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Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties
Optics Communications 281 (2008) 3748–3754
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Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties Sergey A. Podoshvedov a,*, Jaewan Kim a, Juhui Lee b a b
Southern Ural State University, Nonlinear Optics Laboratory, Lenin Avenue 76, 454080 Chelyabinsk, Russian Federation Department of Physics, Sookmyung Women’s University, Seoul, South Korea
a r t i c l e
i n f o
Article history: Received 1 December 2007 Received in revised form 21 March 2008 Accepted 22 March 2008
PACS: 42.50.Dv 03.65.Bz
a b s t r a c t We study optical schemes for generating both a displaced photon and a displaced qubit via conditional measurement. Combining one mode prepared in different microscopic states (one-mode qubit, single photon, vacuum state) and another mode in macroscopic states (coherent state, single photon added coherent state), a conditional state in the other output mode exhibits properties of a superposition of the displaced vacuum and a single photon. We propose to use the displaced qubit and entangled states composed of the displaced photon as components for quantum information processing. Basic states of such a qubit are distinguishable from each other with high fidelity. We show that the qubit reveals both microscopic and macroscopic properties. Entangled displaced states with a coherent phase as an additional degree of freedom are introduced. We show that additional degree of freedom enables to implement complete Bell state measurement of the entangled displaced photon states. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Schrödinger asked what happens when an atom is in an entangled state with an alive and dead cat [1]. Now, this puzzle of quantum mechanics is resolved by realizing that such an object is very sensitive to decoherence and so is unlikely to occur in real life. But nevertheless, it poses the question of whether a classical object on a macroscopic level may be in a state of quantum superposition. If it is possible, then, according to the basic-theoretical principles of quantum mechanics, the superposition of macroscopically distinguishable states (the component states composing such a superposition must give classically distinct measurement outcomes [2]) can give a rise to quantum interference since the superposition is highly nonclassical. This can serve as evident manifestation of the existence of the superposition of classical objects. A typical example of such a superposition, called the ‘‘Schrödinger cat state,” is the superposition of two optical coherent states with sufficiently large amplitudes but with opposite phases [3]. The superposition exhibits typical interference features. Measuring the quadrature-component distribution of a statistical mixture of coherent states with opposite phases, we observe two peaks. In the case of a coherent superposition, the two peaks change their mutual distance depending on the phase difference of the compo-
* Corresponding author. Address: School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, South Korea. Tel.: +82 2 958 3863; fax: +82 2 958 3820. E-mail addresses: [email protected], [email protected] (S.A. Podoshvedov). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.073
nents until they eventually overlap. Thus, the difference between a statistical mixture and a coherent superposition is most evident. Generation of a Schrödinger cat superposition in a realistic experiment is not trivial. A main method to generate such a superposition is the nonlinear interaction of an optical field in a coherent state with a Kerr medium [3]. But for all practically available values of the Kerr nonlinearities, the time needed to generate the Schrödinger cat superposition, is much longer than the decoherence time. Decoherence and phase fluctuations may destroy any inceptive superposition during the propagation of the light inside the medium [4]. Another alternative method to generate a superposition of macroscopically distinguishable states is based upon conditional measurement [5–7]. The Schrödinger-cat-like state so called, by virtue of its definite resemblance with a Schrödinger cat superposition, is conditionally obtained if we launch a squeezed vacuum into a beam splitter and count the photons in one of the output channels [5]. It is also worth noting a number of papers closely related to our consideration [8,9]. In the paper, we want to propose alternative to the Schrödinger cat state, namely, displaced qubit and entangled displaced photon states. The authors in [7] showed that an arbitrary single-mode state can be created by starting from the vacuum and applying a sequence of displacements and single-photon additions. Following the similar method, we use similar treatment to generate both a pure displaced photon state [10–12] and displaced qubit. We show that such a displaced qubit possesses both microscopic and macroscopic properties. The components of the qubit give classically
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distinct measurement outcomes and the itself shows interference properties. Extending our analysis we introduce entangled displaced state composed of the displaced qubits and describe a method to perform for them a complete Bell state measurement. 2. Generation of the displaced qubit It is well-known that the input–output relations at a lossless beam splitter can be characterized by the SU(2) Lie algebra [13] (all needed mathematical formulas for our consideration are presented in Appendix A). Assume that the modes of the beam pffiffiffi splitter pffiffiffi are prepared in a coherent state with amplitude 2aðj 2ai1 Þ (mode 1) and in a state of a single-mode qubit consisting of the vacuum and a single-photon pffiffiffi ! 2 jaj jsi2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j0i j1i : ð1Þ a 2 þ jaj2 2 The experimental generation of an arbitrary superposition of the vacuum and single photon states has been accomplished using parametric down conversion with the input signal mode prepared in a coherent state [14], or conditioning on homodyne measurements on one part of a nonlocal single photon in two spatial modes [15]. Theoretical study of employing the quantum scissors scheme is presented in [16]. If we combine the input modes on a device with Hadamard input–output relations (A3c), the result is given by 0 1 pffiffiffi ! pffiffiffi jaj 2 B C b @j 2ai qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j0i j1i A H 1 a 2 2 þ jaj 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jaj ð2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1; ai1 jai2 þ 1 þ jaj2 jai1 ja; 1i2 ; a 2 þ jaj2 where 0 b j1; ai ¼ DðaÞj1i ¼ a @jai
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ jaj2 a
1 ja; 1iA
ð3Þ
b ^Þ is the uni^þ a a is the displaced photon [10–12] ( DðaÞ ¼ expðaa tary displacement operator [17]), and the one-photon added coherent state is given by [18,19] 1 exp jaj2 =2 X ^þ jai a a1 l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi jli ja; 1i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð4Þ þ ^ ^ 2 hajaa jai l! l¼1 a 1 þ jaj Restrict our attention to the events when no photons are recorded in the second output channel (Eq. (2)). Then, the first output channel of the beam splitter is prepared in the state j1,ai since the onephoton added coherent state does not contain a vacuum state unlike the coherent one. We now mention our notation. One should not mistake the displaced single photon j1,ai state for the single photon added coherent state ja,1i. There is a difference between them since neither the annihilation nor creation operators commute with the displaceb ^; DðaÞ ment operator ð½a 6¼ 0; where a 6¼ 0Þ. In the limiting case a ? 0, the states j1,ai and ja,1i simply reduce to one-photon j1i. Coherent and one-photon added coherent states are not orthogoqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nal to each other ðhaja; 1i ¼ a = 1 þ jaj2 Þ and their scalar product approaches 1 if a ? 1. This means that single photon added coherent states become scarcely distinguishable from coherent states with growing amplitude (a ? 1). On the contrary, the state j1,ai is always orthogonal to the coherent state with the same amplitude a,regardless of the value of a(haj1,ai = 0). Therefore, we can interpret the coherent sate and displaced single photon states as
basic logic states (logical zero and logical one, respectively) in the framework of quantum information. We are going to denote a coherent state by j0,ai jai by analogy with the j1,ai state. The number state is determined by its photon number while the phase of the number state is completely random. By displacing in phase space, the field amplitude is added to the number state. The displaced qubit jWi ¼ Aj0; ai þ Bj1; ai;
ð5Þ 2
2
where A and B are arbitrary amplitudes (jAj + jBj = 1), can also be generated by a photon number conditional measurement. Consider a singe-mode qubit with arbitrary amplitudes, jsi = aj0i + bj1i. Then, superimposing an ancilla coherent state with jsi, one obtains pffiffiffi 0 b Hðj 2ai1 ðaj0i þ bj1iÞ2 Þ ¼ ða0 j0; ai þ b j1; aiÞ1 j0; ai2 þ cj0; ai1 ja; 1i2 ; ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 pffiffiffi 2 where a ¼ a þ ba = 2, b ¼ b= 2, and c ¼ b 1 þ jaj =2. Using the definition of the coherent and one-photon added coherent states and detecting l photons in the second output mode, the state (6) is conditionally transformed to state (5), where A ¼ AðlÞ ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 ða0 lb=a 2Þ= ja0 lb=a 2j2 þ jb j2 is the amplitude of the displaced vacuum, j0,ai, dependent on l, and where B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 b= 2ðja0 lb=a 2j2 þ jb j2 Þ is the amplitude of the j1,ai state independent of l. The number of detected photons influences only the amplitude A(l), leaving other parameter B unaffected.pffiffiffi The success 0 probability of such an event is expðjaj2 Þjaj2l ðja0 lb=a 2j2 þ jb j2 Þ=l! Consider another case of preparation of the state (5). One of the modes is prepared in an ancilla coherent state and the other in a single photon state. Then, the result of the mixing is given by [20,21] 0
pffiffiffi 1 b Hðj 2ai1 j1i2 Þ ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: 2
ð7Þ
From this equation we have that when the number of photons detected in the second mode is l, the mode in the other output channel is prepared in the quantum state in Eq. (6) where A ¼ AðlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a= jaj2 þ jbj2 , B ¼ b= jaj2 þ jbj2 , a = a* (1 l/jaj2), and b = 1. In particular, when the number of detected photons, l, is zero, then the output is a single-photon added coherent state, whose properties were described in detail in Ref. [6]. The success probability of such an event is exp(jaj2) jaj2l (jaj2 + jbj2)/2l! Another method of preparation of state (5) is the following. Let us prepare one of the modes of a beam splitter in a single photon added coherent state, another mode of the same beam splitter in the vacuum state. Then, combining the modes, one obtains [20,21] pffiffiffi b Hðj 2a; 1i1 j0i2 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ jaj2 ðja; 1i1 jai2 þ jai1 ja; 1i2 Þ: 2ð1 þ 2jaj2 Þ
ð8Þ
When the photon number of the mode in the second output channel is measured and l photons are detected, then the mode in the first output channel is prepared in the quantum state (5) with A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðlÞ ¼ a= jaj2 þ jbj2 , B ¼ b= jaj2 þ jbj2 , a = a* + l/a, and b = 1 with success probability exp(jaj2)jaj2l(jaj2 + jbj2)/2(1 + 2jaj2)l! For the number of detected photons, l = 0, the conditional state becomes the single photon added coherent state. Thus, the conditional generation of the displaced qubit (5) needs highly efficient and precise photocounting. Unfortunately, there are no highly efficient photodetectors available which precisely distinguish between m and m + 1 photons, where m > 2. Only recently, Takeuchi et al. [22] developed an avalanches photodetector which can discern 0, 1, and 2 photons with high efficiency. Nevertheless, we can ease requirements imposed on the efficiency of photocounting in conditional preparation of the displaced qubit
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(5). Consider one example, we combine a single photon added coherent state with amplitude a1 in one mode with an ancilla coherent state with amplitude a2 in another mode of a beam splitter with arbitrary parameters T and R. The result of the mixing following from formulas from Appendix A is given by 1 b ðja1 ;1i ja2 i Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 2 1þja1 j2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 B T 1þja1 T a2 R j ja1 T a2 R ;1i1 ja1 Rþa2 T i2 þ C @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: þR 1þja1 Rþa2 T j2 ja1 T a2 R i1 ja1 Rþa2 T ;1i2 ð9Þ *
Suppose the condition a1R + a2T 1 holds for the input parameters. Then, the states ja1R + a2T*i and ja1R + a2T*,1i may be approxiðiÞ ðiÞ ðiÞ mated by the finite superposition c0 j0i þ c1 j1i þ c2 j2i, where the ðiÞ0 cj s are the amplitudes (i = 0,1 and j = 0 2) of the coherent and single-photon added coherent states, respectively, in the case that a1R + a2T* 1 when we neglect higher order photon number states. When the state j0i2 is detected in the second output mode, this detection event projects the first mode into a single photon added coherent state with amplitude a1T a2R*. The state like Eq. (5) is conditionally generated if either the state j1i or j2i is registered in the second mode [22]. It is worth noting the used truncation of the state in measured mode is possible owing to choice of small value of the parameter s = a1R + a2T* 1. Direct calculation shows that the influence of the truncated terms on the fidelity of the conditionally generated state is negligible. So, if we register two-photon state [22] the fidelity of the conditionally prepared superposition state can be estimated as 1 s. 3. Properties of the displaced qubit Let us analyze the readout of the qubit in Eq. (5) in the framework of quantum information processing. Consider a partial case. Somebody gives us either j0,ai or j1,ai, which exactly we do not know. Our task is to determine what state we obtained. Use a simple measurement scheme with a balanced beam splitter with the
Q¼
Distinguishability of the states j1,ai1 and j0,ai1, like many ideas in quantum computation and quantum information, is most easily understood using the metaphor of a game involving a coin that is tossed up. We can always identify whether coin has landed on heads or tails and these probabilities, at least in the ideal case, are equal. Imagine we are concerned with some fixed set of outcomes, for example, if the coin only lands on heads. The success probability of such a set is 1/2n, where n is the number of tosses. This is in direct analogy with the identification of the states j1,ai1 and j0,ai1. We can improve the method to distinguish the states j0,ai1 and j1,ai1 from each other. Prepare a second input channel in an ancilla coherent state with amplitude a1, j0,a1i2 so that condition b Bðj0; b aR + a1T* = 0 is performed. Then, we have P ai1 j0; a1 i2 Þ ¼ j0; a=T i1 j0i2 and b aRþa T ¼0 ðj1; ai j0; a1 i Þ ¼ Tj1; a=T i j0i þ Rj0; a=T i j1i : U 1 2 1 2 1 2 1
Again, if an ideal detector in the second output mode clicks, it means it was the j1,ai1 state. If the detector does not click we can tell nothing definite about the state we obtained. But we can decrease the influence of the T—1,a/T*i1—0i2 term in Eq. (11) by choosing a beam splitter transmittance T as small as possible (jTj 1) satisfying limT !1 a1 T ¼ a. If we take the transmittance T very small, then it is sufficient to manage one trial. Then, the failure probability to mistake the j0,ai1 state for the j1,ai1 state is jTj2 1. The same consideration is applicable to the state (5). Again if a single photon hits a photodetector in the second mode, we definitely know that the j1,ai1 state was measured while if we do not register something we know it was the j0,ai1 state with fidelity F = jAj2/ (jAj2 + jBj2jTj2) 1 jBj2jTj2/jAj2. The fidelity approaches 1 if jTj ? 0, that confirms the possibility of almost definitely distinguishing the states j0,ai1 and j1,ai1 from each other. Thus, the components of the displaced qubit give different measurement outcomes. Let us show other nonclassical properties of the displaced qubit (5). A measure of the deviation of the photon-number distribution from a Poissonian is the Mandel Q parameter which for the displaced photon is given by
jaj2 jBj4 þ jBj2 ð1 þ jaj2 Þ2 2jAj jBjðjBj2 jAj2 Þjaj cosðd hÞ 4jAj2 jBj2 jaj2 cos2 ðd hÞ jaj2 þ jBj2 þ 2jAj jBj jaj cosðd hÞ
unitary Hadamard matrix (A3c). The input state jWi is superimposed with p the ffiffiffi ancilla state j0,ai2. Mixing of the states j0,ai1 and j0,ai2 gives 2ai1 j0i2 , while the same mixing of the states j1,ai1 and j0,ai2 gives a balanced superposition (see Appendix A) pffiffiffi pffiffiffi b H ðj1; ai j0; ai Þ ¼ p1ffiffiffi ðj1; 2ai j0i þ j0; 2ai j1i Þ: U 1 2 1 2 1 2 2
ð11Þ
ð12Þ
:
We display the Q parameter as a function of jAj for different absolute magnitudes of a in Fig. 1. For small values of jaj the statistics of
Q ð10Þ
If the detector registers a single photon in the second mode, which happens with probability 1/2, we definitely know that it was the j1,ai state. If the detector registers nothing then we cannot identify the post-selected outcome. We can repeat again the same measurement procedure by superimposing outcome and another ancilla pffiffiffi the p ffiffiffi coherent state with amplitude 2aðj0; 2ai2 Þ. Again if a single photon hits a photodetector in the second output mode, we definitely know that the state we obtained was j1,ai. If we do not register any photon impacts we do not know what state was measured. By applying a sequence of the same measurements many times, we can conclude that if we obtain a series n of vacuum measurements, then initially it was the j0,ai state with probability to mistake the j0,ai state for the j1,ai state equal to 1/2n, where n is the number of repetitions.
2.5 | α |=2
2 | α |=1
1.5 1 0.5
| α |=0.5
|A| 0.2
0.4
0.6
0.8
1
Fig. 1. Q parameter as a function of the absolute amplitude jAj for different values of amplitude of the displaced qubit a = 0.5, a = 1, and a = 2. The values of a are assumed to be real numbers.
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a balanced superposition of displaced vacuum and single photon states becomes sub-Poissonian (Q < 1). The variance of the quadrature components for the state (5) is given by ðD^ xÞ2 ¼
1 ð1 þ 2ð1 jAj2 ÞjBj2 2jAj2 jBj2 cosð2ðd þ hÞÞÞ; 4
ð13Þ
2
where h is the phase of the local oscillator and d is the relative phase of A and B. Fig. 2 shows the result of numerical calculation of Eq. (13) for different input parameters. Variances of the displaced single photon state j1,ai prevail over the ones for the coherent state [12]. Nevertheless, the variance of the displaced photon can take values less than the variance of coherent state for some values of the input parameters. The input parameters can be found from Fig. 2. The Wigner function of the balanced displaced qubit can be evaluated as 4 ððx xa Þðx xa cos Q a Þ þ ðy ya Þðy ya sin Q a ÞÞ p expð2ððx xa Þ2 þ ðy ya Þ2 ÞÞ; ð14aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cos Q a ¼ xa = jxa j2 þ jya j2 and sin Q a ¼ ya = jxa j2 þ jya j2 . The Wigner function of the statistical mixture described by the density matrix q = (j0,aih0, aj + j1,aih1, aj)/2 is
(Δx)
W jWB i ðx; yÞ ¼
0.5 3
0.4 0.3
2
0 1
φ
1
δ
W q ðx; yÞ ¼
2 3
0
pffiffiffi Fig. 2. Plot of variance versus h and d for the displaced qubit (5) with A ¼ B ¼ 1= 2. There are ranges where variance of the state (5) is less of one of the coherent state (squeezing).
a
4 ððx xa Þ2 þ ðy ya Þ2 Þ expð2ððx xa Þ2 þ ðy ya Þ2 ÞÞ: p ð14bÞ
Fig. 3a–d reveals the Wigner functions of a balanced superposition and statistical mixture, respectively. The Wigner function of a balanced superposition exhibits a single peak unlike the Wigner function of the statistical mixture with two peaks. The difference in the
c
0.6
1
2
0.5
0 2
-0.2 0
0 -1 0
3
0.2
1
0
0.4
1
1
-1 1
2
2 -2
3
b
0
d
0.8 0.6 0.4 0.2 0
2 1 0
-1
0.2 0.15 0.1 0.05 0 0
3 2
1
1
0
-1 2
1 2
-2
30
pffiffiffi Fig. 3. Wigner function of the balanced displaced photon jWi ¼ ðj0; ai expðiua Þj1; aiÞ= 2 (a and c) and statistical mixture described with density operator q = (j0,aih0, aj + j1,ai h1, aj)/2 (b and d, respectively) for a = 0.5, Qa = p/4 (a) and (b) and a = 2, Qa = p/4 (c) and (d).
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Φ ( x)
may become a displacement operator for the components of the entangled state jW(a,a)i12. Another possible state with the same parameter a is given by
2
a
0.8
1 jWþ ða; aÞi12 ¼ pffiffiffi ðj1; ai1 j0; ai2 þ j0; ai1 j1; ai2 Þ: 2
1
0.6
ð16Þ
If we change a parameter a on opposite a, for example, in first mode in used notations we obtain another pair of the entangled displaced states
0.4
2 0.2
x 1
Φ ( x)
2
3
4
2
b
0.8
1 0.6
0.4
2
x 1
2
3
ð17Þ
Each pair of the entangled displaced states jW±(a,a)i12 and jW±(a,a)i12 is orthogonal to each other. But the entangled states with opposite phases are not orthogonal to each other hWþ ; a; ajWþ ; a; ai ¼ ð1 2jaj2 Þ expð2jaj2 Þ;
ð18aÞ
hWþ ; a; ajW ; a; ai ¼ 2jaj2 expð2jaj2 Þ:
ð18bÞ
Nevertheless, the scalar products go fast to zero when a grows since theirs overlaps decrease exponentially with a. For example, when a is as small as 3, the overlap is 107. Thus, the states (18a) and (18b) are almost orthogonal, especially with increase of a. Unlike traditional Bell states, the entangled displaced states are distinguishable from each other due to additional degree of freedom a phase of the coherent state. It can be seen if we combine the states jW±(±a, a)i12 on a beam splitter with a matrix (A3c)
0.2
- 1
1 jW ða; aÞi12 ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: 2
4
Fig. 4. Quadrature distribution jU(x)j2 of the balanced displaced photon and statistical mixture for a = 2 and Qa = 0 (a) and Qa = p/4 (b). The curve 1 corresponds to the superposition state (5) while the curve 2 corresponds to the statistical mixture. Two separated peaks of the statistical mixture transform to one peak of pure superposition.
forms is caused by the interference terms due to correlations inherent to the superposition state (5). The interference terms are proportional to cosQa and sinQa, respectively, and destroy one of the peaks as illustrated in Fig. 3a and c. The marginal distribution for one of the two components, for example x, is obtained by integration over the other variable y rffiffiffi 2 ððx xa Þðx xa cos Q a Þ þ 0:25Þ expð2ðx xa Þ2 Þ: p ð15Þ
jUðxÞj2 ¼ 2
The corresponding dependencies of the quadrature distributions jU(x)j2 are plotted on Fig. 4a and b for the both displaced qubit and the statistical mixture. If we take Qa = p/2, then there is no difference between jU(x)j2 for a pure superposition and statistical mixture. 4. A complete Bell state measurement of the entangled displaced states A problem of generation of the displaced qubit is turned out not to be trivial as it was described above. Although, it may look easy if we already know how to produce a superposition of Fock states j0i and j1i, just applying the displacement operator ðjWi ¼ Aj0; aiþ b Bj1; ai ¼ DðaÞðAj0i þ Bj1iÞÞ. But, there is not pertinent model for such a displacement operator. But nevertheless, such a model of the displacement operator can be easy done if we are interested in generation of entangled state composed of two displaced qubits (5). Indeed, as can be seen from the formula (A6), a beam splitter
b HðjW
ð a; aÞi12 Þ ¼ j1;
pffiffiffi 2ai1 j0i2 if the input state was jWþ ða; aÞi; ð19aÞ
j0;
pffiffiffi 2ai1 j1i2 if the input state was jW ða; aÞi;
ð19bÞ
pffiffiffi j1i1 j0; 2ai2 if the input state was jWþ ða; aÞi;
ð19cÞ
pffiffiffi j0i1 j1; 2ai2 if the input state was jW ða; aÞi:
ð19dÞ
Next step is to discern photon number states —0i and j1i from their displaced analogues j0,ai and j1,ai, respectively. It can be evidently done from intuitive point of view taking into account, for example, the overlap of the coherent and vacuum states (jh0j0,aij2 = exp(jaj2)) fast approaches zero when a grows, so thus, they become almost orthogonal with larger a. In practice, off-the-shelf photon counters can only differentiate between zero and more photons because we insert additional beam splitters with detectors placed behind them b^ 0H b 0H b 12 ðjW ð a; aÞi Þ ¼ p1ffiffiffi ðj1; ai j0; ai 0 þ j0; ai j1; ai 0 Þj0i j0i 0 H 12 12 1 1 1 1 2 2 11 2 if the input state was jWþ ða; aÞi; ð20aÞ 1 pffiffiffi j0; ai1 j0; ai10 ðj1i2 j0i20 þ j0i2 j1i20 Þ 2 if the input state was jW ða; aÞi;
ð20bÞ
1 pffiffiffi ðj1i1 j0i10 þ j0i1 j1i10 Þj0; ai2 j0; ai20 2 if the input state was jWþ ða; aÞi;
ð20cÞ
1 pffiffiffi j0i1 j0i10 ðj1; ai2 j0; ai20 þ j0; ai2 j1; ai20 Þ 2 if the input state wasjW ða; aÞi;
ð20dÞ
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b ij means the Hadamard where 10 and 20 are the ancilla modes and H operation applied to the modes i and j, respectively. Analyzing given formulas, we can conclude the following: (i) If we register the output jnm00i110 220 with n 6¼ 0 and m 6¼ 0 then we definitely know it was a state jWþ ða; aÞi12 . (ii) If we register in sum three clicks ðjnm10i110 220 or jnm01i110 220 , where n 6¼ 0 and m 6¼ 0), then we certainly identifies such a state as jW(a,a)i12. (iii) If we register three clicks ðj10nmi110 220 or j01nmi110 220 , where n 6¼ 0 and m 6¼ 0), then we exactly know that it was jW+(a,a)i12 state. (iv) If only detectors in modes 2 and 20 register coming photons j00nmi110 220 with n 6¼ 0 and m6¼ 0 then we make sure that it was a state jW ða; aÞi12 . All these events are most probable ones in the case of large amplitude a. There are also cases when either n = 0 (m = 0) or both of them n = m = 0. Probability of the events is negligibly small for the large value of a since it drops exponentially with a growing. Thus, we conclude it is possible to distinguish two pairs jW±(a,a)i12 and jW±(a, a)i12 from each other. The success probability becomes exactly equal one if a ? 1. In practice, it is sufficient to deal with an input quantum channel with sufficiently large amplitude a. Failed events not distinguished by the method must be rejected. Assume that somebody has in his possession both qubits, for example, the jW+(a1, a2)i12 state. A question that can be raised whether someone is able to transform the state to the rest three states by local operations. It can be easily done in the case of transformations jW+(a1, a2)i M jW(a1,a2)i and jW(a1, a2)i M b jW+(a1,a2)i by phase shifting unitary transformation on pð PðpÞÞ b applied to one of the modes. Indeed, PðpÞj0; ai ¼ j0; ai and b b b b DðaÞj0i b b DðaÞ b P b þ ðpÞ PðpÞj1i PðpÞj1; ai ¼ PðpÞj1; ai ¼ PðpÞ ¼ PðpÞ b ¼ DðaÞj1i ¼ j1; ai:
ð21Þ
Another type of transformations jW+(a1,a2)i M jW(a1,a2)i and jW+(a1,a2)i M jW+(a1,a2)i is not trivial and requires ancilla coherent states. Combine a mode 1 with mode 3 occupied by the coherent state j0,a3i3 and mode 2 with mode 4 in the coherent state j0,a4i4 on two corresponding beam splitters with matrix (A3a). Then, we have b 34 B b 13 B b 24 ðjWþ ða1 ; a2 Þi j0; a3 i j0; a4 i Þ ¼ cos Q jWþ ða1 ; a2 Þi H 12 3 4 12 pffiffiffi pffiffiffi j0; ða3 a4 Þ= 2i3 j0; ða3 þ a4 Þ= 2i4 i sin Q j0; a1 i1 j0; a2 i2 pffiffiffi pffiffiffi j1; ða3 a4 Þ= 2i3 j0; ða3 þ a4 Þ= 2i4 ; ð22Þ provided that a3 = ia1(1 + cosQ)/sinQ and a4 = ia2(1 cosQ)/sinQ. Use a beam splitter with large transmittance Q ffi 0. Next, one makes a measurement in mode 3 no longer concerning the actual value of measurement that provides us a mixture of states jW+(a1,a2)i12 and j0,a1i1j0,a2i2 with probabilities cos2Q and sin2Q, respectively. As Q ffi 0, cos2Q ffi 1 sin2Q, the contribution of the state jW+(a1, a2)i12 prevails over one of another state. Thus, it is possible to perform all unitary transformations to obtain all four entangled states jW±(a,a)i12 and jW±(a,a)i12 with large success probability. Let us also mention the paper [23] where a scheme to discern four Bell states using linear optics when the Bell states are prepared with coherent states with coherent states of two different amplitudes unlike our work. 5. Discussion We have analyzed a problem of conditional generation of displaced qubit and entangled displaced states. The motivation to deal
with such states comes from the fact that they carry phase of coherent ‘‘shell” as additional degree of freedom that may become key moment in quantum information. Quantum information offers a number of striking applications. But to reach all advantages of these applications one needs to prepare and measure the Bell states. Problem of creating the Bell state is solved via nonlinear interaction of light in parametric down-conversion (polarization entanglement) [24]. The rest three Bell states can be prepared from maximally entangled state by simple local unitary transformations. The problem of performance a complete Bell measurement with linear devices (like beam splitters and phase shifters) encounters serious difficulties. Combination of controlled NOT operation (CNOT) on two interacting systems and Hadamard operation on one of the system allows for one to transform the four Bell states into four disentangled basis states. But the case is not trivial in practice. The Bell state measurement is inherently nonlinear in the optical case and only partial set of Bell measurement can be made in practice with linear optical elements [25,26]. A phase of the displaced photon as an additional degree of freedom gives a possibility to perform complete Bell state measurement of the entangled displaced states. We have shown that both displaced qubit and entangled displaced states can be generated provided that we have corresponding resource of the single-mode qubit, photon added coherent state and a simple photon, respectively. We showed that the components of such a displaced state can be distinguished from each other with high fidelity. The displaced qubit reveals the properties of a mesoscopic state that has both microscopic (interference and superposition) and macroscopic (classically distinguishable) properties. Displacement of the microscopic state is performed via conditional measurement for the displaced qubit and via use of beam splitter for entangled displaced state. Such s mechanism is different from that generating a Schrödinger cat state based on a nonlinear phase shift. Acknowledgement This work was supported by the IT R&D program of MIC/IITA (2005-Y-001-04, Development of next generation security technology). Appendix A. Beam splitter transformations with displaced states Let us to present some useful formulas used during the text and related to the beam splitter transformations [13] of the displaced b ¼a ^þ ^ ^þ ^ states. An evolution operator, where X 1 a2 þ a2 a1 , Q is the some parameter b Þ ¼ expðiQ XÞ; b BðQ
ðA1Þ
defines the beam splitter transformation " # " # " þ # ^þ ^þ ^1 a a a cos Q i sin Q 1 1 ¼B þ ¼ : þ ^þ ^ ^ a a a i sin Q cos Q 2 2 2 out
in
ðA2aÞ
in
A very simple way to describe the action of a beam splitter is to fix the phase relations such that in general case the beam splitter transformation is described with the following input-output relations " # " # " þ # ^þ ^1 ^þ T R a a a 1 1 ¼ PB þ ¼ ; ðA2bÞ þ ^þ ^ ^ a a a T R 2 2 2 out
in
in
where T and R are the transmittance and reflectivity of the beam splitter, respectively. The corresponding evolution operator is b 2 ðuÞ BðQ b Þ, where the phase shifting evolution operator bB b¼P b 1 ðuÞ P P
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b ¼ b ^Þ. We can apply two phase shifters H ^þ a is PðuÞ ¼ expðiua b b b P 2 ðp=2Þ Bðp=4Þ P 2 ðp=2Þ to second mode to deal with Hadamard transformation on optical modes " # ^þ þ þ
a 1 1 1 þ þ
1 ^ ^2 in : ^ ^1 a p ffiffiffi ¼ H ¼ a a a ðA2cÞ 1 2 in ^þ a 2 1 1 2
b Hðj0;
pffiffiffi 2ai1 j1i2 Þ ¼ jW ða; aÞi12
1 ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: ðA5Þ 2 It is worth mentioning the generation of a single photon can be conditionally accomplished using parametric down conversion.
out
Let us now consider output state stemming from the input state j1,ai1j0, ai2 entering the beam splitter with a matrix (A2b). Direct calculations give the following b Bð b D b 1 ða1 Þ D b 2 ða2 Þj10i Þ b Bðj1; b P a1 i1 j0; a2 i2 Þ ¼ P 12 b Bj10i b b 2 ða2 ÞÞð P b BÞ b þP b Bð b D b 1 ða1 Þ D ¼P
12
^þ ^þ ¼ expða 1 ða1 T a2 R Þ þ a2 ða1 R þ a2 T Þ b Bj10i b ^2 ða R þ a TÞÞ P ^1 ða T a RÞ a a 1
2
1
2
12
b 1 ða1 T a2 R Þ D b 2 ða1 R þ a2 T Þ P bB ba bb þ ^þ ¼D 1 ð P BÞ j00i12 b 1 ða1 T a2 R Þ D b 2 ða1 R þ a2 T ÞðTj10i þ Rj01i Þ ¼D 12
12
¼ Tj1; a1 T a2 R i1 j0; a1 R þ a2 T i þ Rj0; a1 T a2 R i1 j1; a1 R þ a2 T i:
ðA3aÞ
All the same is applicable to another ca b Bðj0; b P a1 i1 j1; a2 i2 Þ ¼ R j1; a1 T a2 R i1 j0; a1 R þ a2 T i þ T j0; a1 T a2 R i1 j1; a1 R þ a2 T i:
ðA3bÞ
In particular, applying the Hadamard transformation, we have pffiffiffi pffiffiffi 1 b Hðj1; ai1 j0; ai2 Þ ¼ pffiffiffi ðj1; 2ai1 j0i2 þ j0; 2ai1 j1i2 Þ; 2
ðA4aÞ
pffiffiffi pffiffiffi 1 b Hðj0; ai1 j1; ai2 Þ ¼ pffiffiffi ðj1; 2ai1 j0i2 j0; 2ai1 j1i2 Þ: 2
ðA4bÞ
Let us consider appartial case combining a single photon with a ffiffiffi coherent state j0; 2ai, then we have
References [1] E. Schrödinger, Naturwissenschaften 23 (1935) 807; E. Schrödinger, Naturwissenschaften 23 (1935) 823; E. Schrödinger, Naturwissenschaften 23 (1935) 844. [2] M.D. Reid, Phys. Rev. Lett. 84 (2000) 2765; M.D. Reid, Phys. Rev. A 62 (2000) 022110. [3] B. Yurke, D. Stoler, Phys. Rev. Lett. 57 (1986) 13. [4] B.C. Sanders, G.J. Milburn, Phys. Rev. A 45 (1992) 1919; C.C. Gerry, Phys. Rev. A 59 (1999) 4095. [5] D. Dakna, T. Anhut, T. Opatrny, L. Knöll, D.-G. Welsch, Phys. Rev. A 55 (1997) 3184. [6] M. Dakna, L. Knöll, D.-G. Welsch, Opt. Commun. 145 (1998) 309. [7] M. Dakna, J. Clausen, L. Knöll, D.-G. Welsch, Phys. Rev. A 59 (1999) 1658. [8] J. Furasek, R. Garsia-Patron, N.J. Cerf, Phys. Rev. A 72 (2005) 033822. [9] H. Jeong, A.P. Lund, T.C. Ralph, Phys. Rev. A 72 (2005) 013801. [10] M. Boiteux, A. Levelut, J. Phys. A 6 (1973) 589. [11] S.M. Roy, V. Singh, Phys. Rev. D 25 (1982) 3413. [12] F.A.M. de Oliveira, M.S. Kim, P.L. Knight, V. Buzek, Phys. Rev. A 41 (1990) 2645; M.G.A. Paris, Phys. Lett. A 217 (1996) 78. [13] R.A. Campos, B.E.A. Saleh, M.C. Teich, Phys. Rev. A 40 (1989) 1371. [14] K.J. Resch, J.S. Lundeen, A.M. Steinberg, Phys. Rev. Lett. 88 (2002) 113601. [15] S.A. Babichev, B. Brezger, A.I. Lvovsky, Phys. Rev. Lett. 92 (2004) 047903. [16] D.T. Pegg, L.S. Phillips, S.M. Barnett, Phys. Rev. Lett. 81 (1998) 1604; M. Koniorczyk, Z. Kurucz, A. Gabris, J. Janszky, Phys. Rev. A 62 (2000) 013802. [17] D.F. Walls, G.J. Milburn, Quantum Optics, Springer-Verlag, Berlin, Heidelberg, 1994. [18] G.S. Agrawal, K. Tara, Phys. Rev. A 43 (1991) 492. [19] A. Zavatta, S. Viciani, M. Bellini, Science 306 (2004) 660. [20] S.A. Podoshvedov, J. Kim, Phys. Rev. A 74 (2006) 033810. [21] S.A. Podoshvedov, J. Kim, Phys. Rev. A 75 (2007) 032346. [22] S. Takeuchi, Y. Yamamoto, H.H. Hogue, Appl. Phys. Lett. 74 (1999) 1063. [23] H. Jeong, M.S. Kim, J. Lee, Phys. Rev. A 64 (2001) 052308. [24] P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, V.V. Sergienko, Y.H. Shih, Phys. Rev. Lett. 75 (1995) 4337. [25] D. Bouwmeester, J. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390 (1997) 575. [26] K. Mattle, H. Weinfurter, P.G. Kwiat, A. Zeilinger, Phys. Rev. Lett. 76 (1996) 4656.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Generation of a displaced qubit and entangled displaced photon state via conditional measurement and their properties Sergey A. Podoshvedov a,*, Jaewan Kim a, Juhui Lee b a b
Southern Ural State University, Nonlinear Optics Laboratory, Lenin Avenue 76, 454080 Chelyabinsk, Russian Federation Department of Physics, Sookmyung Women’s University, Seoul, South Korea
a r t i c l e
i n f o
Article history: Received 1 December 2007 Received in revised form 21 March 2008 Accepted 22 March 2008
PACS: 42.50.Dv 03.65.Bz
a b s t r a c t We study optical schemes for generating both a displaced photon and a displaced qubit via conditional measurement. Combining one mode prepared in different microscopic states (one-mode qubit, single photon, vacuum state) and another mode in macroscopic states (coherent state, single photon added coherent state), a conditional state in the other output mode exhibits properties of a superposition of the displaced vacuum and a single photon. We propose to use the displaced qubit and entangled states composed of the displaced photon as components for quantum information processing. Basic states of such a qubit are distinguishable from each other with high fidelity. We show that the qubit reveals both microscopic and macroscopic properties. Entangled displaced states with a coherent phase as an additional degree of freedom are introduced. We show that additional degree of freedom enables to implement complete Bell state measurement of the entangled displaced photon states. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Schrödinger asked what happens when an atom is in an entangled state with an alive and dead cat [1]. Now, this puzzle of quantum mechanics is resolved by realizing that such an object is very sensitive to decoherence and so is unlikely to occur in real life. But nevertheless, it poses the question of whether a classical object on a macroscopic level may be in a state of quantum superposition. If it is possible, then, according to the basic-theoretical principles of quantum mechanics, the superposition of macroscopically distinguishable states (the component states composing such a superposition must give classically distinct measurement outcomes [2]) can give a rise to quantum interference since the superposition is highly nonclassical. This can serve as evident manifestation of the existence of the superposition of classical objects. A typical example of such a superposition, called the ‘‘Schrödinger cat state,” is the superposition of two optical coherent states with sufficiently large amplitudes but with opposite phases [3]. The superposition exhibits typical interference features. Measuring the quadrature-component distribution of a statistical mixture of coherent states with opposite phases, we observe two peaks. In the case of a coherent superposition, the two peaks change their mutual distance depending on the phase difference of the compo-
* Corresponding author. Address: School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, South Korea. Tel.: +82 2 958 3863; fax: +82 2 958 3820. E-mail addresses: [email protected], [email protected] (S.A. Podoshvedov). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.073
nents until they eventually overlap. Thus, the difference between a statistical mixture and a coherent superposition is most evident. Generation of a Schrödinger cat superposition in a realistic experiment is not trivial. A main method to generate such a superposition is the nonlinear interaction of an optical field in a coherent state with a Kerr medium [3]. But for all practically available values of the Kerr nonlinearities, the time needed to generate the Schrödinger cat superposition, is much longer than the decoherence time. Decoherence and phase fluctuations may destroy any inceptive superposition during the propagation of the light inside the medium [4]. Another alternative method to generate a superposition of macroscopically distinguishable states is based upon conditional measurement [5–7]. The Schrödinger-cat-like state so called, by virtue of its definite resemblance with a Schrödinger cat superposition, is conditionally obtained if we launch a squeezed vacuum into a beam splitter and count the photons in one of the output channels [5]. It is also worth noting a number of papers closely related to our consideration [8,9]. In the paper, we want to propose alternative to the Schrödinger cat state, namely, displaced qubit and entangled displaced photon states. The authors in [7] showed that an arbitrary single-mode state can be created by starting from the vacuum and applying a sequence of displacements and single-photon additions. Following the similar method, we use similar treatment to generate both a pure displaced photon state [10–12] and displaced qubit. We show that such a displaced qubit possesses both microscopic and macroscopic properties. The components of the qubit give classically
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distinct measurement outcomes and the itself shows interference properties. Extending our analysis we introduce entangled displaced state composed of the displaced qubits and describe a method to perform for them a complete Bell state measurement. 2. Generation of the displaced qubit It is well-known that the input–output relations at a lossless beam splitter can be characterized by the SU(2) Lie algebra [13] (all needed mathematical formulas for our consideration are presented in Appendix A). Assume that the modes of the beam pffiffiffi splitter pffiffiffi are prepared in a coherent state with amplitude 2aðj 2ai1 Þ (mode 1) and in a state of a single-mode qubit consisting of the vacuum and a single-photon pffiffiffi ! 2 jaj jsi2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j0i j1i : ð1Þ a 2 þ jaj2 2 The experimental generation of an arbitrary superposition of the vacuum and single photon states has been accomplished using parametric down conversion with the input signal mode prepared in a coherent state [14], or conditioning on homodyne measurements on one part of a nonlocal single photon in two spatial modes [15]. Theoretical study of employing the quantum scissors scheme is presented in [16]. If we combine the input modes on a device with Hadamard input–output relations (A3c), the result is given by 0 1 pffiffiffi ! pffiffiffi jaj 2 B C b @j 2ai qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j0i j1i A H 1 a 2 2 þ jaj 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jaj ð2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1; ai1 jai2 þ 1 þ jaj2 jai1 ja; 1i2 ; a 2 þ jaj2 where 0 b j1; ai ¼ DðaÞj1i ¼ a @jai
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ jaj2 a
1 ja; 1iA
ð3Þ
b ^Þ is the uni^þ a a is the displaced photon [10–12] ( DðaÞ ¼ expðaa tary displacement operator [17]), and the one-photon added coherent state is given by [18,19] 1 exp jaj2 =2 X ^þ jai a a1 l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi jli ja; 1i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð4Þ þ ^ ^ 2 hajaa jai l! l¼1 a 1 þ jaj Restrict our attention to the events when no photons are recorded in the second output channel (Eq. (2)). Then, the first output channel of the beam splitter is prepared in the state j1,ai since the onephoton added coherent state does not contain a vacuum state unlike the coherent one. We now mention our notation. One should not mistake the displaced single photon j1,ai state for the single photon added coherent state ja,1i. There is a difference between them since neither the annihilation nor creation operators commute with the displaceb ^; DðaÞ ment operator ð½a 6¼ 0; where a 6¼ 0Þ. In the limiting case a ? 0, the states j1,ai and ja,1i simply reduce to one-photon j1i. Coherent and one-photon added coherent states are not orthogoqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nal to each other ðhaja; 1i ¼ a = 1 þ jaj2 Þ and their scalar product approaches 1 if a ? 1. This means that single photon added coherent states become scarcely distinguishable from coherent states with growing amplitude (a ? 1). On the contrary, the state j1,ai is always orthogonal to the coherent state with the same amplitude a,regardless of the value of a(haj1,ai = 0). Therefore, we can interpret the coherent sate and displaced single photon states as
basic logic states (logical zero and logical one, respectively) in the framework of quantum information. We are going to denote a coherent state by j0,ai jai by analogy with the j1,ai state. The number state is determined by its photon number while the phase of the number state is completely random. By displacing in phase space, the field amplitude is added to the number state. The displaced qubit jWi ¼ Aj0; ai þ Bj1; ai;
ð5Þ 2
2
where A and B are arbitrary amplitudes (jAj + jBj = 1), can also be generated by a photon number conditional measurement. Consider a singe-mode qubit with arbitrary amplitudes, jsi = aj0i + bj1i. Then, superimposing an ancilla coherent state with jsi, one obtains pffiffiffi 0 b Hðj 2ai1 ðaj0i þ bj1iÞ2 Þ ¼ ða0 j0; ai þ b j1; aiÞ1 j0; ai2 þ cj0; ai1 ja; 1i2 ; ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 pffiffiffi 2 where a ¼ a þ ba = 2, b ¼ b= 2, and c ¼ b 1 þ jaj =2. Using the definition of the coherent and one-photon added coherent states and detecting l photons in the second output mode, the state (6) is conditionally transformed to state (5), where A ¼ AðlÞ ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 ða0 lb=a 2Þ= ja0 lb=a 2j2 þ jb j2 is the amplitude of the displaced vacuum, j0,ai, dependent on l, and where B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 b= 2ðja0 lb=a 2j2 þ jb j2 Þ is the amplitude of the j1,ai state independent of l. The number of detected photons influences only the amplitude A(l), leaving other parameter B unaffected.pffiffiffi The success 0 probability of such an event is expðjaj2 Þjaj2l ðja0 lb=a 2j2 þ jb j2 Þ=l! Consider another case of preparation of the state (5). One of the modes is prepared in an ancilla coherent state and the other in a single photon state. Then, the result of the mixing is given by [20,21] 0
pffiffiffi 1 b Hðj 2ai1 j1i2 Þ ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: 2
ð7Þ
From this equation we have that when the number of photons detected in the second mode is l, the mode in the other output channel is prepared in the quantum state in Eq. (6) where A ¼ AðlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a= jaj2 þ jbj2 , B ¼ b= jaj2 þ jbj2 , a = a* (1 l/jaj2), and b = 1. In particular, when the number of detected photons, l, is zero, then the output is a single-photon added coherent state, whose properties were described in detail in Ref. [6]. The success probability of such an event is exp(jaj2) jaj2l (jaj2 + jbj2)/2l! Another method of preparation of state (5) is the following. Let us prepare one of the modes of a beam splitter in a single photon added coherent state, another mode of the same beam splitter in the vacuum state. Then, combining the modes, one obtains [20,21] pffiffiffi b Hðj 2a; 1i1 j0i2 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ jaj2 ðja; 1i1 jai2 þ jai1 ja; 1i2 Þ: 2ð1 þ 2jaj2 Þ
ð8Þ
When the photon number of the mode in the second output channel is measured and l photons are detected, then the mode in the first output channel is prepared in the quantum state (5) with A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðlÞ ¼ a= jaj2 þ jbj2 , B ¼ b= jaj2 þ jbj2 , a = a* + l/a, and b = 1 with success probability exp(jaj2)jaj2l(jaj2 + jbj2)/2(1 + 2jaj2)l! For the number of detected photons, l = 0, the conditional state becomes the single photon added coherent state. Thus, the conditional generation of the displaced qubit (5) needs highly efficient and precise photocounting. Unfortunately, there are no highly efficient photodetectors available which precisely distinguish between m and m + 1 photons, where m > 2. Only recently, Takeuchi et al. [22] developed an avalanches photodetector which can discern 0, 1, and 2 photons with high efficiency. Nevertheless, we can ease requirements imposed on the efficiency of photocounting in conditional preparation of the displaced qubit
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(5). Consider one example, we combine a single photon added coherent state with amplitude a1 in one mode with an ancilla coherent state with amplitude a2 in another mode of a beam splitter with arbitrary parameters T and R. The result of the mixing following from formulas from Appendix A is given by 1 b ðja1 ;1i ja2 i Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 2 1þja1 j2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 B T 1þja1 T a2 R j ja1 T a2 R ;1i1 ja1 Rþa2 T i2 þ C @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: þR 1þja1 Rþa2 T j2 ja1 T a2 R i1 ja1 Rþa2 T ;1i2 ð9Þ *
Suppose the condition a1R + a2T 1 holds for the input parameters. Then, the states ja1R + a2T*i and ja1R + a2T*,1i may be approxiðiÞ ðiÞ ðiÞ mated by the finite superposition c0 j0i þ c1 j1i þ c2 j2i, where the ðiÞ0 cj s are the amplitudes (i = 0,1 and j = 0 2) of the coherent and single-photon added coherent states, respectively, in the case that a1R + a2T* 1 when we neglect higher order photon number states. When the state j0i2 is detected in the second output mode, this detection event projects the first mode into a single photon added coherent state with amplitude a1T a2R*. The state like Eq. (5) is conditionally generated if either the state j1i or j2i is registered in the second mode [22]. It is worth noting the used truncation of the state in measured mode is possible owing to choice of small value of the parameter s = a1R + a2T* 1. Direct calculation shows that the influence of the truncated terms on the fidelity of the conditionally generated state is negligible. So, if we register two-photon state [22] the fidelity of the conditionally prepared superposition state can be estimated as 1 s. 3. Properties of the displaced qubit Let us analyze the readout of the qubit in Eq. (5) in the framework of quantum information processing. Consider a partial case. Somebody gives us either j0,ai or j1,ai, which exactly we do not know. Our task is to determine what state we obtained. Use a simple measurement scheme with a balanced beam splitter with the
Q¼
Distinguishability of the states j1,ai1 and j0,ai1, like many ideas in quantum computation and quantum information, is most easily understood using the metaphor of a game involving a coin that is tossed up. We can always identify whether coin has landed on heads or tails and these probabilities, at least in the ideal case, are equal. Imagine we are concerned with some fixed set of outcomes, for example, if the coin only lands on heads. The success probability of such a set is 1/2n, where n is the number of tosses. This is in direct analogy with the identification of the states j1,ai1 and j0,ai1. We can improve the method to distinguish the states j0,ai1 and j1,ai1 from each other. Prepare a second input channel in an ancilla coherent state with amplitude a1, j0,a1i2 so that condition b Bðj0; b aR + a1T* = 0 is performed. Then, we have P ai1 j0; a1 i2 Þ ¼ j0; a=T i1 j0i2 and b aRþa T ¼0 ðj1; ai j0; a1 i Þ ¼ Tj1; a=T i j0i þ Rj0; a=T i j1i : U 1 2 1 2 1 2 1
Again, if an ideal detector in the second output mode clicks, it means it was the j1,ai1 state. If the detector does not click we can tell nothing definite about the state we obtained. But we can decrease the influence of the T—1,a/T*i1—0i2 term in Eq. (11) by choosing a beam splitter transmittance T as small as possible (jTj 1) satisfying limT !1 a1 T ¼ a. If we take the transmittance T very small, then it is sufficient to manage one trial. Then, the failure probability to mistake the j0,ai1 state for the j1,ai1 state is jTj2 1. The same consideration is applicable to the state (5). Again if a single photon hits a photodetector in the second mode, we definitely know that the j1,ai1 state was measured while if we do not register something we know it was the j0,ai1 state with fidelity F = jAj2/ (jAj2 + jBj2jTj2) 1 jBj2jTj2/jAj2. The fidelity approaches 1 if jTj ? 0, that confirms the possibility of almost definitely distinguishing the states j0,ai1 and j1,ai1 from each other. Thus, the components of the displaced qubit give different measurement outcomes. Let us show other nonclassical properties of the displaced qubit (5). A measure of the deviation of the photon-number distribution from a Poissonian is the Mandel Q parameter which for the displaced photon is given by
jaj2 jBj4 þ jBj2 ð1 þ jaj2 Þ2 2jAj jBjðjBj2 jAj2 Þjaj cosðd hÞ 4jAj2 jBj2 jaj2 cos2 ðd hÞ jaj2 þ jBj2 þ 2jAj jBj jaj cosðd hÞ
unitary Hadamard matrix (A3c). The input state jWi is superimposed with p the ffiffiffi ancilla state j0,ai2. Mixing of the states j0,ai1 and j0,ai2 gives 2ai1 j0i2 , while the same mixing of the states j1,ai1 and j0,ai2 gives a balanced superposition (see Appendix A) pffiffiffi pffiffiffi b H ðj1; ai j0; ai Þ ¼ p1ffiffiffi ðj1; 2ai j0i þ j0; 2ai j1i Þ: U 1 2 1 2 1 2 2
ð11Þ
ð12Þ
:
We display the Q parameter as a function of jAj for different absolute magnitudes of a in Fig. 1. For small values of jaj the statistics of
Q ð10Þ
If the detector registers a single photon in the second mode, which happens with probability 1/2, we definitely know that it was the j1,ai state. If the detector registers nothing then we cannot identify the post-selected outcome. We can repeat again the same measurement procedure by superimposing outcome and another ancilla pffiffiffi the p ffiffiffi coherent state with amplitude 2aðj0; 2ai2 Þ. Again if a single photon hits a photodetector in the second output mode, we definitely know that the state we obtained was j1,ai. If we do not register any photon impacts we do not know what state was measured. By applying a sequence of the same measurements many times, we can conclude that if we obtain a series n of vacuum measurements, then initially it was the j0,ai state with probability to mistake the j0,ai state for the j1,ai state equal to 1/2n, where n is the number of repetitions.
2.5 | α |=2
2 | α |=1
1.5 1 0.5
| α |=0.5
|A| 0.2
0.4
0.6
0.8
1
Fig. 1. Q parameter as a function of the absolute amplitude jAj for different values of amplitude of the displaced qubit a = 0.5, a = 1, and a = 2. The values of a are assumed to be real numbers.
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a balanced superposition of displaced vacuum and single photon states becomes sub-Poissonian (Q < 1). The variance of the quadrature components for the state (5) is given by ðD^ xÞ2 ¼
1 ð1 þ 2ð1 jAj2 ÞjBj2 2jAj2 jBj2 cosð2ðd þ hÞÞÞ; 4
ð13Þ
2
where h is the phase of the local oscillator and d is the relative phase of A and B. Fig. 2 shows the result of numerical calculation of Eq. (13) for different input parameters. Variances of the displaced single photon state j1,ai prevail over the ones for the coherent state [12]. Nevertheless, the variance of the displaced photon can take values less than the variance of coherent state for some values of the input parameters. The input parameters can be found from Fig. 2. The Wigner function of the balanced displaced qubit can be evaluated as 4 ððx xa Þðx xa cos Q a Þ þ ðy ya Þðy ya sin Q a ÞÞ p expð2ððx xa Þ2 þ ðy ya Þ2 ÞÞ; ð14aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cos Q a ¼ xa = jxa j2 þ jya j2 and sin Q a ¼ ya = jxa j2 þ jya j2 . The Wigner function of the statistical mixture described by the density matrix q = (j0,aih0, aj + j1,aih1, aj)/2 is
(Δx)
W jWB i ðx; yÞ ¼
0.5 3
0.4 0.3
2
0 1
φ
1
δ
W q ðx; yÞ ¼
2 3
0
pffiffiffi Fig. 2. Plot of variance versus h and d for the displaced qubit (5) with A ¼ B ¼ 1= 2. There are ranges where variance of the state (5) is less of one of the coherent state (squeezing).
a
4 ððx xa Þ2 þ ðy ya Þ2 Þ expð2ððx xa Þ2 þ ðy ya Þ2 ÞÞ: p ð14bÞ
Fig. 3a–d reveals the Wigner functions of a balanced superposition and statistical mixture, respectively. The Wigner function of a balanced superposition exhibits a single peak unlike the Wigner function of the statistical mixture with two peaks. The difference in the
c
0.6
1
2
0.5
0 2
-0.2 0
0 -1 0
3
0.2
1
0
0.4
1
1
-1 1
2
2 -2
3
b
0
d
0.8 0.6 0.4 0.2 0
2 1 0
-1
0.2 0.15 0.1 0.05 0 0
3 2
1
1
0
-1 2
1 2
-2
30
pffiffiffi Fig. 3. Wigner function of the balanced displaced photon jWi ¼ ðj0; ai expðiua Þj1; aiÞ= 2 (a and c) and statistical mixture described with density operator q = (j0,aih0, aj + j1,ai h1, aj)/2 (b and d, respectively) for a = 0.5, Qa = p/4 (a) and (b) and a = 2, Qa = p/4 (c) and (d).
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Φ ( x)
may become a displacement operator for the components of the entangled state jW(a,a)i12. Another possible state with the same parameter a is given by
2
a
0.8
1 jWþ ða; aÞi12 ¼ pffiffiffi ðj1; ai1 j0; ai2 þ j0; ai1 j1; ai2 Þ: 2
1
0.6
ð16Þ
If we change a parameter a on opposite a, for example, in first mode in used notations we obtain another pair of the entangled displaced states
0.4
2 0.2
x 1
Φ ( x)
2
3
4
2
b
0.8
1 0.6
0.4
2
x 1
2
3
ð17Þ
Each pair of the entangled displaced states jW±(a,a)i12 and jW±(a,a)i12 is orthogonal to each other. But the entangled states with opposite phases are not orthogonal to each other hWþ ; a; ajWþ ; a; ai ¼ ð1 2jaj2 Þ expð2jaj2 Þ;
ð18aÞ
hWþ ; a; ajW ; a; ai ¼ 2jaj2 expð2jaj2 Þ:
ð18bÞ
Nevertheless, the scalar products go fast to zero when a grows since theirs overlaps decrease exponentially with a. For example, when a is as small as 3, the overlap is 107. Thus, the states (18a) and (18b) are almost orthogonal, especially with increase of a. Unlike traditional Bell states, the entangled displaced states are distinguishable from each other due to additional degree of freedom a phase of the coherent state. It can be seen if we combine the states jW±(±a, a)i12 on a beam splitter with a matrix (A3c)
0.2
- 1
1 jW ða; aÞi12 ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: 2
4
Fig. 4. Quadrature distribution jU(x)j2 of the balanced displaced photon and statistical mixture for a = 2 and Qa = 0 (a) and Qa = p/4 (b). The curve 1 corresponds to the superposition state (5) while the curve 2 corresponds to the statistical mixture. Two separated peaks of the statistical mixture transform to one peak of pure superposition.
forms is caused by the interference terms due to correlations inherent to the superposition state (5). The interference terms are proportional to cosQa and sinQa, respectively, and destroy one of the peaks as illustrated in Fig. 3a and c. The marginal distribution for one of the two components, for example x, is obtained by integration over the other variable y rffiffiffi 2 ððx xa Þðx xa cos Q a Þ þ 0:25Þ expð2ðx xa Þ2 Þ: p ð15Þ
jUðxÞj2 ¼ 2
The corresponding dependencies of the quadrature distributions jU(x)j2 are plotted on Fig. 4a and b for the both displaced qubit and the statistical mixture. If we take Qa = p/2, then there is no difference between jU(x)j2 for a pure superposition and statistical mixture. 4. A complete Bell state measurement of the entangled displaced states A problem of generation of the displaced qubit is turned out not to be trivial as it was described above. Although, it may look easy if we already know how to produce a superposition of Fock states j0i and j1i, just applying the displacement operator ðjWi ¼ Aj0; aiþ b Bj1; ai ¼ DðaÞðAj0i þ Bj1iÞÞ. But, there is not pertinent model for such a displacement operator. But nevertheless, such a model of the displacement operator can be easy done if we are interested in generation of entangled state composed of two displaced qubits (5). Indeed, as can be seen from the formula (A6), a beam splitter
b HðjW
ð a; aÞi12 Þ ¼ j1;
pffiffiffi 2ai1 j0i2 if the input state was jWþ ða; aÞi; ð19aÞ
j0;
pffiffiffi 2ai1 j1i2 if the input state was jW ða; aÞi;
ð19bÞ
pffiffiffi j1i1 j0; 2ai2 if the input state was jWþ ða; aÞi;
ð19cÞ
pffiffiffi j0i1 j1; 2ai2 if the input state was jW ða; aÞi:
ð19dÞ
Next step is to discern photon number states —0i and j1i from their displaced analogues j0,ai and j1,ai, respectively. It can be evidently done from intuitive point of view taking into account, for example, the overlap of the coherent and vacuum states (jh0j0,aij2 = exp(jaj2)) fast approaches zero when a grows, so thus, they become almost orthogonal with larger a. In practice, off-the-shelf photon counters can only differentiate between zero and more photons because we insert additional beam splitters with detectors placed behind them b^ 0H b 0H b 12 ðjW ð a; aÞi Þ ¼ p1ffiffiffi ðj1; ai j0; ai 0 þ j0; ai j1; ai 0 Þj0i j0i 0 H 12 12 1 1 1 1 2 2 11 2 if the input state was jWþ ða; aÞi; ð20aÞ 1 pffiffiffi j0; ai1 j0; ai10 ðj1i2 j0i20 þ j0i2 j1i20 Þ 2 if the input state was jW ða; aÞi;
ð20bÞ
1 pffiffiffi ðj1i1 j0i10 þ j0i1 j1i10 Þj0; ai2 j0; ai20 2 if the input state was jWþ ða; aÞi;
ð20cÞ
1 pffiffiffi j0i1 j0i10 ðj1; ai2 j0; ai20 þ j0; ai2 j1; ai20 Þ 2 if the input state wasjW ða; aÞi;
ð20dÞ
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b ij means the Hadamard where 10 and 20 are the ancilla modes and H operation applied to the modes i and j, respectively. Analyzing given formulas, we can conclude the following: (i) If we register the output jnm00i110 220 with n 6¼ 0 and m 6¼ 0 then we definitely know it was a state jWþ ða; aÞi12 . (ii) If we register in sum three clicks ðjnm10i110 220 or jnm01i110 220 , where n 6¼ 0 and m 6¼ 0), then we certainly identifies such a state as jW(a,a)i12. (iii) If we register three clicks ðj10nmi110 220 or j01nmi110 220 , where n 6¼ 0 and m 6¼ 0), then we exactly know that it was jW+(a,a)i12 state. (iv) If only detectors in modes 2 and 20 register coming photons j00nmi110 220 with n 6¼ 0 and m6¼ 0 then we make sure that it was a state jW ða; aÞi12 . All these events are most probable ones in the case of large amplitude a. There are also cases when either n = 0 (m = 0) or both of them n = m = 0. Probability of the events is negligibly small for the large value of a since it drops exponentially with a growing. Thus, we conclude it is possible to distinguish two pairs jW±(a,a)i12 and jW±(a, a)i12 from each other. The success probability becomes exactly equal one if a ? 1. In practice, it is sufficient to deal with an input quantum channel with sufficiently large amplitude a. Failed events not distinguished by the method must be rejected. Assume that somebody has in his possession both qubits, for example, the jW+(a1, a2)i12 state. A question that can be raised whether someone is able to transform the state to the rest three states by local operations. It can be easily done in the case of transformations jW+(a1, a2)i M jW(a1,a2)i and jW(a1, a2)i M b jW+(a1,a2)i by phase shifting unitary transformation on pð PðpÞÞ b applied to one of the modes. Indeed, PðpÞj0; ai ¼ j0; ai and b b b b DðaÞj0i b b DðaÞ b P b þ ðpÞ PðpÞj1i PðpÞj1; ai ¼ PðpÞj1; ai ¼ PðpÞ ¼ PðpÞ b ¼ DðaÞj1i ¼ j1; ai:
ð21Þ
Another type of transformations jW+(a1,a2)i M jW(a1,a2)i and jW+(a1,a2)i M jW+(a1,a2)i is not trivial and requires ancilla coherent states. Combine a mode 1 with mode 3 occupied by the coherent state j0,a3i3 and mode 2 with mode 4 in the coherent state j0,a4i4 on two corresponding beam splitters with matrix (A3a). Then, we have b 34 B b 13 B b 24 ðjWþ ða1 ; a2 Þi j0; a3 i j0; a4 i Þ ¼ cos Q jWþ ða1 ; a2 Þi H 12 3 4 12 pffiffiffi pffiffiffi j0; ða3 a4 Þ= 2i3 j0; ða3 þ a4 Þ= 2i4 i sin Q j0; a1 i1 j0; a2 i2 pffiffiffi pffiffiffi j1; ða3 a4 Þ= 2i3 j0; ða3 þ a4 Þ= 2i4 ; ð22Þ provided that a3 = ia1(1 + cosQ)/sinQ and a4 = ia2(1 cosQ)/sinQ. Use a beam splitter with large transmittance Q ffi 0. Next, one makes a measurement in mode 3 no longer concerning the actual value of measurement that provides us a mixture of states jW+(a1,a2)i12 and j0,a1i1j0,a2i2 with probabilities cos2Q and sin2Q, respectively. As Q ffi 0, cos2Q ffi 1 sin2Q, the contribution of the state jW+(a1, a2)i12 prevails over one of another state. Thus, it is possible to perform all unitary transformations to obtain all four entangled states jW±(a,a)i12 and jW±(a,a)i12 with large success probability. Let us also mention the paper [23] where a scheme to discern four Bell states using linear optics when the Bell states are prepared with coherent states with coherent states of two different amplitudes unlike our work. 5. Discussion We have analyzed a problem of conditional generation of displaced qubit and entangled displaced states. The motivation to deal
with such states comes from the fact that they carry phase of coherent ‘‘shell” as additional degree of freedom that may become key moment in quantum information. Quantum information offers a number of striking applications. But to reach all advantages of these applications one needs to prepare and measure the Bell states. Problem of creating the Bell state is solved via nonlinear interaction of light in parametric down-conversion (polarization entanglement) [24]. The rest three Bell states can be prepared from maximally entangled state by simple local unitary transformations. The problem of performance a complete Bell measurement with linear devices (like beam splitters and phase shifters) encounters serious difficulties. Combination of controlled NOT operation (CNOT) on two interacting systems and Hadamard operation on one of the system allows for one to transform the four Bell states into four disentangled basis states. But the case is not trivial in practice. The Bell state measurement is inherently nonlinear in the optical case and only partial set of Bell measurement can be made in practice with linear optical elements [25,26]. A phase of the displaced photon as an additional degree of freedom gives a possibility to perform complete Bell state measurement of the entangled displaced states. We have shown that both displaced qubit and entangled displaced states can be generated provided that we have corresponding resource of the single-mode qubit, photon added coherent state and a simple photon, respectively. We showed that the components of such a displaced state can be distinguished from each other with high fidelity. The displaced qubit reveals the properties of a mesoscopic state that has both microscopic (interference and superposition) and macroscopic (classically distinguishable) properties. Displacement of the microscopic state is performed via conditional measurement for the displaced qubit and via use of beam splitter for entangled displaced state. Such s mechanism is different from that generating a Schrödinger cat state based on a nonlinear phase shift. Acknowledgement This work was supported by the IT R&D program of MIC/IITA (2005-Y-001-04, Development of next generation security technology). Appendix A. Beam splitter transformations with displaced states Let us to present some useful formulas used during the text and related to the beam splitter transformations [13] of the displaced b ¼a ^þ ^ ^þ ^ states. An evolution operator, where X 1 a2 þ a2 a1 , Q is the some parameter b Þ ¼ expðiQ XÞ; b BðQ
ðA1Þ
defines the beam splitter transformation " # " # " þ # ^þ ^þ ^1 a a a cos Q i sin Q 1 1 ¼B þ ¼ : þ ^þ ^ ^ a a a i sin Q cos Q 2 2 2 out
in
ðA2aÞ
in
A very simple way to describe the action of a beam splitter is to fix the phase relations such that in general case the beam splitter transformation is described with the following input-output relations " # " # " þ # ^þ ^1 ^þ T R a a a 1 1 ¼ PB þ ¼ ; ðA2bÞ þ ^þ ^ ^ a a a T R 2 2 2 out
in
in
where T and R are the transmittance and reflectivity of the beam splitter, respectively. The corresponding evolution operator is b 2 ðuÞ BðQ b Þ, where the phase shifting evolution operator bB b¼P b 1 ðuÞ P P
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b ¼ b ^Þ. We can apply two phase shifters H ^þ a is PðuÞ ¼ expðiua b b b P 2 ðp=2Þ Bðp=4Þ P 2 ðp=2Þ to second mode to deal with Hadamard transformation on optical modes " # ^þ þ þ
a 1 1 1 þ þ
1 ^ ^2 in : ^ ^1 a p ffiffiffi ¼ H ¼ a a a ðA2cÞ 1 2 in ^þ a 2 1 1 2
b Hðj0;
pffiffiffi 2ai1 j1i2 Þ ¼ jW ða; aÞi12
1 ¼ pffiffiffi ðj1; ai1 j0; ai2 j0; ai1 j1; ai2 Þ: ðA5Þ 2 It is worth mentioning the generation of a single photon can be conditionally accomplished using parametric down conversion.
out
Let us now consider output state stemming from the input state j1,ai1j0, ai2 entering the beam splitter with a matrix (A2b). Direct calculations give the following b Bð b D b 1 ða1 Þ D b 2 ða2 Þj10i Þ b Bðj1; b P a1 i1 j0; a2 i2 Þ ¼ P 12 b Bj10i b b 2 ða2 ÞÞð P b BÞ b þP b Bð b D b 1 ða1 Þ D ¼P
12
^þ ^þ ¼ expða 1 ða1 T a2 R Þ þ a2 ða1 R þ a2 T Þ b Bj10i b ^2 ða R þ a TÞÞ P ^1 ða T a RÞ a a 1
2
1
2
12
b 1 ða1 T a2 R Þ D b 2 ða1 R þ a2 T Þ P bB ba bb þ ^þ ¼D 1 ð P BÞ j00i12 b 1 ða1 T a2 R Þ D b 2 ða1 R þ a2 T ÞðTj10i þ Rj01i Þ ¼D 12
12
¼ Tj1; a1 T a2 R i1 j0; a1 R þ a2 T i þ Rj0; a1 T a2 R i1 j1; a1 R þ a2 T i:
ðA3aÞ
All the same is applicable to another ca b Bðj0; b P a1 i1 j1; a2 i2 Þ ¼ R j1; a1 T a2 R i1 j0; a1 R þ a2 T i þ T j0; a1 T a2 R i1 j1; a1 R þ a2 T i:
ðA3bÞ
In particular, applying the Hadamard transformation, we have pffiffiffi pffiffiffi 1 b Hðj1; ai1 j0; ai2 Þ ¼ pffiffiffi ðj1; 2ai1 j0i2 þ j0; 2ai1 j1i2 Þ; 2
ðA4aÞ
pffiffiffi pffiffiffi 1 b Hðj0; ai1 j1; ai2 Þ ¼ pffiffiffi ðj1; 2ai1 j0i2 j0; 2ai1 j1i2 Þ: 2
ðA4bÞ
Let us consider appartial case combining a single photon with a ffiffiffi coherent state j0; 2ai, then we have
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