Nature arXiv:0905.2292

One principle that has been proved useful in deriving some of the properties of quantum mechanics is no-signalling. It states that information

requires a carrier and without sending any physical object information cannot be transferred. No-signalling has been extensively studied and

shown to be enough to formulate no-cloning [1], monogamy of Bell's inequality violations [2] (QAP Research Highlight of October) or security

of quantum key distribution [3]. Yet, we know that this principle alone is too weak to single out quantum theory. The most prominent

example for the case where it fails in this respect is the amount of violation by quantum correlations of the Bell-CHSH inequality (which is

satisfied by all possible classical theories, which are bounded by the value of 2). While quantum mechanics cannot reach value of the CHSH

expression greater than the Tsirelson bound there is an abstract unphysical theoretical construct known as PR-Box [4], which satisfies

no-signaling and allows for the maximal algebraic value for this expression (=4).

This is where our new principle of Information Causality [5] comes into play. It is basically no-signaling condition, albeit with an additional

requirement (see below in italics), which seems natural, nevertheless thus far it has not been formulated. Still, it drastically narrows the

range of possible theories of Nature. We show that it can be used to derive the quantum mechanical Tsirelson bound, without invoking

quantum theory at any stage. Information Causality can be phrased as follows:

If some receiver gets m classical bits of information, his knowledge cannot increase by more than m bits and it is only the sender not the

receiver who can choose (before the bits are sent) what this knowledge is about.

More formally, for a mathematically minded reader. Imagine two parties Alice and Bob. They carefully prepare themselves for a transfer of

encoded information that will be described a bit later. That is, they are allowed to share some data (including random strings of bits),

computer programmes, and even arbitrarily many entangled qubits (or other quantum systems), and perfect measurement devices to detect

them in any states. They may have at their disposal an unbounded number of physically allowable correlated systems, classical or quantum.

Once this preparation stage is over (this is a strict requirement), one party (Alice) gets N (classical) uniformly and independently distributed

regardless of the physical resources (quantum or classical) the parties share. It is Shannon's mutual information describing the degree of

All this points once more that quantum mechanics seems to be describing information that we have, or can gain about physical systems:

quantum laws are, at least partially, deductible form reasonable assumptions on information theory.

Figure 1

References

[1] D. Dieks, Phys. Lett. 92 A 271 (1982); W.K. Wootters, W.H. Zurek, Nature 299 802 (1982).

[2] M. Pawlowski, C. Brukner, Phys. Rev. Lett. 102, 030403 (2009); B. F. Toner, Proc. R. Soc. A 465, 59-69 (2009).

[3] Ll. Masanes, Phys. Rev. Lett. 102, 140501 (2009).

[4] S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994).

[5] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, M. Zukowski, Nature 461, 1101 (2009).

M. Sedlák, M. Ziman (University of Bratislava)
 

Individual clicks can be used to unambiguously compare quantum devices

Imagine we are given a pair of unknown quantum devices and our goal is to decide whether they operate in the same way, or not. A natural

solution would be to completely reconstruct the action of both devices separately and then compare the results. However, it turns out that in

quantum case such comparison procedure is very inefficient. It requires enormous amount of experimental and computational resources.

On the other hand since the comparison is a binary decision problem it seems to be possible to design an experiment such that individual

outcomes can provide us with unambiguous error-free conclusions.

One of the basic lessons of quantum theory is that predictions and conclusions we can make are intrinsically nondeterministic. This feature gives

the quantum theory the hallmark of a statistical theory describing large ensembles of systems. Because of their randomness, the individual

outcomes of the experiment are considered to be of a very little use and sense. However, with the development of quantum information theory

and related experimental progress the individual experimental clicks are becoming of interest and potential use. Under very specific

circumstances the uncertainty of predictions of individual outcomes and conclusions made out of individual clicks can be reduced to zero.

QAP researchers have investigated whether this is the case for the problem of comparison of quantum devices.

The research was focused on the comparison of unitary channels [1] and projective measurements [2]. These specific families of quantum

processes are of particular interest in quantum information processing. The projective measurements represent the final read-out stage of

quantum information processing and both of them can be used to perform individual steps of quantum algorithms (quantum gates).

The statistical nature of quantum theory implies that the perfect comparison of unitary channels and projective measurements is not possible.

However, if we allow imperfections in the form of inconclusive outcomes, than we can make an experiment unambiguously concluding the

difference. It turns out it is impossible to unambiguously conclude that unknown unitary channels, or unknown projective measurements,

are identical. In summary, it is possible to design a two-outcome comparison experiment with one inconclusive outcome and one outcome

implying with certainty that the devices are different.

Optimal solutions using a minimal number of usages of the unknown devices are illustrated on figures. Let us note that the concept of unknown

measurement apparatus can be interpreted in two ways. Either its outcomes are labelled (known), or not. We found that for measurements with

unlabeled outcomes each of the devices must be employed at least twice. Otherwise, for labelled measurement devices and for unitary channels,

a single usage of each apparatus is sufficient to make the unambiguous conclusion. Moreover, the frequency of inconclusive outcome serves

as an operational measure of the difference between the devices.

Figure 1: Comparison of unitary channels. Optimal experiment consists of the preparation of a bipartite state ρasym having support on the antisymmetric subspace and of the symmetry measurement S deciding whether the state is symmetric, or antisymmetric. The outcome symmetric is used to unambiguously conclude that the unitary channels A and B are different. The average success probability equals P=(d+1)/(2d), where d is the dimension of the system.

Figure 2: Comparison of labelled projective measurements. Optimal experiment consists of the preparation of a bipartite state ρasym . If each of the measurement apparatuses A and B gives the same outcome, then we can unambiguously conclude that they are different. This happens with the average success probability P=1/d, where d is the dimension of the system.

Figure 3: Comparison of unlabeled projective measurements. The optimal solution was found only for the case of qubit observables and consist of the preparation of a specific four-qubit state .In this case each of the measurements apparatuses A and B is used twice If the pair of outcomes observed on one of the apparatuses are the same, whereas the ones observed on the second measurement device are different, then we can unambiguously conclude that A and B are different. For qubits the optimal average success probability equals P=4/9.

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