# Fidelity and Fisher information on quantum channels

###### Abstract

The fidelity function of quantum states have been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguish of quantum states. A fidelity function on quantum channel is expected to have same wide applications in quantum information science. In this paper we propose a fidelity function on quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function, for example the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a new Fisher information which quantifies the ultimate precision limit in quantum metrology, the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity on quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

## I Introduction

Fidelity, as a measure of the distinguishability between quantum statesFuchs and Caves (1994); Jozsa (1994); Fuchs (1996), plays an important role in many areas of quantum information science, for example it is related to the precision limit in quantum metrology Braunstein and Caves (1994), serves as a measure of entanglement preservation through noisy quantum channels Schumacher (1996), and a measure of entanglement preservation in quantum memory Surmacz et al. (2006); it has also been used as a characterization method for quantum phase transitions Gu (2010), and a criterion for successful transmission in formulating quantum channel capacities Barnum et al. (2000).

Unlike the fidelity of quantum states which is defined directly on quantum states, most commonly used measures for the distinguishability of quantum channels are defined indirectly through the effects of the channels on the states. For example the diamond norm, which is defined as Kitaev (1997); Kitaev et al. (2002); Watrous (2009)( here , denotes a state on system+ancilla, and denotes the identity operator on the ancillary system), is induced by the trace distance on quantum states ; another measure on quantum channels which is defined as Gilchrist et al. (2005); Belavkin et al. (2005), is induced by the fidelity on quantum states

In this paper we provide a fidelity function defined directly on quantum channels, and show that this fidelity function on quantum channels, together with the classical fidelity on probability distribution and the fidelity on quantum states, form a hierarchy of fidelity functions in terms of optimization. This fidelity function on quantum channels also lead to various distance measures defined directly on quantum channels, in particular we show the Bures angle and the Bures distance can be extended to quantum channels. We then show the distance between quantum channels leads naturally to a new Fisher information on quantum channels which quantifies the ultimate precision limit in quantum metrology. We also show that this fidelity function provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.

## Ii Fidelity function on quantum channels

We start by defining the fidelity function on unitary channels then extend it to noisy channels.

For a unitary matrix , we denote as the eigenvalues of , where for and we call the eigen-angles of . We define(see alsoChau (2011); Fung and Chau (2013, 2014)) and as the minimum of over equivalent unitary operators with different global phases, i.e., . We then define

(1) |

Quantitatively is equal to the maximal angle that can rotate a state away from itselfAcin (2001); D’Ariano et al. (2001); Fung and Chau (2014), i.e., For mixed states it can be written as

If are arranged in decreasing order, then when Fung and Chau (2014). We then define , here and are unitary operators on the same Hilbert space(we can expand the space if they are not the same). It is easy to see that

(2) |

thus corresponds to the maximal angle between the output states of and (however we note that the definition of is independent of the states). We then denote as the fidelity between and . For unitary channels this is equivalent to the fidelity function proposed previously in Acin (2001).

We now generalize this to noisy quantum channels. A general quantum channel , which maps from - to -dimensional Hilbert space, can be represented by Kraus operators, where . Equivalently it can also be written as where denotes some standard state of the environment, and is a unitary operator acting on both system and environment, which we call as the unitary extension of .

We define and where are unitary extensions of , . In Appendix A, we show that the optimization can be taken by fixing one unitary extension and just optimizing over the other unitary extension, i.e.,

(3) |

In terms of it can be written as

(4) |

This can be seen as the counterpart of Uhlmann’s purification theorem on quantum states Uhlmann (1976)(however the proof does not use Uhlmann’s purification theorem Yuan and Fung (2017)). In Appendix B, we show that is a metric and can be computed directly from the Kraus operators of and as Yuan and Fung (2017)

(5) |

here denotes the minimum eigenvalue of with , and denote the Kraus operators of and respectively, denotes the -th entry of a matrix with where is the operator norm which corresponds to the maximum singular value, here arises from the non-uniqueness of the Kraus representations. Thus

(6) |

We emphasize that is defined directly on quantum channels without referring to the states, such direct connection, in contrast to the induced measure, is crucial when applying the fidelity to channel discrimination and quantum metrology as we will show later. Furthermore the fidelity can be formulated as a semi-definite programming and computed efficiently as

(7) |

Analogous to the Bures distance on quantum states , we can similarly define a Bures distance on quantum channels as . In Appendix A, we prove an intriguing and useful connection between and the minimum distances between the Kraus operators of and as

where are the sets of all equivalent Kraus representations of and respectively. This connection is particular useful in studying the scalings of the distance between quantum channels as we will show later.

In which sense we call a fidelity function? It turns out that . To see this, it is proved in the supplemental material of Ref. Yuan and Fung (2017) that

(8) |

which coincides with Eq. (6). From this relationship it is also immediate clear that is stable, i.e., . This result gives an operational meaning to . We emphasize that although we made connections between and the minimum fidelity of the output states, is defined directly on quantum channels and does not depend on the states. The definition and the operational meaning of play distinct roles in applications, the operational meaning provides a physical picture while the direct definition brings insights which enable or ease the proofs and computations, which will be demonstrated in the applications. This is in analogy to how fidelity of quantum states is connected to the classical fidelity , here denotes the classical fidelity with and , denotes a set of Positive Operator Valued Measurements(POVM)Fuchs (1996), here similarly the fidelity between quantum states has the operational meaning as the minimum classical fidelity, however the fidelity between quantum states is defined directly on quantum states which is independent of the measurements and such direct definition has provided numerous insights which would be hindered with just the classical fidelity.

It is known that the trace distance and the fidelity between quantum states have the following relationshipsFuchs1999

(9) |

from which it is straightforward to get the relationships between the diamond norm and the fidelity of quantum channels. This can be obtained by substituting and , then optimizing over

(10) |

which gives

(11) |

Since can be computed directly from the Kraus operators, this also provides a way to bound the diamond norm using the Kraus operators.

In Raginsky the Choi matrices of the quantum channels are used to compute the fidelity between the channels, which corresponds to the fidelity between the output states of two quantum channels when the input state is taken as the maximal entangled state. As the maximal entangled state is in general not the optimal input state, the fidelity thus defined does not have operational meaning as the minimum fidelity of the output states, thus can not be related to the ultimate precision limit in quantum metrology etc(instead related to the precision limit when the probe state is taken as the maximally entangled state).

## Iii A unified framework for quantum metrology and perfect channel discrimination

Next we demonstrate the applications in quantum information science, in particular we show how the fidelity provides a unified platform for the ultimate precision in quantum metrology and the minimum number of uses needed for perfect channel discrimination.

The task of quantum metrology, or quantum parameter estimation in general, is to estimate a parameter encoded in some channel , this can be achieved by preparing a quantum state and let it go through the extended channel with the output state . By performing POVM, , on one gets the measurement result with probability . According to the Cramér-Rao boundHelstrom (1976); Holevo (1982); Cramer (1946); Rao (1945), the standard deviation for any unbiased estimator of is bounded below by where is the standard deviation of the estimation of , is the classical Fisher information and is the number of times that the procedure is repeated. The classical Fisher information can be further optimized over all POVMs, which gives

(12) |

where the optimized value ) is usually called the quantum Fisher informationHelstrom (1976); Holevo (1982); Braunstein and Caves (1994); Braunstein et al. (1996), here for distinguish we will call it the quantum state Fisher information.

We first recall established connections between the fidelity functions and the Fisher information. Given and its infinitesimal state , for a given POVM , the classical fidelity between and is given by which defines an angle as . The classical Fisher information is related to the classical fidelity as up to the second order of Braunstein and Caves (1994), this can also be written as

(13) |

If we optimize over the classical fidelity then leads to the fidelity between quantum states as Braunstein and Caves (1994)

(14) |

and the classical Fisher information leads to the quantum state Fisher information and up to the second order of Braunstein and Caves (1994); Braunstein et al. (1996)

(15) |

If we denote , then

(16) |

The precision can be further improved by optimizing over the probe states, which leads to the ultimate local precision limit of estimating from . Intuitively, this ultimate precision limit should be quantified by the distance between and its infinitesimal neighboring channel , in a way analogous to how Bures distance of quantum states quantifies the precision limit of estimating from the state Braunstein and Caves (1994). However although much progress has been made on calculating the ultimate precision limitFujiwara and Imai (2008); Escher et al. (2011); Tsang (2013); Demkowicz-Dobrzanski et al. (2012); Knysh et al. (2011, 2014); Kolodynski and Demkowicz-Dobrzanski (2013); Demkowicz-Dobrzański and Maccone (2014); Alipour et al. (2014), such a clear physical picture has still not been established after more than two decades since Braunstein and Caves’s seminal paperBraunstein and Caves (1994), this is mainly due to the lack of proper tools on quantum channels. Here we show that the fidelity between quantum channels can be used to establish such a physical picture, which also leads naturally to a new Fisher information on quantum channel.

Further optimizing over the probe states

(17) |

this leads naturally to a quantum channel Fisher information which is similarly related to the distance on quantum channels as

(18) |

The quantum channel Fisher information quantifies the ultimate precision limit upon the optimization over the measurements and probe states

(19) |

This connects the precision limit directly to the distance between quantum channels which provides a clear physical picture for the ultimate precision limit. The scaling of the ultimate precision limit can now be seen as a manifestation of the scaling of the distances between quantum channels as we now show.

Two schemes on multiple uses of quantum channels are usually considered in quantum parameter estimation, the parallel scheme and the sequential scheme as shown in Fig.1. We will show that for both schemes, the scaling of the distances between two quantum channels are at most linear, which underlies the scaling for the Heisenberg limit.

For parallel scheme with uses of a channel as shown in Fig.3, the total dynamics can be described by . If we denote as one unitary extension of , then is a unitary extension of as shown in Fig.3. Given two channels and , we choose and as the unitary extension for and respectively which satisfies . Now as and are unitary extensions of and respectively, we then have

(20) |

For the sequential scheme, we consider the general case that controls can be inserted between sequential uses of the channels. Any measurements that are used in the control can be substituted by controlled unitaries with ancillary systems, the controls interspersed between the channels can thus be taken as unitaries, which is shown in Fig.5. Parallel scheme can be seen as a special case of the sequential scheme by choosing the controls as SWAP gates on the system and different ancillary systemsDemkowicz-Dobrzański and Maccone (2014). We show that with uses of the channel, the distance is still bounded above by .

We present the proof for the case of , same line of argument works for general . For , one unitary extension of is , similarly is a unitary extension of , here denote a unitary extension of , , with as the environment. We can choose such that , here all operators are understood as defined on the whole space so the multiplication makes sense, for example the control , which only acts on the system and ancillaries, is understood as , an operator on the whole space including the environment. We then have

(21) |

i.e., with two uses of the channel, the distance is bounded above by . With the same line of argument it is easy to show that with uses of the channel the distance is bounded above by .

Substitute with and with , we have for both schemes. From Eq.(19) the ultimate precision limit is then bounded by

(22) |

the scaling is called the Heisenberg scaling, which, as we showed, is just a manifestation of the fact that the distance between quantum channels can grow at most linearly with the number of channels.

For uses of the channels under the parallel scheme we can also obtain a tighter bound as

(23) |

here as previously defined, and the inequality holds for any with (see Appendix C). In the asymptotical limit, is the dominating term, in that case we would like to choose a minimizing to get a tighter bound. This can be formulated as semi-definite programming with

(24) |

If we let and , then Eq.(23) provides bounds on the scalings in quantum parameter estimation, which is consistent with the studies in quantum metrologyFujiwara and Imai (2008); Escher et al. (2011); Demkowicz-Dobrzanski et al. (2012); Kolodynski and Demkowicz-Dobrzanski (2013); Demkowicz-Dobrzański and Maccone (2014) but here with a more general context (see also Ref. Yuan and Fung (2017)).

Next we show how the tools unify quantum parameter estimation and the perfect quantum channel discriminationAcin (2001); Duan et al. (2007, 2008); Lu et al. (2012); Chiribella et al. (2008); Duan et al. (2009); Harrow et al. (2010).

Given two quantum channels and , they can be perfectly discriminated with one use of the channels if and only if there exists a such that and are orthogonal, i.e., , which is the same as . When and can not be perfectly discriminated with one use of the channel, finite number of uses may able to achieve the taskDuan et al. (2009). This is in contrast to the perfect discrimination of non-orthogonal states which always requires infinite number of copies. The minimum number of uses needed for perfect channel discrimination should satisfy . The perfect channel discrimination is thus determined by the distances between quantum channels, and the scalings of obtained before can be used to determine the minimum . For example, from we can obtain a lower bound on as

(25) |

where is the smallest integer not less than . This bound is tighter than existing bounds for noisy channelsLu et al. (2012) and for unitary channels it reduces to the formula which is known to be tightAcin (2001). For noisy channels under the parallel scheme we can also substitute into the inequality (23) to get a tighter bound.

The lower bound on minimum can also be obtained via a connection to quantum metrology. Given two channels and , let as a path connecting and . With uses of the channel under the parallel strategy we have . From the triangular inequality

(26) |

This connects the prefect channel discrimination to the ultimate precision limit. By choosing different paths various useful lower bounds on the minimum number of uses for perfect channel discrimination can be obtained.

For example, given and and are Pauli matrices and assume . For the parallel strategy the lower bound given by Eq.(25) is . If we choose a simple path , , which is a line segment connecting to , then with the connection provided by Eq.(26) we obtain . Other paths may be explored to further improve the bound. By using the inequality (23) with the obtained from the semi-definite programming that minimizes , we get . For any we can also choose the to minimize , it turns out the minimum such that is , thus . For comparison we also explicitly computed the actual distance with the increasing of , it turns out that the minimum such that is actually . All computations here are done with the CVX package in MatlabGrant and Boyd (2011). , where

## Iv Summary

A fidelity function defined directly on quantum channels is provided, which leads to various distance measures defined directly on quantum channels, as well as a new Fisher information on quantum channel. This forms another hierarchy for fidelity functions and Fisher information as shown in the table:

where and , . In this table the functions on quantum states equal to the optimized value over all measurements of the corresponding functions on probability distribution, and the functions on quantum channels equal to the optimized value over all probe states of the corresponding functions on quantum states. This framework connects quantitatively the ultimate precision limit and the distance between quantum channels, which provided a clear physical picture for the ultimate precision limit in quantum metrology. It also provide a unified framework for the continuous case in quantum parameter estimation and the discrete case in perfect quantum channel discrimination, with this framework the progress in one field can then be readily used to stimulate the progress of the other field. We expect these tools will find wide applications in many other fields of quantum information science.

## Appendix A Formula to compute

We show that the distance between two quantum channels can be computed from the Kraus operators of and as here are unitary extensions of , and denotes the minimum eigenvalue of with , , denotes the Kraus operators of and , denotes the -th entry of a matrix with ( is the operator norm which equals to the maximum singular value), is the number of the Kraus operators. Furthermore the minimization on both and can be reduced to the minimization of just one

(27) |

We start by a general unitary extension for any given channel with , which maps from a - to - dimensional Hilbert space,

(28) |

where only acts on the environment and can be chosen arbitrarily, here denotes the set of unitary operators with as zero Kraus operators can be added. Here only the first columns of are fixed, the freedom of other columns can be represented as

(29) |

where can be any unitary.

For two channels and , with and , the unitary extensions can be written as