† Corresponding author. E-mail:

Project supported by the National Basic Research Program of China (Grant No. 2013CB338002) and the Foundation of Science and Technology on Information Assurance Laboratory (Grant No. KJ-14-001).

In this paper, from the original definition of fidelity in a pure state, we first give a well-defined expansion fidelity between two Gaussian mixed states. It is related to the variances of output and input states in quantum information processing. It is convenient to quantify the quantum teleportation (quantum clone) experiment since the variances of the input (output) state are measurable. Furthermore, we also give a conclusion that the fidelity of a pure input state is smaller than the fidelity of a mixed input state in the same quantum information processing.

Recently, much attention has been devoted to investigating the use of continuous variable quantum information processing (CV-QIP). Continuous-spectrum quantum variables may be easier to manipulate than discrete quantum bits in order to perform various quantum information processes. For the case of Gaussian state of light, e.g., squeezed coherent beams,^{[1]} it can be created by means of linear optical circuits and homodyne detection, which can be used to perform quantum teleportation,^{[2]} quantum error correction,^{[3]} and even a quantum nonlocality test. The similarity or distinguishability of CV quantum states is an important resource in quantum information processing and there are several measures widely used to characterize it, such as trace distance^{[4]} and fidelity.^{[5–8]}

The fidelity is an average value which is often used in CV-QIP.^{[8]} The experiments with a coherent state input have been performed by assuming that the input is a pure state. For such a state, there have been a lot of studies of fidelity as a success criterion, and its value and the classical limit are well understood.^{[9–14]}

In theory, the fidelity between any two mixed states is^{[6,7]}

In this paper, based on the original definition of fidelity, we first give a well-defined universal fidelity between two Gaussian mixed states. It is related to the variances of output and input states in quantum information processing. We can use it to quantify the quantum teleportation (quantum clone) experiment conveniently since the variances of input (output) state are measurable. Furthermore, we also give a conclusion that the fidelity of a pure input state is smaller than the fidelity of a mixed input state in the same quantum information processing, which means it is harder to preserve the quantum character of a pure state than a mixed state.

The fidelity for a pure input state,^{[8]}

*x*

_{in}(

*y*

_{in}) the expectation value of the amplitude (phase) quadrature, is defined by Schumacher

*F*= 〈

*α*

_{in}|

*ρ*

_{out}|

*α*

_{in}〉, where

*ρ*

_{out}is the density operator of the output state, and the fidelity is equivalently expressed with the Wigner function

^{[15]}by

*W*

_{in}(

*α*) and

*W*

_{out}(

*α*) are the Wigner functions of the input and output states, respectively.

In this paper the states we considered are Gaussian states, which are at the heart of quantum information processing with continuous variables. They can also be generated and manipulated experimentally in a variety of physical systems, ranging from light fields to atomic ensembles.

A state of a continuous variable system is called Gaussian if its Wigner function, or equivalently its characteristic function, is Gaussian. If the input is a Gaussian state, we can denote the input state Wigner function as

*x*

_{in}and

*y*

_{in}are the displacement (also named the expectation value in theory) of the input state in the phase space and 〈

*δ*

^{2}

*X̂*

_{in}〉 and 〈

*δ*

^{2}

*Ŷ*

_{in}〉 are the quadrature component variances of the input state.

In the CV-QIP, the Gaussian character of the input state does not need be altered. Even when a Gaussian state suffers from some losses in the CV-QIP and becomes a mixed state, the state is just transformed into another Gaussian state. So we can also denote the Wigner function

*x*

_{out}(

*y*

_{out}) and 〈

*δ*

^{2}

*X̂*

_{out}〉 (〈

*δ*

^{2}

*Ŷ*

_{out}〉) have the same meaning with Eq. (

If the input is prepared in a pure state, the fidelity is calculated^{[6,7]} by Eq. (

The displacement *x*_{in} (*y*_{in}) of an input state can be easily reconstructed at the output station by setting the gains of classical channels to unity, i.e., *x*_{out} = *x*_{in}, *y*_{out} = *y*_{in}, where the fidelity is peaked,

When the input state *δ*^{2}*X̂*_{in}〉 = 〈*δ*^{2}*Ŷ*_{in}〉 = 1, equation (

*et al*.

^{[14]}used this expression to investigate the fidelity of continuous-variable quantum teleportation for a pure coherent state.

However, the input state cannot always be a pure state but is a mixed state due to some inevitable losses and imperfection in real experiments. For example, squeezed states used in successful experiments of quantum teleportation or the generation of entanglement are commonly generated by using an optical parametric oscillator^{[14,18–22]} or a Kerr medium.^{[23]} In these experiments, each squeezed state suffers from some inevitable losses, mainly in its generation process, having excess noise in the antisqueezing component. Such squeezed states are not pure, but mixed states. So we cannot directly qualify the similarity between the input and output state.

Here we can assume the ideal input

*X̂*

_{p}and the phase

*Ŷ*

_{p}quadrature with the canonical commutation relation [

*X̂*

_{p},

*X̂*

_{p}] = 2i, and 〈

*δ*

^{2}

*Â*

_{p}〉 are the quadrature component variances with

*Â*=

*X̂*=

*Ŷ*.

*x*

_{p}and

*y*

_{p}are the expectation values of the

*X̂*

_{p}and

*Ŷ*

_{p}. Since

*δ*

^{2}

*X̂*

_{p}〉·〈

*δ*

^{2}

*Ŷ*

_{p}〉 = 1.

Due to some inevitable noise coupled in the ideal input state

*δ*

^{2}

*Â*

_{in}〉 are the quadrature variances with

*Â*=

*X̂*or

*Ŷ*.

*x*

_{in}and

*y*

_{in}are the expectation values of

*X̂*

_{in}and

*Ŷ*

_{in}. Since the prepared input

Due to the no-cloning theorem, the state transport process inevitably induces noise in comparison with the input state

*x*

_{out}and

*y*

_{out}are the expectation values of

*X̂*

_{out}and

*Ŷ*

_{out}. Since the output state

Since the prepared input state

First, because the ideal input is a pure state, we can calculate the fidelity between

Similarly, we can also calculate the fidelity between

A well-defined fidelity must qualify the similarity between quantum states. In mathematics, the value of the fidelity must belong to [0,1]. So the fidelity between mixed states

Considering two ideal cases, firstly, when

^{[18–20,24]}the displacement

*x*

_{in}(

*y*

_{in}) of an input state can be easily reconstructed at the output station by setting the gains of classical channels to unity, i.e.,

*x*

_{out}=

*x*

_{in},

*y*

_{out}=

*y*

_{in}. Since

*x*

_{p}=

*x*

_{in}=

*x*

_{out},

*y*

_{p}=

*y*

_{in}=

*y*

_{out}, and the fidelity is strongly peaked and changed into

^{[24]}Considering two ideal cases, first, when

*F*= 1; second, when 〈

*δ*

^{2}

*X̂*

_{out}〉 ≫ 〈

*δ*

^{2}

*X̂*

_{in}〉 and 〈

*δ*

^{2}

*Ŷ*

_{out}〉 ≫ 〈

*δ*

^{2}

*Ŷ*

_{in}〉, the fidelity

*F*= 0. It only depends on the variances of the input state and output state, which are just what we measured in the experiment. It is very useful to qualify the Gaussian state quantum teleportation (quantum clone) experiment.

When the prepared input state *δ*^{2}*X̂*_{p}〉 = 〈*δ*^{2}*Ŷ*_{p}〉 = 1, equation (

^{[14,18–20,25]}with a coherent state input have been performed by assuming that the prepared input state

*δ*

^{2}

*X̂*

_{in}〉 = 〈

*δ*

^{2}

*Ŷ*

_{in}〉 = 1. So equation (

^{[14]}Equation (

^{[26,27]}

The real squeezed input state

*δ*

^{2}

*X̂*

_{p}〉 = 〈

*δ*

^{2}

*Ŷ*

_{p}〉 = 1 and coth (2

*β*)/4 is the variance of an initial thermal state.

*β*is the inverse temperature 1/2

*k*

_{B}

*T*, where

*k*

_{B}is the Boltzmann constant and

*T*is temperature. Accordingly a squeezed thermal state is no longer the minimum-uncertainty state 〈

*δ*

^{2}

*X̂*

_{in}〉·〈

*δ*

^{2}

*Ŷ*

_{in}〉 > 1. If such a squeezed thermal state is generated as an input of a quantum clone (or quantum teleportation), we can obtain the fidelity by Eq. (

Considering the ideal pure state

*δ*

^{2}

*X̂*

_{1}〉 (〈

*δ*

^{2}

*X̂*

_{1out}〉) and 〈

*δ*

^{2}

*Ŷ*

_{1}〉 (〈

*δ*

^{2}

*Ŷ*

_{1out}〉) are the quadrature component variances of the pure input state

*δ*

^{2}

*X̂*

_{T}〉 (〈

*δ*

^{2}

*Ŷ*

_{T}〉) is inevitable noise induced by the quantum transport process. This inevitable noise is the same for all input states during the same quantum transport process. This is up to the character of the quantum transport process. In unity gains, we do not care about the mean value between the input and output state. We can obtain the fidelity of the pure input state

However, in experiments the prepared input state is not always an ideal pure state

*δ*

^{2}

*X̂*

_{N}〉 (〈

*δ*

^{2}

*Ŷ*

_{N}〉) is additional noise on the mixed state

Similarly, when the input state

So we can obtain an inequality by simple calculation

Based on the original definition of fidelity, we give a universal fidelity between two Gaussian mixed states with quantum state quadrature variances. It is convenient to quantify the quantum teleportation (quantum clone) experiment since the variances of the quadrature components of the input and output states are measurable. Furthermore, we also give a conclusion that the fidelity of a pure input state is smaller than the fidelity of a mixed input state in the CV-QIP due to the fragility of the pure state.

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