Importance and Applications of Infinite Dimensional Non-Orthogonal Quantum State

Non-orthogonal quantum states in infinite dimensional space are playing a special role in foundation of quantum mechanics. For example, the Gaussian state is a typical state that was considered at beginning of history of quantum theory. The explicit importance of Gaussian quantum states such as coherent state was certified by Glauber [1], Sudarshan et al. [2] in quantum optics for understanding a nature of laser. More progress has been given by Yuen [3] who discovered a special properties of generalized coherent state known as two-photon coherent state (squeezed state), and a method to verify them experimentally, and its applications were opened up by Hirota [4] and Shapiro [5]. Then entanglement of non-orthogonal quantum state such as two-mode squeezed state and quasi-Bell entangled coherent state [6] stimulate another interesting subject in physics.


Introduction
Non-orthogonal quantum states in infinite dimensional space are playing a special role in foundation of quantum mechanics. For example, the Gaussian state is a typical state that was considered at beginning of history of quantum theory. The explicit importance of Gaussian quantum states such as coherent state was certified by Glauber [1], Sudarshan et al. [2] in quantum optics for understanding a nature of laser. More progress has been given by Yuen [3] who discovered a special properties of generalized coherent state known as two-photon coherent state (squeezed state), and a method to verify them experimentally, and its applications were opened up by Hirota [4] and Shapiro [5]. Then entanglement of non-orthogonal quantum state such as two-mode squeezed state and quasi-Bell entangled coherent state [6] stimulate another interesting subject in physics.
On the other hand, many fundamental theorems in quantum information science were developed based on non-orthogonal quantum state. In fact, a problem of discrimination of non-orthogonal quantum states through quantum measurement, that was pioneered by Helstrom [7], is a typical example. To formulate quantum communication theory, several researchers generalized from Bayes criterion to Neyman-Peason , Minimax criteria, and Shannon mutual information which play different roles in each other. Thus, theory of non-orthogonal state is a foundation for quantum information science.
In this paper, I describe a survey of theory of non-orthogonal quantum state in the sections II and III, and discuss a generation method for such states including entangled coherent state and Schrodinger cat state in the section IV.
Then progress of a basic theory for quantum information science such as quantum communication theory is introduced. In the section VI, I discuss potential applications of theoretical achievements on non-orthogonal quantum state to Quantum Methodology such as quantum illumination, quantum reading and quantum imaging. These performances in a real situation may be drastically improved by entangled state based on infinite dimensional state. Furthermore, it will be suggested for future works that quantum imaging has a potential of a real application for a new type of camera by connecting the original theory and Volterra-Wiener theory. In the section VII, the concept of a new physical cipher so called Quantum Enigma Cipher is described, in which the security is ensured by a combination of a mathematical encryption and physical randomization of its ciphertext.

Basis of Quantum Optics Quantum optical field
Glauber unified a formulation on classical and quantum optical field that is described by where the electric field is given by Here uk (r) is mode function. This mode is called Q-mode which corresponds to infinite dimensional Hilbert space H S . Observables in the mode are described by photon annihilation and creation operators as follows:  (4) where whole state vectors are normalized.
In the subsequent sections, I will provide useful applications of the above theories.

Basis of entanglement
Entanglement and its information-theoretic aspects have been studied by many authors [9][10][11][12][13]. Here I give a short survey of the theory of entanglement that I will later apply to quasi-Bell states. For a pure entangled state of a bipartite system | AB Ψ〉 , the measure of entanglement defined as [9,14]: is called "entropy of entanglement". This quantity enjoys two kinds of information-theoretic interpretations. One is that E n () gives the entanglement of formation, which is defined as the asymptotic number k of standard singlet states required to faithfully locally prepare n identical copies of a system in the bipartite state | AB Ψ〉 for very large k and n. The other is that E n () gives the amount of distillable entanglement, which is the asymptotic number of singlets k that can be distilled from n identical copies of | AB Ψ〉 . With either of these definitions of k and n, E n () satisfies For pure states we can rewrite where . The above expression is valid for mixed states of two qubit systems as well [12]. For mixed states of qubits one may also define an expression for the entanglement of formation [9,12,13]. It is defined as the average entanglement of the pure states of a decomposition = of the density operator ρ, minimized over all decompositions [9].
In general, it is difficult to find the exact amount of entanglement of formation except for special cases. However, there is a lower bound which is expressed in terms of a quantity called "fully entangled fraction", which we denote by f (ρ) and is defined as where the maximum is over all completely entangled states | e〉 . A lower bound on the entanglement of formation is [9] Usually in order to construct entangled states one writes superpositions of orthogonal states. For instance the standard Bell basis uses states like | 〉  and |↔〉 , and of course its properties are well known. Our concern here is what kind of properties appear if we have superpositions of non-orthogonal states. In the following, I will clarify properties of entangled states of non-orthogonal states such as coherent states based on the above basic theory.

Entanglement of orthogonal states
Let us assume that two states |+>,|−> are orthogonal each other such as <+|−>=0. One can form the following entangled states so called Bell state.

Entangled state of non-orthogonal states
Let us consider entangled states based on two non-orthogonal states 1 |ψ 〉 and 2 |ψ 〉 such that 1 2 where k is real. We can define a set of 4 entangled states as follows [7]: where {h i } are normalization constants: We call these states "quasi-Bell states". They are not orthogonal each other. In fact, for κ real their Gram matrix where D=2k/(1+k2). If the basic states are orthogonal (k=0), then these states reduce to standard Bell states. Let us discuss the entropy of entanglement for the above states. We first calculate the reduced density operators of the quasi-Bell states. They are The eigenvalues of the above density operators A ρ ) are given in terms of the Gram matrix elements G ij as follows, Hence, the entropy of entanglement is | AB Ψ 〉 are maximally entangled, even though the entangled states consist of non-orthogonal states in each subsystem. So, these states are called H-state: These results are true for arbitrary non-orthogonal states with

Quasi-Bell states based on bosonic coherent states
Let us consider two coherent states of a bosonic mode {| ,| } α α 〉 − 〉 , e.g., let α ± be the coherent amplitude of a light field. Using previous notation, we have Tombesi et al. [15]and Sanders [16] discussed properties of the following entangled state: where | ,| ,| ,| α β γ δ 〉 〉 〉 〉 are coherent states, and h is a normalization constant. This is called entangled coherent state. Here we apply the previous discussion to entangled coherent state. Then one can construct the quasi Bell entangled coherent states as follows [7]: Here we have that On the other hand, are maximally entangled states only in the limit α→∞.
The average photon numbers of the reduced states of the quasi Bell entangled coherent states read This is shown in the following way. The characteristic functions of the quasi Bell entangled coherent states are given by † * They can be calculated, with the result mode frequency ω. An eigenstate of the annihilation operator can alternatively be written as a single mode coherent state by introducing a new mode operator by , and this, with c 1 properly defined, is what we mean by a coherent state | α 〉 from now on. Equivalently, we may rewrite all of this in terms of time dependent functions, operators and modes by introducing Fourier transforms This way one can define independent modes by introducing a complete orthogonal set of functions Let us consider here the ideal version of a scheme to produce a propagating entangled coherent state. Consider an optical Fabry-Perot cavity with two identical mirrors characterized by a decay rate r. The cavity mode is described by a single mode annihilation operator ( ) c t . If that mode is weakly coupled to a single two level atom, and the cavity is driven by two input fields (1) ω , a standard calculation shows that the Fourier-transformed annihilation operator of the cavity mode, c(ω), is Where ω c is the resonance frequency of the cavity, ω A is the resonance frequency of the atom, g is the coupling between atom and cavity mode, and τ is the spontaneous decay rate of the atom. The output fields emanating from the cavity are, according to standard input-output theory, determined by for k=1,2. For an atom far off-resonance with the cavity, or to be more precise, when |ωc −ωA|≫g 2 /r, all light at ω=ω c will be transmitted: On the other hand, if the atomic resonance frequency is chosen such that ω A =ω c+ τ/2, and we are in the strong coupling regime where g 2 ≫rτ, then the cavity resonance is shifted by so much that all light at ω=ω c will be reflected: Now we consider an atom in an equal superposition 1 of two non-degenerate ground states, where there is a resonant coupling from 1 | g 〉 at the cavity frequency, while 2 | g 〉 is off-resonant. Given two coherent input fields | k α〉 for k-1,2 at a frequency ω=ω c , the atom-cavity system will then enact the transformation The characteristic functions are indeed not Gaussian.

Mixtures of quasi Bell entangled coherent states
We can construct a quasi-Werner mixed state based on quasi-Bell states by | AB Ψ 〉 are Bell states, then the above equation gives a standard Werner state [17]. It is known that the fully entangled fraction of the Werner state is F, and the entanglement of formation of the Werner state is given by The fully entangled fraction of the quasi Werner state is analogously given by because the quasi Bell states are orthogonal to each other, except for the pair of states 1 | AB Ψ 〉 and 3 | AB Ψ 〉 , as one can see from the Gram matrix G. However, the quasi Werner state and Werner state are completely different states. In particular, the eigenvalues of quasi Werner states are different from those of Werner states. The eigenvalues of the density operator are given by those of the modified Gram matrix [18]. For the quasi Werner state, the Gram matrix is As a result, we have F, as the eigenvalues of the quasi Werner state.
Thus the lower bound of the entropy of formation of quasi Werner state is the same as that of Werner state.

Application of entangled coherent state
After discovery of the complete entanglement of H-state, van Enk and Hirota opened up applications [19] such as teleportation and quantum repeater. Especially, serial works of van Enk stimulated an investigation of entangled coherent state by Wang [20], and Jeong, et al. [21].

Generation of Propagating Entangled Coherent State
Let us survey a generation of entangled coherent state which was proposed by van Enk et al. [22]. Entangled state in propagating mode is essential for applications which involve the above example. For a propagating mode, the state | α 〉 should be understood as a continuous mode coherent state. In a simple one-dimensional picture one can define creation and annihilation operators † ( ) b ω and b(ω) for each If one subsequently performs a measurement on the atom in the Another method is a generation based on cat state [19], and it was demonstrated experimentally by Grangier's group in 2009 [23]. Since this issue is one of fundamental problems in quantum optics, results by Grangier's group should be appreciated.
On the other hand, superposition of macroscopic coherent state is an important subject before that one discusses on entangled coherent state. A generation method based on photon subtraction was proposed in 1997 [24], and it was experimentally demonstrated by Grangier's group [25]. Furthermore, theoretical and experimental progress have been given by NICT group [26] and Scully's group [27].
Thus, subject of quantum states based on coherent state should be focused for development of quantum optics and quantum information science.

Main Theorems in Quantum Information Science
A role of quantum information science is to verify potential applications of fundamental nature of quantum mechanics. To do so, quantum communication theory was developed. In this section, I will describe the fact that non-orthogonal states play a very important role in such a theory.

Quantum detection theory for non-orthogonal states
Quantum detection theory makes clear the fundamental limit for the discrimination among quantum states. Basically, if a set of quantum states is non-orthogonal each other, no one can discriminate without error. In the following, the formulations are shown.
Let us first describe the theory of quantum Bayes criterion.
Theorem: {Roman type} [7] The quantum limitation (average error probability) for the discrimination for two quantum states ρ 1 and ρ 2 is given by where ρ 1, and ρ 2 are a priori probabilities.
Let us generalize to M-ary case. That is, a set of quantum states is given as {ρ i ,i=1,2,3…M}. The criterion of quantum Beys strategy is as follows: where, π={π j } is POVM(positive operator valued measure). As usual, we define the risk operator as follows: In general, we consider C ji =1(i ̸ =j), C ji =0 (i=j). Then the criterion becomes average error probability P e .
In case of quantum minimax strategy, the criterion is given by where C ji =1(i≠j), C ji =0 (i=j).

Classical capacity for quantum Gaussian channel
When Shannon mutual information is employed as a criterion for evaluation of communication performance, collective quantum measurement effect provides the following capacity formula for lossy Gaussian noise channel, which can be realized by coherent state signals or two-photon coherent state. This is quantum version of well known Shannon-Wiener formula in classical Gaussian channel. In fact, when S=<n>, it reduces to classical one. In addition, several features on difference between quantum and classical of classical capacity for several quantum channels were clarified by author's group.

Applications of Non-orthogonal State to Quantum Methodology Quantum illumination
The model of the quantum illumination was given by Lloyd [40] to improve the performance of the optical radar. One photon of an entangled photon is transmitted to a target, and another photon is retained in the receiver. The receiver operates joint measurement on the received light and the retained photon. He showed that quantum illumination scheme achieves an effective signal to background noise ratio. In any case, single photon scheme is useless. So a scheme with Gaussian states was proposed [41]. It employs an entangled signal and idler mode pair obtained from cw SPDC (Spontaneous parametric down-conversion). The output state is given as follows: The optimum average error probability is given by applying Helstrom receiver.

Quantum reading
Formulation: The model of the quantum reading was proposed by Pirandola based on the concept of quantum illumination [42]. "It is to discriminate quantum states which are parametrized by reflection target". As an example, he examined the model for discrimination of quantum states affected by the different reflections r 0 and r 1 . This corresponds to the model of ASK(amplitude shift keying) scheme. If reflection coefficients are 0 = 1 r and 1 << 1 r , it imposes great energy loss effect. The typical systems in quantum information science suffer the degradation from the energy loss effect. If one employs the nonclassical state in the above model, the loss effect has to be taken into account.
On the other hand, PSK(phase shift keying) scheme to realize the classical digital memory does not have a loss effect. We will employ PSK scheme such that the memory on the classical disk consists of the flat and concave which correspond to "0" and "1" , respectively.
The difference of two schemes ASK and PSK is not important in the case of classical state, but they make great difference in the case of nonclassical states. Thus, PSK scheme was introduced by Hirota [43]. This model can be described by an unitary operator as follows: where a and † are the annihilation and creation operator in bosonic system, respectively.
The reading method in the current PSK scheme is to illuminate the laser light (coherent state) on a disk, and to read the phase difference of returned light. That is, the signal states of light to discriminate are where I is the identity operator.
Error free quantum reading [43]: Let us consider the detection problem of the above model. Here we restrict the problem to the binary detection. So detection targets are two quantum states. Since the phase difference will be π, the problem is to read the phase shift π from the steady state or input state.
The coherent states are prepared in the current classical disk system, and the target signal model becomes as follows: If one employs the conventional homodyne receiver to discriminate the above quantum states, the limitation is imposed by so called quantum shot noise limit. So this is the standard quantum limit in the modern information technology. If one employs the injection back effect [44] to a laser diode to detect the phase delay of the reflection from a disk as the conventional system, the sensitivity is inferior to a homodyne receiver, though it is a typical scheme. However, in general, one can enjoy the Helstrom bound to overcome the standard quantum measurement which is achieved by a homodyne receiver.
Finally, let us employ the quantum optimum receiver for the binary pure state of two modes. The ultimate detection performances of systems with the coherent state and H-state are given, respectively, as follows: Thus, one can see that the property of H-state provides an attractive improvement, and this property can be obtained only by the combination of H-state and the quantum optimum receiver. Although there are many non-classical states, almost all is not changed from the input state to the orthogonal state to the original input state just by reflection. So we interpret that such an effectiveness of the H-state comes from a special symmetry which is a feature of H-state.
On the other hand, different applications of entangled coherent state are developed by Munro's group [45].

Quantum ghost imaging
Quantum imaging is one of attractive applications of new physical phenomena. Especially ghost imaging is a technology to synthesize target image by means of certain correlation between signal beam and reference beam. In original proposal [46] and experiment [47], an entanglement light was employed in the system. By lively investigation, it was clarified that ghost imaging does not represent a true quantum, and that the function can be realized by semi-quantum or classical resource. This fact is reasonable in the world of science and technology, because any useful technologies in real world should be realized by classical way, and these functions may be enhanced by quantum nature. The important fact is that this function can be realized only at optical field. In addition, a special feature of ghost imaging is to have immunity against atmospheric turbulence. To analyze it, one can employ the extended Huygens-Fresnel principle. is a complex valued random process due to turbulence.
Thus, based on Glauber's coherence theory and the above formula, unified theory has been presented by Erkmen and Shapiro [48], Hardy and Shapiro [49]. The basic formula is given by following the average cross correlation function.
Their works give a great contribution towards a real development of this technology. However, one can see easily the fact that their model includes highly non-linear random process in which Volterra-Wiener theory is applicable. Based on the above, a general design theory as space-time quantum Wiener receiver theory has been developed by the present author, and it will be reported in the subsequent paper.

Quantum Enigma Cipher
In this section, I will give a general framework of a physical cipher as application of quantum detection theory and no cloning theorem for non-orthogonal states.

Concept
The general network systems need to be protected from interception by unauthorized parties. The most serious attack is "Cyber attack against Layer-1 (physical layer such as optical communication line)", because technologies of coupler for tapping have been developed by several institutes. In addition, there are many optical monitor ports for network maintenance. In fact, physical layer of high speed data link is a defenseless. To date, that protection has been provided by classical encryption systems. However, such technologies cannot ensure the provable security, and also the eavesdropper can obtain the correct ciphertext: C of mathematical cipher for payload at Layer-2, and she can store it in memory devices. Thus, we cannot rule out the possibility that the cipher may be decrypted by future development of algorithm and computer science.
The best way to protect high speed data is to physically randomize signals as the ciphertext of a mathematical cipher. This is called physical random cipher. The most important feature of this physical random cipher is that the eavesdropper cannot get the correct ciphertext of mathematical cipher, for example a stream cipher by PRNG (pseudo random number generator), from communication lines, while the legitimate user can get it and he can decrypt based on a knowledge of secret key of PRNG.
First example was proposed by Yuen as "Keyed communication in quantum noise(KCQ)"in 2000 [50]. In his scheme, the mathematical cipher is used to select optical communication basis to transmit binary bit data. So the optical signals correspond to ciphertext of the mathematical cipher. Thus a modulation scheme becomes an encryption box for electric data sequence. A legitimate user can get the correct ciphertext, but an eavesdropper cannot get the correct ciphertext, because she does not know which communication basis is used and her received signals are randomized by large effect of quantum noise due to mismatch in communication basis.
cipher, which may be a generalization of KCQ. Let us describe here the ideal quantum enigma cipher system.
The quantum enigma cipher consists of an integration of mathematical encryption box and physical randomization box. Here, the physical randomization means that optical signals as the ciphertext of a mathematical cipher are randomized by quantum noise when the eavesdropper observes optical signals with coherent states or another non-orthogonal state. Along with this concept, Quantum Enigma Cipher allows a secure high speed data transmission by means of the quantum noise randomization by a mathematical encryption box and signal modulation systems. Thus, we will define the quantum enigma cipher.
Definition Quantum Enigma Cipher is defined as a scheme which has the following property [55][56][57][58][59][60]: Optical signals correspond to ciphertext of a mathematical cipher. The observation signals of legitimate's receiver are error free with a priori knowledge in communication systems. An eavesdropper's receiver suffer a serious error without a priori knowledge.
Examples of the implementation are as follows: (a) Communication basis for transmission of data is scrambled by PRNG with a secret key [50].
(b) Mapping scheme between data for communication system and optical signal is scrambled by PRNG [61].
(c) Fusion of (a) and (b) [61] (d) A priori probability for a set of coherent states is hidden as secret key against eavesdropper [56].
(e) A difference of error performance between legitimate and eavesdropper's receiver is created by entanglement resource in transmitter and receiver [59,62].
If the data for communication system is not encrypted by a mathematical cipher, one has to employ (a),(b) and (c). If the data is already encrypted by a mathematical cipher, one can employ one of {(d), (e)}.
In any scheme, the quantum enigma cipher has a mathematical encryption box and physical encryption box. The mathematical encryption box has a secret key of the length |K S | bits and PRNG for expansion of the secret key. The physical encryption box has a mechanism to create ciphertext as signal and it has a function to induces an error when the eavesdropper's receiver receives the ciphertext as signal. Consequently different ciphertext sequences are observed in the legitimate's receiver and the eavesdropper's receiver, respectively. A requirement for the physical randomization is This means that the error performance P e of the eavesdropper becomes worse than that of the legitimate user, when they observe the ciphertext as signal in communication lines.

Security analysis
In the investigation for the quantitative evaluation of information theoretically secure scheme, Shannon mutual information, trace distance (statistical distance), and Holevo quantity are not appropriate as measure of security. In the following, I will give a guide for such a purpose.
Model: Let us describe a standard symmetric key encryption. A general symmetric key encryption ᴧ can be given by where [P k ] is key generation algorithm and it provides key sequence K K ∈ depending on the probability P k , Enc is an encryption algorithm which generates ciphertext C=Enc(K,M) where M is plaintext, Dec is a decryption algorithm which produces plaintext = ( , ) M Dec K C . In the case of symmetric key cipher, the secret key is fixed.
When ᴧ cannot be decrypted by means of computational resource, its security is evaluated by "Guessing probability" [58][59][60].
(i) Ciphertext only attack on data: (ii) Ciphertext only attack on key: On the other hand, when some plaintext M K and ciphertext corresponding to them are known, it is called known plaintext attack. It is easy to generalize the above formula as follows: (iii) Known plaintext attack on data: (iv) Known plaintext attack on key: If one needs an average, then one can define average guessing probability as follows: In order to apply the above concept to quantum enigma cipher, one can employ the quantum detection theory for observation of ciphertext, and easily modify the formula of guessing probability. These are sometimes called maximum "a posteriori probability" guessing.

Evaluation of security:
A mathematical encryption box produces the ciphertext of length at most | | 2 K s bits. Because the key length is |K S | bits, when the eavesdropper gets the known plaintext of the length |K S | bits and ciphertext corresponding to them, she can pindown the secret key by the Brute force attack (trying | | 2 K s key candidates). That is, the guessing probability is one. In addition, the sequence of the ciphertext has certain correlation because of the structure of PRNG. So the eavesdropper can investigate several mathematical algorithms to estimate the secret key.
In the ideal quantum enigma cipher, the eavesdropper's observation of the cipertext as signal in communication lines suffers error completely by quantum noise randomization, while the legitimate user does not. So the legitimate user can decrypt with the secret key, but the eavesdropper does not even if she gets the secret key after her observation of ciphertext as signal.
Thus, the guessing probability is Recently, 1 Gbit/sec physical random cipher as a first generation of quantum enigma cipher was demonstrated and Y-00 cipher of 100 Gbit/sec by wave length division multiplex was also demonstrated [54].

Conclusion
The contents of this paper was presented in the International conference on Quantum Physics and Nuclear Engineering held in London. However, I extended the contents based on historical flow on development of quantum information science. From my survey paper, it is expected that further applications of non-orthogonal states as macroscopic quantum state are discovered.