{N}-D complex 1-vector by a general 2^{N}×2^{N} complex matrix requires 2^{3N+2} real multiplies. For N=8 this is 2^{26} which would take a 1 GigaHz Pentium Pro roughly 1/3 second for the multiplications alone. Obvious optimisations exist for particular unitary matrices, in particular we could multiply by A S_{j}(q) with just ? multiplies but some readers might consider ? multiplies merely to initialise a byte to a given nonzero value, and with a ?% failure rate at that, somewhat outrageous. The idea is of course that such "matrix multiplies" are done for us effectively instantaneously and "for free" by the "quantum hardware". However the fact remains that anybody considering emulating a quantum register on a traditional computer faces serious computational overheads increasingly exponentially with N.
The idea is to "embed" classical functions "into" the "matrix" operators of quantum states in such a way that the quantum evolution y(t) ® y(t+dt) allows potentially infinite number of classical function evaluations to be deduced from a single quantum register observation.
Consider a 2-quantum register (aka. biqubit) and let us replace clumsy-but-traditional labels ↑,↓ with dull-but-saner 0,1. We represent superposition z_{1}ï00ñ+z_{2}ï10ñ+z_{3}ï10ñ+z_{4}ï11ñ by the complex column vector (z_{1},z_{2},z_{3},z_{4})^{T} .
Let ¦ be one of the two possible invertible functions mapping {0,1}®{0,1} and let U_{¦} be the unitary matrix
W = ¼ | æ | +1 | +1 | +1 | +1 | ö |
ç | +1 | -1 | +1 | -1 | ÷ | |
ç | +1 | +1 | -1 | -1 | ÷ | |
è | +1 | -1 | -1 | +1 | ø |
U_{¦} = | æ | ¦_{0} | 0 | 1-¦_{0} | 0 | ö |
ç | 0 | 1-¦_{1} | 0 | ¦_{1} | ÷ | |
ç | 1-¦_{0} | 0 | ¦_{0} | 0 | ÷ | |
è | 0 | ¦_{1} | 0 | 1-¦_{1} | ø |
WU_{¦} = ¼ | æ | +1 | +1 | +1 | +1 | ö |
ç | +1 | -1 | 1 | -1 | ÷ | |
ç | +1 | +1 | 1-2¦_{0} | 2¦_{1}-1 | ÷ | |
è | +1 | -1 | 2¦_{1}-1 | 1-2¦_{1} | ø |
We have presented quantum bits and regsiters as abstract mathematical entities;
the programmers' model rather than the engineers'.
Experiments strongly suggest, however that quantum registers
exist in nature.
Welcome to the bizzare realm of quantum mechanics.
Next : Multivector Quantum Mechanics