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Unambiguous discrimination of two squeezed states using probabilistic quantum cloning Devendra Kumar Mishra Physics Department, V. S. Mehta College of Science, Bharwari, Kaushambi-212201, U. P., India

a r t i c l e

i n f o

Article history: Received 16 May 2011 Received in revised form 15 August 2011 Accepted 20 October 2011 Available online 3 November 2011

a b s t r a c t 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. © 2011 Elsevier B.V. All rights reserved.

Keywords: POVM State discrimination Unambiguous measurement Wigner function Squeezed states Probabilistic quantum cloning

1. 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

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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

D.K. Mishra / Optics Communications 285 (2012) 1560–1565

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. 2. 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^ i P^ j ¼ δij P^ i that applies n o to the projectors, though the elements of a POVM, E^ k , must satisfy ^ k jψ〉≥0 and the conditions that they be positive and complete: 〈ψjE ^ k ¼ ^I. ∑k E n o A measurement described by a POVM, E^ k , is called an Unambiguous State Discrimination Measurement (USDM) [5] on a set of ^ i g iff the following conditions are satisﬁed: (i) the POVM states fρ n o contain the elements E^ ? ; E^ 1 ; ………; E^ k where k is the number of different signals in the set of states. The element E^ ? is associated to ^ i , i = 1, …, k, correan inconclusive result, while the other elements E spond to an identiﬁcation of the state ρ^ i . Existence of the inconclusive measurement element, E^ ? , is due to the fact that the quantum mechanics forbids us to perfectly distinguish the nonorthogonal quantum states when only ﬁnite number of copies is provided and thus the presence of inconclusive results is the price we have to pay for the unambiguous measurement; and (ii) No states are mis takenly identiﬁed, that is Tr ρ^ i E^ k ¼ 0, ∀ i ≠ k, i, k = 1, …, n. The rate of inconclusive results (i.e., failure probability) corresponding to each USDM is given by hn oi ^ P E ¼ ∑ ηi Tr ρ^ i E^ ? : ð1Þ k i n o opt ^ is called an optimal A measurement described by a POVM E k USDM [5] on a set of states fρ^i g with the corresponding a priori

1561

n o ^ opt is a USD measurement on probabilities ηi iff (i) the POVM E k fρ^i g; and (ii) the probability of inconclusive results is minimal, that is hn oi hn oi ^ opt ¼ min P E^ ; P E k k

ð2Þ

where the minimum is taken over all USDM. Let us think about the situation where the two parties, say Alice and Bob, desires to communicate. Alice prepares a quantum system in a state, member of a set of states known to Bob. Alice, in general, prepares each state with some a priori probability. Alice sends the quantum system to Bob who performs a measurement in order to take the information he needs, i.e., a state ensemble of a quantum system is given and we want to determine the state of that system. Given a strategy, we ﬁnally have to optimize the measurement with respect to some criteria, i.e., we look for to minimize the probability of inconclusive outcomes. The optimal measurement can be constructed as follows: First, one observes that the relevant part of the Hilbert space is only a plane in which states |ψ1〉 and |ψ2〉 lays because the support of the measurement does not affect the overlap with |ψi〉 and implies the same positivity conditions for the POVM elements. Secondly, require ^ 1 ¼ c1 ψ⊥ 〉〈ψ⊥ , E^ 2 ¼ c2 ψ⊥ 〉〈ψ⊥ , where ment of unambiguity implies E 2 2 1 1 ⊥ ci ∈R, |ψi 〉 is orthogonal to |ψi〉 and lays in the span of |ψ1〉 and |ψ2〉. Here, ⊥

⊥

⊥

⊥

⊥

⊥

〈ψ1 jψ1 〉 ¼ 0; 〈ψ2 jψ2 〉 ¼ 0; 〈ψ1 jψ1 〉 ¼ 1; 〈ψ2 jψ2 〉 ¼ 1:

ð3Þ

Finally, maximization of the probability of unambiguous discrimination PD can be done explicitly, because it is constrained only by the pos^ k (k= 0, 1, 2), which have rank at most two. The itivity of elements E ﬁnal form of the optimal measurement depends on the relation of the overlap σ ≡ |〈ψ1|ψ2〉| to the prior probability η1 = 1 − η2. When the two a priori probabilities are equal (η1 = η2 = 1/2), the optimal failure probability to unambiguously discriminate the states |ψ1〉 and |ψ2〉 is opt

Pfail ¼ j〈ψ1 jψ2 〉j ¼ σ:

ð4Þ

This solution is known as the Ivanovic–Diesk–Peres (IDP) limit [4,8,9]. The modulus of the overlap between the two pure states is symmetric and takes value in [0, 1]. When the modulus of the overlap between the two pure states is 1, the two states are identical and when it is 0, the two states have orthogonal supports. Then, the optimal probability of successfully unambiguous discrimination of the two states |ψ1〉 and |ψ2〉 is opt

PD ¼ 1−Pfail :

ð5Þ

Therefore, in order to study the unambiguous discrimination of two known pure squeezed states, we will have to ﬁrst calculate their scalar product. 3. Wigner function of squeezed states and their scalar product ^ (with ℏ = 1)), A quantum state, described by a density operator ρ can be represented in terms of the Wigner phase space distribution [21], ∞ 1 i 1 1 ^ jq− ξ〉: Wðq; pÞ≡ ð6Þ ∫ dξ exp − pξ 〈q þ ξjρ 2π −∞ ℏ 2 2 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 property ∞

∞

∫ dq ∫ dpWðq; pÞ ¼ 1: −∞

−∞

ð7Þ

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D.K. Mishra / Optics Communications 285 (2012) 1560–1565

Trace of the product of two density operators is determined by the product of the two corresponding Wigner functions, Wρ^ j ðq; pÞ, of the density operators ρ^ j (j = 1 and 2) integrated over phase space, ∞

∞

Trðρ^ 1 ρ^ 2 Þ ¼ 2π ∫ dq ∫ dpWρ^ 1 ðq; pÞWρ^ 2 ðq; pÞ: −∞

ð8Þ

−∞

This trace product rule allows us to gain deeper insight into the shape of quantum states. For the case of two pure states ρ^ j ¼ ψj 〉〈ψj , (j= 1 or 2), we ﬁnd Trðρ^ 1 ρ^ 2 Þ ¼ Tr½jψ1 〉〈ψ1 jjψ2 〉〈ψ2 j ¼ j〈ψ1 jψ2 〉j

2

ð9Þ

and the trace product rule reduces to ∞

2

∞

j〈ψ1 jψ2 〉j ¼ 2π ∫ dq ∫ dpWjψ1 〉 ðq; pÞWjψ2 〉 ðq; pÞ: −∞

ð10Þ

−∞

1 2 †2 S^ ðζÞ ¼ exp ζ a^ −ζa^ ; 2

Wigner function for a vacuum state, |0〉, is concentrated within circular region around the origin in the phase space:

f

g

1 2 2 exp ðq þ p Þ : π

ð11Þ

This is localized within the circle at the origin in phase space: q 2 + p 2 = 1. 3.2. Coherent state ^ In quantum optics, the displacement operator, D ðαÞ, is deﬁned as ^ ðαÞ ¼ exp αa^ † −α a^ . In the phase space composed of two conjugate D ^ ðαÞdisplaces a state by (Re[α], Im[α]) and variables q and p, operator D ^ † ðαÞa^ D ^ ðαÞ ¼ a^ þ α. Action of displacement operhence the name [23], D ^ ðαÞ, upon a quantum state |ψ〉 can be best visualized in terms of the ator, D Wigner function. Wigner function is invariant under displacement of position and momentum coordinates, q and p, respectively, i.e., a displace ^ ðαÞψ〉, ment of the state |ψ〉 of the particle changes according to ψ〉↦D ^ ðαÞ is the displacement operator, results in W(q,p)↦ W(q−a, where D pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ p−b), where a ¼ 2 ReðαÞ and b ¼ 2 ImðαÞ, i.e., α ¼ ða þ ibÞ= 2. Thus, W(q,p) is simply shifted in the direction (a, b) in the q–p plane. Ap^ ðαÞ, on the vacuum state |0〉, we get the plying a displacement operator, D ^ ðαÞ0〉 in the phase space and the corresponding coherent state α〉≡D Wigner function is n o 1 2 2 exp ðq aÞ ðp bÞ : π

ð12Þ

No two coherent states are orthogonal to each other, since the square of the modulus of scalar product of two coherent states, |α〉 and |β〉, is |〈α|β〉| 2 = exp(− |α − β| 2). The Wigner function for a coherent state, Eq. (12), is localized within the circle centered at (a, b) in the phase-space: (q-a) 2 + (p-b) 2 = 1.

ð13Þ

is called the single-mode squeezing operator and its effect is to transform the coordinates of phase space, expanding one axis at the expense of contracting the other, where ζ = reiϕ is the complex squeeze parameter [24]. Here, r (0≤ r b ∞) is called ‘squeezing coefﬁcient’ which is a characteristic interaction time and is a logarithmic measure of the degree of squeezing. Squeezing level is equal to − 10 log 10e − 2r [dB] and from this formula we can see that, for example, r = 0.35 corresponds to 3 dB and r = 0.70 corresponds to 6 dB squeezing. The squeezing phase Φ (0≤ ϕ ≤ 2π) equal to that of the pump ﬁeld, determines the phase-space orientation of the squeezing transformation and the direction of squeeze is along ϕ/2. The unitary transformation of the bosonic operator is ð14Þ

A more general squeezed state may be obtained by applying the dis^ ðαÞ, and then we can write the single-mode placement operator, D squeezed state of light, ^ ðαÞS^ ðζÞ0 : α; ζ ¼D

〉

3.1. Vacuum state

Wjα〉 ðα; q; pÞ ¼

A single-mode squeezed vacuum state is denoted by S^ ðζÞj0〉. Here the unitary evolution operator,

† † S^ ðζÞa^ S^ ðζÞ ¼ a^ coshζ þ a^ sinhζ:

Hence the modulus square of the scalar product between the quantum states |ψ1〉 and |ψ2〉 is the product of the Wigner function W|ψ1〉 and W|ψ2〉 of the two states integrated over the phase space. This relation can be generalized for mixed states as well.

Wj0〉 ð0; q; pÞ ¼

3.3. Squeezed state

〉

ð15Þ

Here, α = 0 corresponds to the squeezed vacuum state whereas ζ = reiϕ = 0 corresponds to the coherent state. Unlike the case of coherent states, the distribution for a squeezed state, |α, ζ〉, is elliptic but the area in the phase space of this ellipse is the same as that of circle for the coherent state (ζ = 0). In the phase space, every squeezed state is a translated squeezed vacuum, and we can write the Wigner function, Wjα;ζ〉 ðα; ζ; q; pÞ ¼ ð1=πÞ expfer ½ðq aÞ cosϕ=2 þ ðp bÞ sin ϕ=22 −er ½ðq aÞ sinϕ=2−ðp bÞ cosϕ=22 g;

ð16Þ

for a squeezed state |α, ζ〉 and the direction of squeeze is along ϕ/2 [25]. Therefore, the most general squeezed state is centered at a point other than the origin, and its principal axes are not necessarily parallel to the coordinate axes. We intend to make unambiguous discrimination of two squeezed states. For simplicity, we consider two squeezed states with the same amount of squeezing but their direction of squeeze is different. Such a light beam can be generated in degenerate parametric ampliﬁer [26] where a pump photon is converted to two nearly degenerate photons of frequency and the two simultaneously generated signal and idler photons starting from a single pump photon have conjugate phases, so they interfere with each other to suppress one quadrature amplitude noise and enhance the other quadrature amplitude noise. This can be achieved in interferometric arrangements where amplitude ﬂuctuations are canceled out and phase modulations are turned into modulation of the intensity at the output. In such an arrangement only the noise of the light in the phase quadrature will contribute to the ﬁnal signal to noise ratio [27]. For, let us consider the two squeezed states, |ψ1〉 and |ψ2〉, squeezed along ϕ1/2 and ϕ2/2 directions (Fig. 1), respectively with respect to the q axis and their respective Wigner functions (using Eq. (16)) are Wjψ1 〉 ðα; ζ1 ; q; pÞ

¼ ð1=πÞ exp fe−r ½ðq aÞ cosϕ1 =2 þ ðp bÞ sinϕ1 =22 −er ½ðq aÞ sinϕ1 =2−ðp bÞ cosϕ1 =22 g; ð17Þ

D.K. Mishra / Optics Communications 285 (2012) 1560–1565

Fig. 2. Two squeezed vacuum states.

Fig. 1. Two squeezed states.

h i2 Wjψ2 〉 α′ ; ζ2 ; q; p ¼ ð1=πÞ exp e−r ðqa′ Þ cosϕ2 =2 þ ðpb′ Þ sinϕ2 =2

f

h i2 −er ðqa′ Þ sinϕ2 =2−ðpb′ Þcosϕ2 =2 :

g

ð18Þ The modulus square of the scalar product of these two states is

h i pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 ; j〈ψ1 jψ2 〉j ¼ 4= 4AC−E2 exp B C þ AD þ BDE þ 4ACF−E F = 4AC−E

ð19Þ if Re(E 2) b 4AC. Here, h i −r 2 2 A ¼ 2 e þ sinhr sin ϕ1 =2 þ sin ϕ2 =2 ;

ð20aÞ

′ −r 2 ′ 2 B ¼ 2 a þ a e þ 4 sinhr⋅ a sin ϕ1 =2 þ a sin ϕ2 =2 ′ − 2 sinhr⋅ b sinϕ1 þ b sinϕ2 ;

ð20bÞ

h i r 2 2 C ¼ 2 e − sinhr sin ϕ1 =2 þ sin ϕ2 =2 ;

ð20cÞ

′ r 2 ′ 2 D ¼ 2 b þ b e −4 sinhr b sin ϕ1 =2 þ b sin ϕ2 =2 ′ −2 sinhr a sinϕ1 þ a sinϕ2 ; E ¼ 2 sinhrð sinϕ1 þ sinϕ2 Þ;

1563

states |ψ1〉 and |ψ2〉 with the help of Eq. (5). We can consider different simple special cases, using Eq. (19), for the two squeezed states: Scalar product of the two squeezed vacuum states, |ψ1〉 and |ψ2〉, squeezed (i) along p and q directions, respectively, is |〈ψ1|ψ2〉|2 =sechr; (ii) along ϕ1/2 and ϕ2/2 directions (Fig. 2), respectively, with reqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 spect to the q axis is j〈ψ1 jψ2 〉j2 ¼ 1= 1 þ sinh r⋅ sin2 12 ðϕ1 −ϕ2 Þ; and (iii) along p and q directions displaced by a and b, respectively, h i 2 is j〈ψ1 jψ2 〉j2 ¼ sechr exp − 12 a2 þ b sechr . We can verify easily that the modulus of the overlap between the two pure squeezed states is symmetric and takes value in [0, 1]. When the modulus of the overlap between the two pure states is 1, the two states are identical and when it is 0, the two states have orthogonal supports. Signiﬁcance of the present study for unambiguous discrimination of the two squeezed states can be seen easily from the Figs. (3)–(5). Let us consider pﬃﬃﬃ the simplest case ′ Reα =Imα =|α| ≡ x, real and a ¼ b ¼ a′ ¼ b ¼ 2x. For x = 2, ϕ2 = π/ 2, probability of unambiguous discrimination, PD, increases as r increases (Fig. 3). For x =2, ϕ2 = 0, we plot ϕ1 vs. PD and ﬁnd that the probability of unambiguous discrimination, PD, increases and attains maximum value (=1) at ϕ1 = π (Fig. 4), i.e., when the two states are orthogonal. For x= 2, r= 10, we plot ϕ1 vs. ϕ2 and ﬁnd that the probability of

ð20dÞ

ð20eÞ

2 2 2 F ¼ − a2 þ a′ e−r − b2 þ b′ er −2 sinhr a2 sin2 ϕ1 =2 þ a′ sin2 ϕ2 =2 2 þ2 sinhr b2 sin2 ϕ1 =2 þ b′ sin2 ϕ2 =2 ð20fÞ ′ ′ þ2 sinhr ab sinϕ1 þ a b sinϕ2 ;

and 2

2

2

4AC−E ¼ 16 þ 16 sinh r sin ðϕ1 −ϕ2 Þ=2:

ð20gÞ

Eq. (19) is the modulus square of the scalar product of two squeezed states expressed by Eqs. (17) and (18) and using this result, we can ﬁnd optimal unambiguous discrimination of two squeezed

Fig. 3. For Re α = Im α = |α| = 2, ϕ2 = π/2, variation of probability of unambiguous discrimination, PD, with r and ϕ1 (i. e., phi).

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D.K. Mishra / Optics Communications 285 (2012) 1560–1565

output consists of perfect copies. If it goes on, which it does with a certain nonzero probability if the input set contains nonorthogonal states, then the copying process has failed, and we discard the output. Such a probabilistic cloning machine [32] is a procedure combining unitary evolution and measurement to get perfect clones with success probability less than 1. In this strategy, given a system where the states are known to be linearly independent, a cloning operation can be followed by a measurement which will collapse the ﬁnal state into either a series of perfect copies or a series of states which possess no information. The outcome of the measurement will reveal which has occurred. The states chosen from a set S = {|ψ1〉,|ψ2〉,......,|ψn〉} can be probabilistically cloned iff the states |ψi〉 are linearly independent [32,33]. In this case a unitary transformation of the following form exists: ﬃ ^ jψ 〉j0〉jA〉 ¼ pﬃﬃﬃﬃ pi jψi 〉jψi 〉jA0 〉 þ U i

n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X 1−pi jΦj 〉jAj 〉;

ð21Þ

j¼1

Fig. 4. For Re α= Im α= |α| =2, ϕ2 = 0, variation of probability of unambiguous discrimination, PD, with ϕ1 (i. e., phi) for different r.

unambiguous discrimination, PD, is zero at ϕ1 = ϕ2 = 0, then increases and attains maximum at ϕ1 = π (Fig. 5), i.e., when the two squeezed states are orthogonal. Now, we will look at about the theoretical model for unambiguous discrimination of the two known states with the help of probabilistic quantum cloning machine in order to realize experimentally. 4. 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 as inputs. Let us think about a condition where we relax the constraint that a copier produces perfect copies and works every time (i.e., copier acts as a perfect cloner, albeit probabilistic). We require that the copies of our allowed input set be perfect, and we permit the copier to have a certain probability of failure, but impose the constraint that the copier lets us know when it has failed and when it has succeeded just like an alarm bell. We put in one of our allowed input states, and the copier produces an output. If the alarm bell stays off, the copying process has succeeded, and our

^ being a with i = 1, 2,……, n and 〈Ak|Al〉 = δkl for k, l = 0, 1, 2,……, n, U ^† ¼ U ^ −1 , or equivalently, U ^ is said to be unitary opunitary operator if U ^ †U ^ ¼ ^I. The state |Aj〉 of the ancillary system is orthogonal to erator if U the state |A0〉 and the states |Φj〉 are normalized, linearly dependent collective states of original and copy systems. These states are not necessarily orthogonal. Measuring the ancilla state in the basis {|Ak〉} then leads with probability pi to the desired clones, |ψi〉|ψi〉, of the state |ψi〉. Otherwise, with probability (1− pi) original and copy systems are left in the state |Φj〉 which cannot be unambiguously discriminated. Using this method for the case of a 1 → 2 cloner, the maximum average probability of successful copying is P ¼ 1=ð1 þ j〈ψ1 jψ2 〉jÞ;

ð22Þ

where |ψ1〉and |ψ2〉 are the two linearly independent states [33,34]. If the two states are orthogonal, this can be seen to yield a perfect clone 100% of the time and decreases as their overlap increases. For the unambiguous discrimination of the two known states with the help of probabilistic cloning machine, we are given one of the two possible states |ψ1〉 or |ψ2〉 and we have to know then the state |ψ?〉. Restriction is that we are given only single copy of the state |ψ?〉 = |ψ1〉 or |ψ2〉. For the case of 1 → 2 cloning, |ψ?〉|ψ?〉, namely two output copies are produced from a single input, cloning transformation is described by a unitary operation acting on the state we want to copy and the “blank” ancilla, |0〉, to which we want the copied state. On the ﬁrst ^ † which produces U ^ † jψ 〉 and copy, we perform unitary operation U 1 1 ? ^ † (j= 1 or 2) acts on the respective then perform photo detection. Here U j state |ψj〉 (j= 1 or 2). If we register a signal (click) at the detector, then for sure the state |ψ?〉 is not |ψ1〉. On the second copy we perform unitary ^ † which produces U ^ † jψ 〉 and then perform photo detection. operation U 2 2 ? If we register a signal (click), we can unambiguously say that the state ^ ^ † ψ 〉 ¼ U ^†U |ψ?〉 is not in |ψ2〉 state. For, if |ψ?〉 = |ψ1〉, U ? 1 1 1 0〉 ¼ j0〉, † † † ^ U ^ U ^ jψ 〉 ¼ U ^ 1 j0〉 ≠ |0〉, since U ^ 1 ≠1 and as soon as we perform a U ?

2

2

2

photon detection, we have a signal (i.e., click). Probability of the unambiguous discrimination, PD, is conditioned by the maximum average probability of successful copying P = 1/(1+ |〈ψ1|ψ2〉|), i.e., h ′ ′ ′ ′ PD ¼ P η1 ψ1 ^I−0 0 ψ1 þ η2 ψ2 ^I−0 0 ψ2 :

〉〈 jÞj 〉

〈

〈 j

〉〈 jÞj 〉

ð23Þ

^ ¼ ^I (j = 1 or 2), ^ j, U ^ †U Since for any unitary transformation U j j

j〈ψ jψ 〉j ¼ j〈ψ jU^ U^ jψ 〉j 2

1

2

1

† 2

2

2

2

′ ¼ ψ 1 0

〈

〉j ; 2

ð24Þ

^ † ¼ 〈ψ′ j and U ^ jψ 〉 ¼j0〉. Similarly, when 〈ψ1 jU 2 2 1 2 Fig. 5. For Re α=Im α=|α|=2, r=10, variation of probability of unambiguous discrimination, PD, with ϕ1 (i. e., phi1) and ϕ2 (i. e., phi2).

j〈ψ jψ 〉j ¼ j〈ψ jU^ U jψ 〉 2

2

1

2

† 1

2

1

1

′ ¼ ψ 2 0

〈

〉j : 2

ð25Þ

D.K. Mishra / Optics Communications 285 (2012) 1560–1565

Then h i PD ¼ P η1 1−〈ψ′1 0〉 2 þ η2 1−〈ψ′2 0〉 2 h i ¼ P η1 1−j〈ψ1 jψ2 〉j2 þ η2 1−j〈ψ2 jψ1 〉j2 :

j

j

ð26Þ [14]

If η1 ¼ η2 ¼ 12, then 2 2 PD ¼ P 1−j〈ψ1 jψ2 〉j ¼ 1−j〈ψ1 jψ2 〉j =ð1 þ j〈ψ1 jψ2 〉jÞ ¼ 1−j〈ψ1 jψ2 〉j:

ð27Þ

[15]

This general result describes the optimal probability of successful unambiguous discrimination of any two known pure states with equal a priori probability. In this way, we have found the optimal unambiguous measurement for the two copies using probabilistic cloning machine. 5. Conclusions [16]

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 networks and a higher level of precision control in comparison with that for the case of approximate cloning machines [35]. Recently, Chen et al. [36] designed an efﬁcient quantum network with a limited amount of resources and performed the ﬁrst experimental demonstration of probabilistic quantum cloning in a NMR quantum computer and the optimal cloning efﬁciency is achieved. Therefore, after this experimental demonstration the present study is expected to be experimentally realizable with the probabilistic quantum cloning in optical systems in the near future. 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 M. Sedlák for productive discussions on this work. This work was partially supported by the ‘Exchange Programme with Foreign Academies and Organisations’ of Indian National Academy of Sciences, New Delhi (Letter No. IA/Misc. 2009-2010/ 4237) and Slovak Academy of Sciences, Bratislava, Slovakia.

[17]

[18]

[19] [20] [21] [22] [23]

[24] [25]

[26]

[27] [28] [29] [30] [31] [32] [33] [34]

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