Elsevier

Optics Communications

Volume 285, Issue 6, 15 March 2012, Pages 1560-1565
Optics Communications

Unambiguous discrimination of two squeezed states using probabilistic quantum cloning

https://doi.org/10.1016/j.optcom.2011.10.048Get rights and content

Abstract

Problem of unambiguous state discrimination of two squeezed states of light beam has been investigated. Wigner function of the two squeezed states is used to calculate their scalar product in order to determine optimal success probability of unambiguous discrimination. We propose a general scheme for unambiguous state discrimination using probabilistic quantum cloning for any two known pure quantum states.

Introduction

Capability of discriminating among different outcomes [1] is a fundamental problem in many protocols such as quantum teleportation or quantum cryptography. These outcomes, in general, appear as nonorthogonal quantum states (lying in a space spanned by the logical qubits), which can either be spatially close together or be located at widely separated places of a quantum network. Whenever the possible quantum states are nonorthogonal, perfect discrimination of the states becomes unattainable. Discrimination problem plays a key role in the quantum communication [2] where only a single copy of the system is given and only a single shot-measurement is performed. Precision with which the state of the system can be determined with a single-shot measurement is limited by quantum physics.

A general unknown quantum state cannot be determined completely by a measurement performed on a single copy of the system. But the situation is different if a priori knowledge is available [3], [4], e.g., if one works only with states from a certain discrete set. Even quantum states that are mutually nonorthogonal can be distinguished with a certain probability provided they are linearly independent (for a review see Ref. [5], [6]).

Discrimination of quantum states was first studied by Helstrom [7], who considered the problem to measure the state of the system which is guaranteed to be in one of the two known states with some prior probabilities. The task is to conclude after each single measurement which of the two states we were given. If the states are not orthogonal we certainly sometimes guess incorrectly by minimizing the probability of making an error. This approach is called minimum error state discrimination. On the other hand, we can increase the reliability of some outcomes at the expense of one totally inconclusive result and this approach is called unambiguous state discrimination (USD). USD can be used, e.g., as an efficient attack in quantum cryptography [2]. Realistic implementations of quantum key distribution (QKD) mostly use signal states which are nonorthogonal but linearly independent. This fact enables an eavesdropper to perform unambiguous state discrimination and to get some information on the key without disturbing the transmission [2].

Problem of USD of a pair of pure states was solved by Ivanovic [4], Dieks [8] and Peres [9]. They recognized that both states can be unambiguously determined, but the price to pay is the possibility of getting an inconclusive result, i.e., the measurement failed to respond unambiguously. The task is to maximize the probability of exact discrimination of the states which corresponds to the minimization of the failure probability. The original Ivanovic, Dieks and Peres's solution for two pure states [4], [8], [9] appearing with equal prior probability was generalized in several ways. Jaeger and Shimony [10] extended the solution to arbitrary a priori probabilities. Chefles and Barnett [11] generalized Peres's solution to an arbitrary number of equally probable states which are related by a symmetry transformation. Raynal's reduction theorems [12] simplified the problem of discrimination of a pair of mixed states by reducing its dimension or splitting it into more pieces, which are USD of two pure states. Kleinmann et al. [13] found commutators revealing two dimensional block diagonal structure in the reduced states.

Study of USD started the investigation of tasks where certain measurement can lead to unambiguous knowledge about some property of the system, viz., unambiguous discrimination of quantum channels. Sedlák and co-workers investigated unambiguous tasks (discrimination, comparison and identification) [6], [14] for states, channels and measurements, respectively and studied in detail about the unambiguous identification of coherent states (pure classical continuous variable states) of single-mode optical fields for the two cases: (i) two unknown states are arbitrary pure states of qudits; (ii) alternatively, they are coherent states of single-mode optical fields. For this case, they proposed simple and optimal experimental setup composed of beam-splitters and a photodetector. Nonclassical features of light [15] are now playing important role in continuous variable quantum information processing and so one can ask about the possible discrimination strategy for nonclassical states such as squeezed states. Recently, the unambiguous quantum state comparison of two unknown squeezed vacuum states (i.e., to unambiguously determine whether two unknown squeezed-vacuum states are the same or not) has been reported and found an optimal probability of the equivalence of the compared squeezed states [16]. In the present paper, we will focus our attention to the unambiguous discrimination for two known pure squeezed states [17]. As the beam splitter introduces entanglement when the input modes are in nonclassical states [18], we exploit the approach of probabilistic cloning for discrimination of squeezed states.

The paper is organized as follows: in Section 2, we summarize the basics of POVM and we discuss the fundamentals of discrimination strategies from the very beginning. In Section 3, we discuss Wigner function of squeezed states and calculate their scalar product. In Section 4, we derive a relation for probability of successful unambiguous discrimination of any two known pure states using the theoretical model of probabilistic quantum cloning. This relation is, then, used for unambiguous discrimination for two known pure squeezed states. We discuss about the results in Section 5.

Section snippets

POVM and unambiguous quantum state discrimination

POVMs (positive operator-valued measure) are a specific form of a general measurement, but more general than projective measurements [3], [19], [20]. They are not bound by the condition P^iP^j=δijP^i that applies to the projectors, though the elements of a POVM, {E^k}, must satisfy the conditions that they be positive and complete: ψ|E^k|ψ0 and kE^k=I^.

A measurement described by a POVM, {E^k}, is called an Unambiguous State Discrimination Measurement (USDM) [5] on a set of states {ρ^i} iff

Wigner function of squeezed states and their scalar product

A quantum state, described by a density operator ρ^ (with  = 1)), can be represented in terms of the Wigner phase space distribution [21],W(q,p)12πexpiq+12ξ|ρ^|q12ξ.

Negative regions in the Wigner function of a given state can be seen as the signature of nonclassical behavior [22]. The Wigner function of a pure normalizable state cannot take on values larger than 1/π (i.e., |W(q, p)|  1/π) and the normalization factor 1/(2π) ensures the propertydqdpW(q, p)=1.

Trace of the

Probabilistic quantum cloning and unambiguous state discrimination

The no-cloning theorem [28], [29], which states that unlike classical information, an arbitrary quantum state cannot in general be perfectly copied, demonstrates one of the fundamental differences between quantum and classical information processing (see for review [30]). Let us design a device which will copy only states from a particular set of allowed input states [31]. A copier of this type can achieve higher fidelities for its copies than one which is required to accept all possible states

Conclusions

In conclusion, we addressed the issue of unambiguous discrimination of two known squeezed states and we found that there are different situations where we can get maximum success probability of unambiguous discrimination. We proposed a theoretical model for the optimal probability of successful unambiguous discrimination of any two known pure states using probabilistic quantum cloning machine. However, probabilistic quantum cloning is an experimental challenge, as it requires more complicated

Acknowledgments

Author is grateful to the learned referee for valuable comments and suggestions that helped in improving the manuscript. Author would like to express his gratitude to Prof. Vladimir Bužek, Head, Research Center for Quantum Information, Slovak Academy of Sciences, Bratislava, Slovakia, for the opportunity to be taught the field of quantum information processing and for our fruitful collaborative work as well as his encouragements. Author would also like to acknowledge M. Ziman, M. Hillery, and

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