Coherent Processing of a Qubit Using One Squeezed State

We use a single squeezed state to represent a qubit, which can be coherently processed in a deconvolution picture (DP) in the presence of noise. We avail ourselves of the fact that when evolution is governed by a quadratic dissipative equation, there exists a basis of squeezed states that evolves to another basis of such states in the DP. An operator acts as an impurity filter, restoring the coherence lost from the inexorable interactions of the qubit with its surroundings.


Introduction
In quantum computing, errors arising from quantum noise may be treated using quantum error-correcting codes, such as the Shor code [1]. The basic idea is to convert a qubit that one is trying to shield from noise to a higher dimensional state containing redundant information. After being subjected to noise, a recovery procedure is implemented to rehabilitate the state. Departing from this approach, the work presented below exploits a unique factorization of quadratic dissipative equations to leave squeezed states coherent and accessible to quantum processing after a deconvolution at a privileged time * t . This is accomplished by using a single Gaussian state to represent a qubit. Quantum noise is thus eliminated without having to increase the dimensions of the system under study, albeit at the expense of having to perform the deconvolution at * t . We suggest that this limitation may be used advantageously to help send information securely. Only authorized persons would be apprised of * t . An intruder accessing the information at any other time encounters noisy data.
Gaussian states have been widely used for continuous variable quantum information processing (see [2] for a review). Experimental examples include teleporting a single coherent state [3] and investigating transmitted states of light represented by a mixture of two coherent states [4,5]. Theoretically, a superposition of Gaussian states has been used to represent states of a two-dimensional code space [6]. Another application of Gaussian states employs a superposition of two substantially non-overlapping coherent or squeezed states as the computational basis to represent a qubit [7,8,9]. Because of the small overlap, this superposition is prone to decoherence, a bane of information processing. Error correction codes for continuous variables have been developed to reduce the deleterious noise effects [2,4,5,6,10,11]. Here, we abandon superpositions of Gaussians and instead utilize a single squeezed state to represent a qubit, reversing decoherence with an impurity filter. We encode information using the phase space position of the state, as in quantum key distribution protocols [2] that use coherent [12] or squeezed states [13,14]. The application of the impurity filter amounts to performing a Wigner function deconvolution, which is in line with the general use of deconvolutions to eliminate noise, such as instrumental quantum noise [15,16]. In quantum homodyne tomography, for example, deconvolutions have been used to reconstruct Wigner functions from noisy data [17].

Review of Quadratic Dissipative Equations
As our testing ground for handling quantum noise, we will utilize general quadratic dissipative equations (QDEs), describing the evolution of an oscillator subject to fluctuations and dissipation. QDEs are a class of master equations with time dependent coefficients, also known as non-autonomous equations. In this section, we will briefly review such equations and mention a few well-known master equations that are examples of QDEs. Some lessons from these examples will also guide us when we attempt to eliminate noise in general QDEs. The reader interested in more details of QDEs can consult Reference [18].
A QDE is the following equation governing the evolution of the density operator, ρ : where the system Hamiltonian is given by 22 b , 1 k and 2 k being real functions, 3 k being a complex function, As motivation for studying QDEs, we note that most one-dimensional quantum master equations describing harmonic motion in the literature, both exact and approximate, are of this form. For example, a subset of QDEs that are identified with Brownian motion has been solved exactly as a Wigner function convolution [19,20,21]. Another well known, autonomous example of a QDE, is the rotating wave optical master equation for an oscillator, which, when written as it usually is in terms of lowering and raising operators, is given in the interaction picture by (see, e.g., [22] and references therein) n  a  a  n  dt   d   I  I   I   ,  ,  ,  ,  1 where γ is a measure of the coupling strength between the oscillator and its environment, and n is the average number of oscillator quanta in a state of thermal equilibrium.
Although Equation (2) is one of the simplest QDEs, certain features of this equation foreshadow our approach for reducing quantum noise in more general cases. While at all temperatures the pure states that maximize the initial rate of change of the purity 2 ρ Tr predicted by Equation (2) are the ordinary coherent states [23,24], it is only at absolute zero, when 0 = n and the { } a a I , , ρ term disappears, that these coherent states-and only these states-remain pure as they evolve [24]. The key to this result is that at absolute zero, only one dissipative operator of the form { } † † , , a a I ρ remains in Equation (2), an idea that we will use below when investigating more general QDEs.
With the goal of determining when states can remain pure, Hasse has examined other autonomous master equations [25]. In this last reference, non-linear Schrodinger equations that describe the evolution of the corresponding wave function were formulated.
These examples of decoherence-free evolution motivate us to inquire about similar behaviour in more general, non-autonomous QDEs. We will see that in the general case, we have to content ourselves with less. In particular, while it is not generally possible for a coherent state to remain coherent for all times, below we will show that by making use of a deconvolution, we can ensure that a squeezed state evolves to another squeezed state at a privileged time.
Before leaving this section, we will provide the solution of Equation (1) that we will need for our work below. In operator form, Equation (1) , and where the coefficients in the preceding exponent are solutions to the following inhomogeneous system of differential equations: , and initial conditions and i k give rise to physical solutions; to ensure that ρ is normalized, Hermitian and positive, we shall assume that these coefficients yield

Noise Reduction in Quantum Dissipative Equations
A state evolving under a generic QDE loses coherence as pure states map to mixed states. Preferred states that minimize entropy production have been investigated for some master equations [26,27]. Better, we show that under a simple transformation, which we dub the deconvolution picture (DP), we can eliminate decoherence altogether: for any QDE, there exists a pair of time-dependent generalized lowering operators, and having squeezed eigenstates and complex eigenvalues satisfying such that every eigenstate of ( ) * t C evolves after an arbitrary but fixed time * t to an eigenstate of ( ) * t B in the DP. Above and throughout this paper, we use Dirac ket notation ψ to represent a state vector ψ . In addition to obeying the foregoing commutation relation, generalized lowering and raising operators are linear combinations, with complex coefficients, of the usual lowering and raising (or position and momentum) operators, and have been extensively studied in the literature [28].
In one example, the DP (denoted by a superscript D) is defined by When we compare this expression with Equation (7), we see that This generalized lowering operator satisfies the eigenvalue Equation (6). The propagator given by Equation (3) can be factored (see [18] for a related factorization) so that the only operator appearing in the non-Hamiltonian part in the DP is † B : where the DP is defined by The operator and satisfies the eigenvalue Equation (5). Letting the initial state be ( ) where * t is an arbitrary but fixed time and β is variable, and using the general formula for 0 ≥ f and B any generalized lowering operator, we obtain at time * t With this evolution, we can associate an operator For any QDE, the density operator corresponding to the state

An Example Using the Wehrl Entropy
To further elucidate our work and characterize the privileged time and where this last expression becomes an equality if α α ρ a a = .
We investigate a simple example, in which B is time independent and given by  (22) and Using these special values in Equation (10) (24) The squeezed state The foregoing relations impose restrictions on the QDE that is supposed to yield solution (24) and it is helpful to summarize what these are. Our initial time is taken to be zero and the privileged (27) Differentiating with respect to τ yields the differential equation (29) Conveniently, a solution of Equation (28) with an initial squeezed state has been computed in the literature [32]. That solution, applied to our particular values, is where we note that for the subset of squeezed states characterized by one squeezing parameter, this last Wehrl entropy appears in Reference [33]. As a check of our work, we note that when we take the noise free case with 0 4 = w (36) This result is consistent with the observation in Reference [24] that for the rotating wave optical master equation (Equation (2)) at absolute zero, a coherent state remains a coherent state.
To sum up, for the example treated in this section in the DP, we start off at time If we take the Wehrl entropy as a measure of noise, we can provide a rough estimate of the time interval during which the noise will be less than some tolerance. In other words, we seek an

Coherent Processing of a Qubit
A qubit, approximated as a superposition of substantially non-overlapping coherent states, is prone to decoherence. Thus, we avoid such a superposition and instead represent a qubit by a single squeezed state whose position in phase space encodes information. The impurity filter enables processing without decoherence. In the following formalism, unitary operators that keep the set of squeezed coherent states invariant are candidates for single qubit gates. . According to convention [1], a qubit ψ is a vector in a two-dimensional state space that is given by where the kets 0 and 1 , constituting an orthonormal basis, are fixed and the coefficients, satisfying the normalization condition where now, opposite to the conventional formalism, the last two kets can vary by changing β and the coefficients are fixed to unity. States differing by only a global phase factor are assumed to be equivalent. Therefore, only two parameters, θ and φ , are needed to specify a and b , which then fix the qubit ψ [1]. We may parametrize the qubit, as follows: ( ) ( )1 erfc 2 1 0 erfc 2 where ( ) x erfc is the complementary error function. For fixed variances and to within a global phase factor, a squeezed state is in one-to-one correspondence with its two first moments, q and p . Thus, with the help of the equations ( ) where q Δ and p Δ are variances of this state, we can uniquely map a squeezed state Turning to two qubits, we may take the tensor product of two squeezed states, B B β α ⊗ , as the fiducial input. However, because gates for multiple qubits generally convert product states to correlated states, we must admit entangled states into the formalism. A CNOT gate [1,10] serves as an example. Conventionally, this gate may be written as

Discussion
It is tempting to think of the impurity filter given by Equation (7) as the inverse of state except in the unlikely event that he or she happens to apply the filter at precisely * t . An authorized person would be apprised of this time, and therefore could recover such a state.

Conflicts of Interest:
The author declares no conflict of interest.