Quantum Information is physical too

Abstract

We discuss the physical nature of quantum information, in particular focussing on tasks that are achievable by some physical realizations of qubits but not by others.

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arXiv:quant-ph/9904037v1 8 Apr 1999
Quantum Information is physical too...
Terry Rudolph∗
Department of Physics, University of Toronto,
Toronto M5S 1A7,
Canada
(February 9, 2008)
We discuss the physical nature of quantum information, in particular focussing on tasks that are
achievable by some physical realizations of qubits but not by others.
One of the most interesting developments of modern
quantum mechanics has been the realization that quan-
tum information is extremely useful stuff, despite the fact
that to quantitatively define exactly what it is turns out
to be a little tricky . This is in contrast to classical in-
formation theory, which has a very elegant and abstract
formulation. From classical information theory we learn
that the medium is irrelevant to the message, in the sense
that our theory can be constructed without requiring a
different formulation for every particular physical mech-
anism of transmitting the classical information. (This is
not to deny, however, the necessity of broad considera-
tions which can be traced back to the physical method
of transmission, such as the error rate).
It is perhaps because of the high degree of abstract-
ness attainable in classical information theory, that it
was many years before it was realized that considering
the physical nature of information carriers, in particu-
lar allowing them to be quantum particles obeying the
laws of quantum mechanics, could lead to novel effects
that lie outside the realm encompassed by classical infor-
mation theory. Such new effects include quantum dense
coding [1], quantum teleportation [2] and quantum com-
puting [3] to name a few.
example in the preceding list, from the perspective of
defining quantum information, is quantum teleportation.
In teleportation a quantum state is transmitted in two
completely seperate parts; the first part, which is trans-
mitted instantaneously via a measurement incorporating
one member of an entangled pair of particles, consists
of purely quantum information; the second part is com-
prised of two bits of purely classical information. The
instantaneous nature of quantum information transmis-
sion is one of several features that distinguish it from its
classical counterpart. Another is the fact that it cannot
be copied and hence amplified [4].
Perhaps the most striking
Despite the generality of classical information theory,
it is not true that the theory captures all possible tasks
that can be acheived by two parties who are allowed to
exchange arbitrary classical physical objects. To see this
consider a situation where the two communicating par-
ties, Alice and Bob are seperated in space and do not
share a common co-ordinate system (set of x,y,z axes).
If they are only allowed to communicate within the con-
fines of classical information theory i.e. by transmisson
of 1’s and 0’s, they will be unable to establish a set of
parallel axes. One could imagine this classical commu-
nication as occurring through the exchange of black and
white ping-pong balls say. On the other hand, if they
were to exchange precisely oriented pencils, then the task
would be easily achievable. Therefore if we want to char-
acterize all the tasks achievable with classical resources,
we must take into consideration the physical nature of
those resources.
Of course the preceding example is not meant to
be a profound insight into classical information theory.
Rather it is to set us thinking along the track that if we
are going to attempt to quantify quantum information
as a resource then we should consider the physical prop-
erties of the quantum information too. At the heart of
almost all the new effects mentioned previously lies quan-
tum entanglement. For this reason much attention has
focussed on quantifying entanglement in a manner which
reflects both its usefulness as a resource, as well as the
ease of producing or transferring it [5]. These methods
of quantifying entanglement have managed to attain an
impressive degree of abstractness, and different measures
are available depending on what particular problems you
are interested in. In general these measures of entangle-
ment do not depend on which particular type of physical
systems are entangled. They depend only on the dimen-
sions of the Hilbert space involved and the form of the
state vector in a particular choice of factorization of the
Hilbert space. Thus a pair of photons with maximally
entangled polarizations i.e. in the state
1
√2(|H?|V ? − |V ?|H?),
is considered equivalent (from the perspective of current
|ψ? =
(1)
∗email: tez@physics.utoronto.ca
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measures of entanglement) to a pair of entangled spin-1/2
particles in the state
|φ? =
1
√2(| ↑z?| ↓z? − | ↓z?| ↑z?).(2)
Here {|H?,|V ?} refer to horizontally and vertically polar-
ized photons, while {| ↑z?,| ↓z?} refer to spin eigenstates
in the z direction.
To see that the assumed equivalence is not always justi-
fied, imagine a situation where Charlie supplies Alice and
Bob with N maximally entangled spin-1/2 particles, as
well as a classical communication channel, and instructs
them to construct a set of parallel axes. It is not hard to
see that they can accomplish the task to a high level of ac-
curacy given enough entangled particles. Alice need only
pick a preferred direction Az, and make measurements
on a subset of her N particles with a Stern-Gerlach de-
vice oriented in this direction. She then uses the classical
communication channel to inform Bob as to the outcome
of her measurements. He may then rotate his own Stern-
Gerlach device until he finds an anti-correlation with her
results (although in general there will be some more ef-
ficient joint measurement to determine the direction). It
must be emphasized that the information as to the special
direction Az, is transmitted purely by the instantaneous
quantum information transfer, the classical channel car-
ries no directional information. The other directions may
be transmitted using the remaining particles.
The ability to complete this task is not particularly
mysterious, since it is a consequence of the fact that
spin-1/2 particles are physical objects whose description
requires some specification of directions in space. It also
is not hard to see that such a task is not achievable by
two parties who share N maximally entangled scalar par-
ticles. Interestingly it also appears that N photons in the
state (1) do not suffice either, since photon polarizations
are constrained to lie in a plane perpendicular to their
direction of propagation. Thus unless we have special
knowledge about the preparation of the photons (for ex-
ample that they are emitted in opposite directions or 60
apart), we cannot deduce a correlation between the mea-
sured polarizations and “absolute” directions in space.
At this stage a clarifying remark regarding the physi-
cal nature of different possible qubit realizations should
probably be made. Obviously there exist some two-state
quantum systems which have “natural” orthogonal bases,
for example the ground and excited states of an atom, or
the zero and one mode excitations of a scalar field. These
are energy eigenstates. Other two-state systems however
have no such universally preferred basis, spin being a case
in point. The difference lies purely in the Hamiltonian of
the universe - if the universe was permeated by a huge
static magnetic field then there would be a “natural” ba-
sis for spin measurements! The distinction is not purely
academic however. Quantum teleportation, for example,
requires the receiver to perform one of four unitary trans-
forms to a qubit in order to recover the teleported state.
If the sender and receiver do not share knowledge of an
orthogonal basis then this is not possible.
Finally we give a rather cute example of “Quantum
Navigation”, which applies the preceding ideas to a prob-
lem which one hopes will not be faced by the reader, but
which is worth bearing in mind when visiting politically
unstable countries. Consider the following (rather con-
trived) scenario: Bob’s friend Alice has been abducted.
She regains consciousness to find herself locked in a win-
dowless room, presumably somewhere on Earth. Check-
ing her pockets she finds that she still has her cellular
telephone, and so she calls Bob. After being reassured
that she’s unhurt, Bob naturally asks “Where are you?”.
Of course Alice has no idea, and they discover that the
telephone company is singularly unhelpful and refuses to
co-operate in Bob’s attempt to locate Alice. (This part
of course is the most realistic part of the story). Fortu-
nately Alice realizes that she still has in her pocket the
box of N spin-1
2particles that are entangled (in singlet
states) with particles in Bob’s possession. The question
we are interested in is how may these particles be used
to locate Alice? (In what follows we ignore any possible
effects on the particle’s spin from Alice’s journey through
the Earth’s magnetic field.)
The simplest solution is the following: Alice alignes
her mini Stern-Gerlach device (which will soon be stan-
dard on all Swiss-Army knives) in her vertical direction
(Az). She then proceeds to measure the spin of each of
her particles in this direction, announcing to Bob over
the phone what the result (↑,↓) of each measurement is.
Bob is now left with N particles, n+of which are in the
state | ↑Az?, while the remaining n− = N − n+ are in
the state | ↓Az?. In general we expect n+≈ N/2. If Bob
can determine the direction Az, then he can determine
where on Earth Alice is, since each position on earth has
a unique z-direction defined by gravity. Of course since
they only share a finite number of particles he will only
be able to do this to a certain accuracy, but hopefully it
is good enough that he can locate her and save her from
her nefarious imprisoners.
An interesting question this now raises is what mea-
surement strategy should Bob follow in order to obtain
the best estimate as to the direction Az? One strategy
is to make joint measurements on the n+and n−sets of
particles seperately (since it is well known that a joint
measurement on all the particles in each set would be
preferable for estimating their state [6]), and then to av-
erage the result somehow. However Gisin and Popescu
recently showed in [7], that a joint measurement on a pair
of anti-parallel spins is in fact more efficient than that on
a pair of parallel ones, given the task of determining as
accurately as possible the direction the spin is pointing!
If the result generalizes to more than two particles, then
in fact Bob would do better to apply a joint measurement
to all N particles. This of course is not so surprisng.
What is really amazing however, is that he would do bet-
ter than if Alice had been allowed to send him a carefully
prepared box of spins all pointing in the same direction.
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The randomness of the outcomes of her measurements
would actually be helping the task at hand! This ap-
pears to be a clear violation of the Law of Conservation
of Trouble. (While no precise formulation of the Law ex-
ists in non-relativistic quantum mechanics, the quantum
field theory version can be found in [8], and of couse we
are all familiar with the Law in the macroscopic limit,
whence it is more often called Murphy’s Law.)
In conclusion it is has been shown that the physical
nature of quantum information should be taken into ac-
count when trying to quantify it as a resource. A task was
presented which can be completed successfully by holders
of maximally entangled spin-1/2 particles, which cannot
be completed successfully by holders of some other types
of maximally entangled particles. Finally it was shown
how these ideas could be used to allow an abducted per-
son to communicate their position on earth, even though
they have no knowledge of where they are themselves.
ACKNOWLEDGMENTS
The author wishes to acknowledge many useful dis-
cussions and several stimulating arguments with R.
Spekkens, as well as useful discussions with both A.
Steinberg and J. Sipe.
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quant-ph/9803033; W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998); M. A. Nielson, quant-ph/9811053.
[6] R. Derka et al, quant-ph/9707028; see also S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995); A. Peres and W.
K. Wootters, Phys. Rev. Lett. 66,1119 (1991); K. R. W. Jones, Phys. Rev. A. 50, 3682 (1995).
[7] N. Gisin and S. Popescu, quant-ph/9901072.
[8] S. Weinberg, The Quantum Theory of Fields I (Cambridge University Press,1995), page 377.
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