Purification-based metric to measure the distance between quantum states and processes
aa r X i v : . [ qu a n t - ph ] M a y Purification-based metric to measure the distance between quantum states and processes
Trist´an M. Os´an and Pedro W. Lamberti
Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba and CONICET,Medina Allende s/n, Ciudad Universitaria, X5000HUA C´ordoba, Argentina. (Dated: July 28, 2018)In this work we study the properties of an purification-based entropic metric for measuring thedistance between both quantum states and quantum processes. This metric is defined as the squareroot of the entropy of the average of two purifications of mixed quantum states which maximize theoverlap between the purified states. We analyze this metric and show that it satisfies many appealingproperties, which suggest this metric is an interesting proposal for theoretical and experimentalapplications of quantum information.
I. INTRODUCTION
Quantum Information Processing is intended to develop newforms and procedures of computation and cryptography be-yond the possibilities of classical devices. Thus, a significa-tive quantity of new algorithms, communications protocols,and suggestions for physical implementations of theoreticalconcepts has been proposed [1–12]. As a consequence, quan-tum information continues to be a topic of major interest forcurrent research.Most important noiseless quantum communication protocolssuch as teleportation, super-dense coding, including their co-herent versions, and entanglement distribution rely on theassumption that noiseless resources are available. For ex-ample, the entanglement distribution protocol assumes thata noiseless qubit channel is available to generate a noiselessentangled bit (ebit). This idealization allows to develop themain principles of the protocols without the need to takeinto account more complicated issues. However, in practice,quantum protocols do not work as expected in the presenceof noise.In order to protect quantum information from noise somestrategies have been proposed, like quantum error-correctingcodes and fault-tolerant quantum computation [12–15]. Inthis regard, a large number of error-correcting codes havebeen developed. For example, a promising approach is to usetopological error-correcting codes to store quantum informa-tion safely by associating it with some topological propertyof the system [16–18]. This strategy works in such a wayto make quantum information resilient against the effects ofnoise. A recent proposal in this area can be found in refer-ences [19, 20]. Another particularly fruitful strategy seemsto be the group-theoretical structure known as “stabilizercodes” [15].Despite the existence of these strategies to protect quantuminformation from noise, in many practical cases it is desir-able to have the means to quantify how much a quantumsystem is effectively affected by a disturbance, no matterhow small. In other words, it is important to have a pro-cedure to determine how close to expected a real quantumsystem is working. The simplest way to do so is to comparethe output state of the quantum system, thought as ideal,with the output state of the real system using a distance measure between them. For example, suppose that a quan-tum information processing protocol should ideally producesome quantum state represented by a density operator ρ , butthe actual output of the protocol is a mixed quantum staterepresented by a density operator σ , then, a distance mea-sure D ( ρ, σ ) should be provided to indicate how close theideal output of a quantum process is to the actual output.One of the most important features of quantum mechanicsis that, in general, two arbitrary quantum states cannot bedetermined with certainty. For example, if two pure statesare non-orthogonal they cannot be perfectly distinguished.Only orthogonal states can be discriminated unambiguously.Therefore, in order to provide a way to determine how wella quantum protocol is working, distance measures need tobe devised to allow us to determine how close two quantumstates or two quantum processes are to each other.A variety of distance measures have been developed forthis purposes, like trace distance , Fidelity , Bures distance , Hilbert-Schmidt distance , Hellinger distance and
QuantumJensen-Shannon divergence , just to name a few [12, 21–27].Quantum processes can be represented by means of positiveand trace-preserving maps E defined on the set of densityoperators belonging to B ( H ) +1 , that is, the set of positivetrace one operators ρ on a Hilbert space H .We say that the map E is monotonous under quantum opera-tions with respect to a given distance D ( ρ, σ ), or contractive for short, if D ( E ( ρ ) , E ( σ )) ≤ D ( ρ, σ ) (1)In particular, when E = E t is a completely positive quantumdynamical semigroup such that ρ ( t ) = E t ρ (0), then contrac-tivity means that D ( E ( ρ ( t )) , E ( σ ( t )) ≤ D ( ρ ( t ′ ) , σ ( t ′ )) , for t > t ′ (2)The physical meaning of last equation is that the distancebetween two quantum processes cannot increase in time andthe distinguishability of any pair of states cannot increasebeyond an initial value.A case of particular interest is a quantum open system[28, 29]. A real quantum system Q , like a system intendedto perform a quantum information processing task, is alwaysin interaction with its environment E . This interaction in-evitably has an influence on the state of the quantum sys-tem, causing losses on the information encoded in the sys-tem. The quantum system Q can not be treated as a closedsystem anymore when interactions with the outside worldare occurring. This kind of systems is known as open quan-tum system . Time evolution of the open system Q cannotin general be described by an unitary operator acting on theHilbert space H Q of Q . As the total system is assumed tobe closed, it will evolve with a unitary operator U ( t ) actingon the total Hilbert space H QE = H Q ⊗ H E . In many cases,as we are interested in extracting information on the state ofthe system Q at some later time t >
0, we perform a partialtrace over the environment E to obtain the reduced state ofthe system Q alone, ρ Q ( t ) = Tr E (cid:2) U ( t ) ρ QE U † ( t ) (cid:3) (3)Given two initial states ρ QE and σ QE of the composite sys-tem, the distance between the corresponding reduced states ρ Q and σ Q at a given time t > Q and its environment E are initially prepared in an uncor-related state, the reduced dynamics is completely positiveand contractive, therefore, the distance D ( ρ Q , σ Q ) betweentwo states can approximate to zero when the open systemis reaching a unique steady state (This is the case, for ex-ample, when dynamics is of the relaxing type). However,contractivity of quantum evolution can show a breakdownwhen system and environment are initially correlated. Ef-fects induced by such correlations have been studied in dif-ferent contexts [30–39]. As a result, contractivity turns outto be not an universal feature but rather depends on the cor-relations between the system and its environment and also,in general, on the particular choice of the distance measure.Experiments on initial system-environment correlations canbe found in Refs. [40] and [41]. Examples of an exact reduceddynamics which fail contractivity with respect to the tracedistance are presented in Refs. [42] and [43]. An increaseof the distance between the states of the reduced system Q can be interpreted in terms of an exchange of informationbetween the system Q and its environment E . For example,an increment of the distance above its initial value can beinterpreted as information locally inaccessible for the system Q at the beginning which was transferred to it later. As a re-sult, this flow of information increases the distinguishabilitybetween reduced system states. Possibly, this process couldbe used to devise experimental schemes for detection of ini-tial correlations between an open quantum system and itsenvironment.Dajka, Luczca and H¨anggi [44] performed a comparativestudy of different distances measures between quantumstates in the presence of initial qubit-environment correla-tions. In that work they show that the correlation-induced distinguishability growth is not generic with respect to dis-tance measures, but distinctly depends on the particularchoice of the distance measure. Their results indicate that anincrease of a distance measure above its initial value consti-tutes no universal property. Dynamics behavior upon evolv-ing time strongly depends on the employed distance measure.To the present, there is not a unique or ideal measure of dis-tinguishability between quantum states or quantum process.Moreover, different distance measures can be useful depend-ing on the particular application, whether a theoretical one,like a bound of what can be physically feasible for a givenprocess, or the measurement of a quantum protocol experi-mentally implemented.In a previous work [45], a metric D E based on the physicalconcepts of entropy and purification of a mixed state wasintroduced [46]. Some useful properties of D E were studiedand, in addition, it was demonstrated that D E is a true met-ric between quantum states.In this work we extend the study of the properties of D E and we also derive an alternative fidelity measure F E for thedegree of similarity between quantum states. We investigatethe properties of F E and show that it shares the main prop-erties of Uhlmann-Jozsa fidelity F [22, 23]. In addition, asa main result, we derive from D E a distance measure ∆ E between quantum processes which turns to have many inter-esting properties for applications in quantum information.This paper is organized as follows. In Sec. II, we brieflyintroduce the criteria that should be satisfied for a suitablemetric between quantum processes. In Sec. III we outlinetwo approaches to describe quantum processes, operator-sumrepresentation and Jamio lkowski isomorphism. These de-scriptions will allow us later to derive from D E a distancemeasure ∆ E between quantum processes and to study itsproperties. In Sec. IV we describe the distance D E and westudy its properties. In Sec. V we introduce the alternativefidelity measure F E . Next, in Sec. VI, we show how a mea-sure of distance between quantum processes can be derivedfrom D E and we study its properties. Finally, we summa-rize our main results in Sec. VII. In the appendix, with thepurpose of making this work self-contained, we survey someimportant properties of the Uhlmann-Jozsa fidelity F thatwill be used in order to prove some properties of D E . II. DISTANCE MEASURES IN QUANTUMINFORMATION PROCESSING
As stated before, there is not a unified criterium to chose ameasure of distance between quantum states and quantumprocesses. However, some guidelines can be provided basedon physical grounds. In this work we have chosen ourselvesto follow the work of Gilchrist, Langford and Nielsen [47]as a guideline of what criteria a good measure of distancebetween quantum processes should satisfy.Suppose ∆ is a good measure of the distance between twoquantum processes. Such processes are described by mapsbetween input and output quantum states, e.g., ρ out = E ( ρ in ), where the map E is a completely positive trace-preserving map (CPTP-map) also known as a quantum op-eration [48]. Physically, ∆( E , F ) may be thought of as ameasure of error in quantum information processing whenit is desired to perform an ideal process E and the actualprocess F is obtained instead. In addition, ∆( E , F ) can beinterpreted as a measure of distinguishability between theprocesses E and F .Bearing in mind the work of Gilchrist, Langford and Nielsen[47], we will look for a measure ∆ between quantum processeswhich should satisfy the following criteria, motivated by bothphysical and mathematical matters [49]:1. Metric : ∆ should be a metric, i.e., for any quantumprocesses E , F and G the following properties shouldbe satisfied:(i) Non-negativity , ∆( E , F ) ≥ E , F ) = 0 ifand only if E = F (ii) Symmetry , ∆( E , F ) = ∆( F , E )(iii) Triangle inequality ∆( E , F ) ≤ ∆( E , G ) + ∆( G , F ).2. Physical interpretation : ∆ should have a well-motivated physical interpretation.3.
Stability [50]: ∆(
I ⊗ E , I ⊗ F ) = ∆( E , F ) where I rep-resents the identity operation on an extra Hilbert of ar-bitrary dimension. This ancillary Hilbert space couldbe associated to a quantum system or to a convenientmathematical construct. The physical meaning behindthis property is that unrelated ancillary quantum sys-tems do not change the value of ∆.4. Chaining : ∆( E ◦ E , F ◦ F ) ≤ ∆( E , F ) + ∆( E , F ).This property just means that for a process composedof several steps, the total error is bound by the sum ofthe errors originated in the individual steps.From a mathematical viewpoint, it is evident that a charac-ter of true Metric is a basic requirement for a suitable dis-tance measure. Besides, the metric character of a distancecould be considered as essential to check on the convergenceof iterative algorithms in quantum processing [51]. In ad-dition, chaining and stability criteria are key properties toestimate the error in complex tasks of quantum informationprocessing which can be split into sequences of simpler com-ponent operations. In this case, a bound on the total errorcan be found by analyzing each single step of a process. III. DESCRIBING QUANTUM PROCESSESIII.1. Operator-sum representation
Quantum operations describe the most general physical pro-cesses that may occur in a quantum system [12, 21, 52], including unitary evolution, measurement, noise, and deco-herence. Any quantum operation can be expressed by meansof an operator-sum representation relating an input state ρ with the output state E ( ρ ) in the form [12, 21, 52–54] E ( ρ ) = X j K j ρK † j (4)where the operators K j are known as Kraus operators or operation elements , and satisfy the condition P j K † j K j ≤ I .Particularly, when Kraus operators satisfy the equation X j K † j K j = I (5)the process E ( ρ ) is a completely positive trace-preservingmap (CPTP-map) and maps density matrices into densitymatrices. Physically, this corresponds to the requirementthat E represents a physical process without post-selection[55]. An important remark is that the operation elements { K j } completely describe the effect of the quantum processon the input state ρ .Relation 5 is a completeness relation because K j and K † j do not necessarily commute. If additionally, the operationelements K j satisfy X j K j K † j = I (6)then, the CPTP-map is said to be a unital map, this means,a map for which E ( I ) = I . One example of such a map isthe qubit-depolarizing channel whereas a negative exampleis provided by the amplitude-damping channel [12, 21, 52].If the operator decomposition of a CP-map satisfies boththese conditions the map is doubly stochastic. The operatordecomposition of a quantum operation is not unique. Inparticular, any two sets of operators K j related to each otherby unitary transformations equally well represent the sameoperation E ( ρ ). III.2. The Jamio lkowski isomorphism
Jamio lkowski isomorphism relates a quantum operation E toa quantum state, ρ E , by the following equation [21, 56, 57] ρ E = [ I ⊗ E ] ρ Ψ (7)where ρ Ψ = | Ψ ih Ψ | and | Ψ i = 1 √ d X j | j i ⊗ | j i (8)is a maximally entangled state of the ( d -dimensional) systemwith another copy of itself, and {| j i} is some orthonormalbasis set. Jamio lkowski isomorphism works bidirectionally,i.e., the map E → ρ E is invertible. Therefore, the knowledgeof ρ E is equivalent to knowledge of E [58]. As a consequence,this isomorphism allows to treat quantum operations usingthe same tools usually used to treat quantum states. IV. PURIFICATION-BASED ENTROPIC METRIC D E Given two pure quantum states | ψ i and | ϕ i , the distance D E ( | ψ i , | ϕ i ) introduced in Ref. [45], is defined as [46]: D E ( | ψ i , | ϕ i ) ≡ s H N (cid:18) | ψ ih ψ | + | ϕ ih ϕ | (cid:19) (9)where H N ( ρ ) represents the von Neumann entropy given by: H N ( ρ ) = − Tr ( ρ log ( ρ )) = − X i λ i log ( λ i ) (10)with { λ i } being the set of eigenvalues of the density operator ρ .The distance D E emerges from the quantum Jensen-Shannondivergence D JS defined as [27]: D JS ( ρ, σ ) = H N (cid:18) ρ + σ (cid:19) − H N ( ρ ) − H N ( σ ) (11)Indeed, due to von Neumann entropy vanishes when eval-uated in pure states ρ = | ψ ih ψ | and σ = | ϕ ih ϕ | , the D JS reduces to D JS ( | ψ ih ψ | , | ϕ ih ϕ | ) = H N (cid:18) | ψ ih ψ | + | ϕ ih ϕ | (cid:19) (12)As a consequence, the distance D E verifies the identity D E ( | ψ ih ψ | , | ϕ ih ϕ | ) = D JS ( | ψ ih ψ | , | ϕ ih ϕ | ) (13) After some algebra, it is possible to write D E in the form[45] D E ( ρ, σ ) = p Φ ( |h ψ | ϕ i| ) (14)whereΦ( x ) ≡ − (cid:18) − x (cid:19) log (cid:18) − x (cid:19) − (cid:18) x (cid:19) log (cid:18) x (cid:19) (15)with Φ( x ) being the Shannon entropy of a probability vectorof size 2 and x = |h ψ | ϕ i| . From equation 15 it is easy to seethat Φ( x ) is a bounded and monotonic decreasing functionof x with 0 ≤ Φ( x ) ≤
1. Figure 1 shows a plot of Φ( x ) as afunction of x . ( x ) x FIG. 1. Plot of Φ( x ) given by Eq. 15 The definition of the metric D E can be extended to the caseof mixed states. Given two arbitrary mixed quantum statesrepresented by density matrices ρ and σ belonging to B ( H ) +1 ,the metric D E ( ρ, σ ) is defined as follows [45]: D E ( ρ, σ ) ≡ min | ϕ i s H N (cid:18) | ψ ih ψ | + | ϕ ih ϕ | (cid:19) (16)In this last expression, | ψ i represents any fixed purificationof ρ , and the minimization is taken over all purifications | ϕ i of σ .In order to derive some appealing properties of D E it is usefulto write it down in terms of the Uhlmann-Jozsa fidelity F (see appendix): F ( ρ, σ ) = max | ϕ i |h ψ | ϕ i| (17)where | ψ i is any fixed purification of ρ and maximization isperformed over all purifications | ϕ i of σ . Thus, taking intoaccount equations 16, 14 and 17, it is straight forward to seethat D E can be expressed as D E ( ρ, σ ) = r Φ (cid:16)p F ( ρ, σ ) (cid:17) (18) IV.1. Properties of the distance D E To easily see that D E is a metric we can write D E as afunction of the Bures distance D B taking into account thatboth distances can be expressed in terms of the fidelity F [cf. Eqs. 72 (see appendix) and 18]. Thus, we have D E ( D B ) = s Φ (cid:18) − D B (cid:19) (19)where Φ( . ) is given by Eq. 15.Figure 2 shows a plot of D E as a function of D B . As thisfunction is concave, D E satisfies the properties of a metric. D E D B FIG. 2. Distance D E as a function of Bures distance D B From its definition [cf. Eqs. 9, 16 and 18], it can be formallyproved that D E satisfies the following properties:1. Normalization : 0 ≤ D E ( ρ, σ ) ≤ Identity of indiscernibles D E ( ρ, σ ) = 0 if and only if ρ = σ (21)For pures states D E vanishes if and only if | ψ i = e ı a | ϕ i (i.e., the two states belong to the same ray in theHilbert space).3. Symmetry : D E ( ρ, σ ) = D E ( σ, ρ ) (22) 4. Triangle inequality : For any arbitrary density matrices ρ , σ and ξD E ( ρ, σ ) ≤ D E ( ρ, ξ ) + D E ( ξ, σ ) (23)For proofs of properties 1 to 4 refer to Refs. [45],[59, 60] and [27].5. Joint convexity : For p i ≥ P i p i = 1, ρ i and σ i arbitrary density matrices D E ( X i p i ρ i , X i p i σ i ) (24) ≤ X i p i D E ( ρ i , σ i ) (25) Proof:
It follows from Eq. 13 and the
Joint concavity property of D JS [cf. Eq. 11] [27]. Remark:
Note that joint convexity implies separateconvexity , but not the converse. For example, the separate convexity of D E can be obtained from jointconvexity by setting σ i = σ and using the fact that P i p i = 1.6. Restricted additivity : For any arbitrary density matri-ces ρ , σ and τD E ( ρ ⊗ τ, σ ⊗ τ ) = D E ( ρ , σ ) (26) Proof:
To prove this property we observe Eq. 18 whichrelates D E with the Uhlmann-Jozsa fidelity F . Then,we use properties 6 and 2 of F (cf. appendix). Bysetting ρ = σ = τ in property 6 of F it follows that F ( ρ ⊗ τ, σ ⊗ τ ) = F ( ρ , σ ) (27)This fact completes de proof of the Restricted additiv-ity property of D E . Comment:
An immediate consequence of this propertyis that for two physical systems, described by densitymatrices ρ and σ , a measure of their degree of simi-larity determined by means of D E remains unchangedeven after appending to each system an uncorrelatedancillary state τ .7. Unitary invariance : For any unitary operation U D E ( U ρ U † , U σ U † ) = D E ( ρ, σ ) (28) Proof:
It follows from Eq. 18 and the
Unitary invari-ance property of F (cf. appendix). Comment:
This is a quite natural property to be sat-isfied by a distance, because a unitary transformationrepresents a rotation in the Hilbert space and the dis-tance between two states should be invariant under arotation of the states.8.
Monotonicity under quantum operations : If E is aCPTP-map, then for any arbitrary density matrices ρ and σ D E ( E ( ρ ) , E ( σ )) ≤ D E ( ρ, σ ) (29) Proof:
It follows from Eq. 18 and the
Monotonicity un-der quantum operations property of F (cf. appendix)taking into account that the function Φ( . ) is a mono-tonically decreasing function of √ F [cf. Eq. 15]. Comment:
This is a very important property becauseit qualifies D E as a monotonically decreasing measureunder CPTP maps and can be considered the quantumanalog of the classical information-processing inequal-ity which states that the amount of information shouldnot increase via any information processing. For exam-ple, the dynamics of an open quantum system can bedescribed by means of a CPTP-map using an operator-sum representation of the form of Eq. 4. Thereforethe meaning of Eq. 29 is that nonunitary evolution de-creases distinguishability between states. Of course, aunitary evolution is a particular case of a CPTP-map.In this case, equality is satisfied in Eq. 29 in com-plete agreement with property 7. Another example ofa CPTP-map is given by E ( ρ ) = X i P i ρP i (30)with P i being a complete set of orthogonal projectors(i.e., P † i = P i , P i = P i and P i P i = I ). In this case,property 8 directly implies D E ( X i P i ρP i , X i P i σP i ) (31)= X i D E ( P i ρP i , P i σP i ) (32) ≤ D E ( ρ, σ ) (33)Therefore, due to the monotonic character of thesquare root we have D E ( X i P i ρP i , X i P i σP i ) ≤ D E ( ρ, σ ) (34) IV.2. Physical interpretation of D E Quantum Jensen-Shannon divergence D JS [cf. Eq.11] can begeneralized as a “measure of distance” between the elementsof an ensemble { q i , ρ i } ( P i q i = 1) [27] D JS ( { q i , ρ i } ) = H N ( X i q i ρ i ) − X i q i H N ( ρ i ) (35)In the context of quantum transmission processes this quan-tity represents the Holevo quantity, which bounds the mu-tual information between the sender of a classical messageencoded in quantum states and a receiver.In a recent paper [61], ˙Zyczkowski and co-workers showedthat the square of the distance D E provides a finest boundfor the Holevo quantity for a particular ensemble { q =1 / , q = 1 / , ρ , ρ } . In this way, the distance D E turnsout to be endowed with an important physical meaning.Another point to be remarked about D E is related to thefact that this distance could be implemented operationally.Indeed, Ricci et al. [62] reported an experimental implemen-tation of a theoretical protocol for the purification of singlequbits sent through a depolarizing channel previously pro-posed by Cirac and co-workers [63]. V. ALTERNATIVE FIDELITY DEFINITION
A very interesting and neat feature of the metric D E is thata fidelity F E for both pure and mixed quantum states can bedefined which fulfills the most important properties satisfiedby the usual (Uhlmann-Jozsa) fidelity F . Bearing in mindRef. [27], we define an alternative fidelity measure F E asfollows: F E ( ρ, σ ) ≡ (cid:2) − D E ( ρ, σ ) (cid:3) (36)The most important properties of F E are the following:1. Normalization : 0 ≤ F E ( ρ, σ ) ≤ F E ( ρ, σ ) = 0 if ρ and σ have supports on orthogonalsubspaces Proof:
It follows straight forward from definition of F E [cf. Eq. 36] and the Normalization property of D E (cf.Sec. IV.1).2. Identity of indiscernibles : F E ( ρ, σ ) = 1 if and only if ρ = σ (38) Proof:
It follows straight forward from definition of F E [cf. Eq. 36] and the Identity of indiscernibles propertyof D E (cf. Sec. IV.1).3. Symmetry : F E ( ρ, σ ) = F E ( σ, ρ ) (39) Proof:
It follows straight forward from definition of F E [cf. Eq. 36] and the Symmetry property of D E (cf. Sec.IV.1).4. Joint Concavity : For p i ≥ P i p i = 1, ρ i and σ i arbitrary density matrices F E ( X i p i ρ i , X i p i σ i ) (40) ≥ X i p i F E ( ρ i , σ i ) (41) Proof:
It follows immediately from definition of F E [cf.Eq. 36] and the property of Joint Convexity of D E (cf.Sec. IV.1). Remark:
While Uhlmann-Jozsa fidelity F has theproperty of being separate concave in each of its argu-ments, F E turns out to have the enhanced Joint Con-cavity property. Therefore, separate concavity on eachof its arguments is also satisfied.5.
Restricted additivity : F E ( ρ ⊗ τ, σ ⊗ τ ) = F E ( ρ, σ ) (42) Proof:
It follows straight forward from definition of F E [cf. Eq. 36] and the Restricted additivity property of D E (cf. Sec. IV.1). Comment:
As a consequence of this property, a mea-sure of the degree of similarity between two physi-cal systems described by density matrices ρ and σ bymeans of F E remains unchanged even after appendingto each system an uncorrelated ancillary state τ .6. Unitary invariance : For any unitary operation U F E ( U ρ U † , U σ U † ) = F E ( ρ, σ ) (43) Proof:
It follows straight forward from definition of F E [cf. Eq. 36] and the Unitary invariance propertyof D E (cf. Sec. IV.1).7. Monotonicity under quantum operations : F E ( E ( ρ ) , E ( σ )) ≥ F E ( ρ, σ ) (44)where E is a CPTP-map. Proof:
It follows from definition of F E [cf. Eq. 36] asa direct consequence of the property of Monotonicityunder quantum operations of D E (cf. section IV.1). Comment:
Physically, this property means that, as F E serves as a kind of measure for the degree of similar-ity between two quantum states ρ and σ , one mightexpect that a general quantum operation E will makethem less distinguishable and, therefore, more similaraccording to F E . Thus, this property qualifies F E as amonotonically increasing measure under CPTP-maps. VI. METRIC TO MEASURE DISTANCESBETWEEN QUANTUM PROCESSES BASED ONTHE METRIC D E From the distance D E it is possible to introduce a distance∆ E between quantum processes. Following Gilchrist, Lang-ford and Nielsen [47], we define the distance ∆ E between thequantum processes E and F as:∆ E ( E , F ) ≡ D E ( ρ E , ρ F ) (45)where ρ E and ρ F are the Jamio lkowski isomorphismscorresponding to the quantum processes E and F [cf. Eq.7].The fundamental properties of ∆ E are presented below. Itis easy to see that the properties of Normalization and
Sym-metry of ∆ E are inherited from the corresponding propertiesof D E .1. Normalization : 0 ≤ ∆ E ( E , F ) ≤ Identity of indiscernibles :∆ E ( E , F ) = 0 if and only if E = F (47) Proof:
It can be proved recalling the Jamio lkowski iso-morphism (cf. Sec. III.2) and definition of ∆ E ( E , F )[cf. Eq. 45]. Thus, there exist an univocal relationshipbetween a quantum process E and the Jamio lkowskistate ρ E . Therefore, if E 6 = F it follows that ρ E = ρ F and ∆ E satisfies property 2.3. Symmetry : ∆ E ( E , F ) = ∆ E ( F , E ) (48)4. Triangle inequality : For any three quantum processes E , F and G ∆ E ( E , G ) ≤ ∆ E ( E , F ) + ∆ E ( F , G ) (49) Proof:
We start from the metric character of D E (cf.Sec. IV.1). Thus, for given processes E , F and G withtheir corresponding Jamio lkowski states ρ E , ρ F and ρ G (cf. Sec. III.2), we have:∆ E ( E , F ) + ∆ E ( F , G ) − ∆ E ( E , G ) = (50) D E ( ρ E , ρ F ) + D E ( ρ F , ρ G ) − D E ( ρ E , ρ G ) ≥ Stability :∆ E ( I ⊗ E , I ⊗ F ) = ∆ E ( E , F ) (52)where I represents the identity operation on an extraHilbert of arbitrary dimension. Proof:
We start from definition of ∆ E ( E , F ) [cf. Eq.45] and use the property of restricted additivity of D E (cf. Sec. IV.1). Thus, we have:∆ E ( I ⊗ E , I ⊗ F ) = D E ( ρ I⊗E , ρ
I⊗F ) (53)= D E ( ρ I ⊗ ρ E , ρ I ⊗ ρ F ) (54)= D E ( ρ E , ρ F ) = ∆ E ( E , F ) (55)In last equation we used the useful property ρ E⊗F = ρ E ⊗ ρ F [47].6. Chaining : For any quantum processes E , E , F and F ∆ E ( E ◦ E , F ◦ F ) ≤ ∆ E ( E , F ) + ∆( E , F ) (56) Proof:
We use the contractivity property of D E and, addi-tionally, we assume that F is doubly stochastic, i.e., F istrace-preserving and satisfies F ( I ) = I (cf. Sec. III.1). Thisis not a significant assumption, since in quantum informa-tion science we are typically interested in the case when F and F are ideal unital processes, and we want to use ∆ E tocompare the composition of these two ideal processes to theexperimentally realized process E ◦ E .The proof of the chaining property starts by applying prop-erty 4, i.e., triangle inequality , so we have:∆ E ( E ◦ E , F ◦ F ) = D E ( ρ E ◦E , ρ F ◦F ) ≤ D E ( ρ E ◦E , ρ E ◦F ) + D E ( ρ E ◦F , ρ F ◦F ) (57)Then, we note the easily verified indentity ρ E◦F = [ F T ⊗E ] ρ Ψ [47], where ρ Ψ = | Ψ ih Ψ | with | Ψ i being the the maximallyentangled state [cf. Eq. 8]. Next, we define F T ( ρ ) = P j F Tj ρF ∗ j , where F j are the operation elements for F [cf.Eq. 4]. Applying this identity to both density matrices inthe second term on the right-hand side of Eq. 57 we obtain∆ E ( E ◦ E , F ◦ F ) ≤ D E ( ρ E ◦E , ρ E ◦F )+ D E (cid:0) [ F T ⊗ E ] ρ Ψ , [ F T ⊗ F ] ρ Ψ (cid:1) (58)The double stochasticity of F implies that F T is a trace-preserving quantum operation. Then, to complete the proof, we can apply the contractivity property of D E to both thefirst and the second terms on the right-hand side of Eq. 58.∆ E ( E ◦ E , F ◦ F ) ≤ D E ( ρ E , ρ F ) + D E ( ρ E , ρ F )(59)= ∆ E ( E , F ) + ∆( E , F ) (60)In addition, since unitary processes are also doubly stochas-tic, it follows that chaining holds for ∆ E in most cases ofusual interest. Remark : Some interesting properties can be derived from thepreceding ones. For example, from the metric and chainingproperties it is possible to show that [47]∆ E ( R ◦ E , R ◦ F ) ≤ ∆ E ( E , F ) (61)where R is any quantum operation. Physically, this meansthat postprocessing E by R cannot increase the distinguisha-bility of two processes E and F . Another interesting conse-quence of the metric and chaining criteria is the property of unitary invariance , i.e.,∆ E ( U ◦ E ◦ V , U ◦ F ◦ V ) = ∆ E ( E , F ) (62)where U and V are arbitrary unitary operations [47]. VII. CONCLUDING REMARKS
The main results of this paper are concerned with theproperties of the entropic metric D E between quantumstates proposed in Ref. [45]. Our results indicate that D E and the derived metric ∆ E show interesting and usefulproperties to measure distances between quantum statesand quantum processes, respectively. These properties, ingeneral, do not depend on the particular quantum systemor process to be considered (as it was emphasized in Sec.VI). In addition, we derived an alternative measure offidelity F E between quantum states which present themost important properties of the Uhlmann-Jozsa fidelity F such as normalization , Symmetry and
Monotonicityunder quantum operations . Moreover, the derived fidelity F E shows the enhanced property of Joint Concavity withrespect to the fidelity F which present the property of Separate Concavity . Regarding practical calculations ofthe metric D E , in Ref. [45] it is shown how to apply thismetric to calculate the distance between a mixed qubitand the resulting state when this qubit is sent througha depolarizing channel. Besides, from an experimentalviewpoint, it is important to mention that an experimentalrealization of a theoretical purification protocol [63] hasbeen already achieved in the case of photons sent through adepolarizing channel [62]. These results are very promisingbecause they open a window to think of the possibilityof using D E directly from purifications of quantum statesexperimentally obtained. Furthermore, it is important tomention that Roga, Fannes and ˙Zyczkowski already founda finest bound for the Holevo quantity which turns out tobe the square of the D E metric [61]. These facts encourageus to continue investigating how to apply this metric todifferent cases of interest beyond the depolarizing channel.This task is currently in progress. Certainly, the possibilityof evaluating D E in as many contexts as possible is ofcentral importance. Particular applications to quantumnoisy channels represented by sums of operators belongingto Pauli group will be also an interesting matter of study[12, 19]. Applications to topological insulators will be alsomatter of consideration [66–68]. Some advances in thecontext of quantum operations written in the operator-sumrepresentation [12] have been made and the results will bepresented elsewhere. APPENDIX: UHLMANN-JOZSA FIDELITY
In literature, the Uhlmann-Jozsa fidelity F is a celebratedand widely used measure of the degree of similarity betweentwo general density matrices. Fidelity F is given by [22, 23] F ( ρ, σ ) = (cid:20) Tr (cid:18)q √ ρσ √ ρ (cid:19)(cid:21) (63)where ρ and σ are arbitrary density matrices.An equivalent definition of F can be provided in terms ofpurifications of the states ρ and σ [23] F ( ρ, σ ) = max | ϕ i |h ψ | ϕ i| (64)where | ψ i is any fixed purification of ρ and maximization isperformed over all purifications | ϕ i of σ .For easy access, we summarize below the most appealingproperties of Uhlmann-Jozsa fidelity F and adequateproperties names and definitions according to the contextof the present work. These properties are used in Secs. IV.1and VI to analyze the properties of the distance D E andthe distance between quantum processes ∆ E that we willintroduce in this work. For proofs of the properties listedhere see, for example, Refs. [12, 23].1. Normalization: ≤ F ( ρ, σ ) ≤ Identity of indiscernibles: F ( ρ, σ ) = 1 if and only if ρ = σ (66)3. Symmetry: F ( ρ, σ ) = F ( σ, ρ ) (67)4. If ρ = | ξ ih ξ | represents a pure state then F ( ρ, σ ) = h ξ | σ | ξ i = Tr( ρ σ )5. Separate Concavity:
For p , p ≥ p + p = 1, andarbitrary density matrices ρ , ρ and σF ( p ρ + p ρ , σ ) ≥ p F ( ρ , σ ) + p F ( ρ , σ ) (68)By symmetry property 3, concavity in the second ar-gument is also fulfilled.6. Multiplicativity under tensor product:
For arbitrarydensity matrices ρ , ρ , σ and σ F ( ρ ⊗ ρ , σ ⊗ σ ) = F ( ρ , σ ) F ( ρ , σ ) (69)7. Unitary invariance:
For any arbitrary unitary process U , F ( ρ, σ ) is preserved i.e., F ( U ρ U † ) , U σ U † ) = F ( ρ, σ ) (70)8. Monotonicity under quantum operations:
For a generalquantum operation E described by a CPTP-map (cf.section III.1) F ( E ( ρ ) , E ( σ )) ≥ F ( ρ, σ ) (71) Remark 1 : The fidelity F serves as a generalized measure ofthe overlap between two quantum states but is not a metric.However, the fidelity can easily be turned into a metric. Forexample, the Bures distance is a metric which can be definedin terms of the fidelity F as [21]: D B ( ρ, σ ) = q − p F ( ρ, σ ) (72) Remark 2 : While F satisfies separate concavity it can beshown that √ F is jointly concave [64, 65] i.e., p F ( p ρ + p ρ , p σ + p σ ) ≥ p p F ( ρ , σ ) + p p F ( ρ , σ ) (73)where p , p ≥ p + p = 1, and ρ , ρ , σ and σ arearbitrary density matrices. Remark 3 : Clearly, by extension, √ F satisfies all propertiesof the fidelity F but property 4. Remark 4 : It is important to realize that any measure M which is unitarily invariant , jointly concave (convex) , and invariant under the addition of an ancillary system is alsomonotonically increasing (decreasing) under CPTP-maps[65], therefore, it turns out to be a suitable measure of thedegree of similarity between quantum states.0 ACKNOWLEDGMENTS
T.M.O and P.W.L are fellows of the National Research Coun-cil of Argentina (CONICET). The authors are grateful toSecretaria de Ciencia y T´ecnica de la Universidad Nacionalde C´ordoba (SECyT-UNC, Argentina) for financial support. [1] A. O. Pittenger,
An Introduction to Quantum ComputingAlgorithms , (Birkh¨auser, Boston, 2000)[2] D. Bouwmeester, A. K. Ekert and A. Zeilinger (Eds.),
ThePhysics of Quantum Information: Quantum Cryptography,Quantum Teleportation, Quantum Computation , (Springer,2000)[3] J. Kempe and T. Vidick, T.,
Quantum Algorithms , Lect.Notes Phys. 808, 309342 (Springer, 2010)[4] D. Deutsch, Proc. R. Soc. Lond. A , 97 (1985).[5] D. Deutsch and R. Jozsa, Proc. R. Soc. Lond. A , 553(1992).[6] P. W. Shor, Proceedings of the 35th IEEE Symposium onFoundations of Computer Science, 124 (1994).[7] L. K. Grover, Proceedings of the Twenty-Eighth AnnualACM Symposium on Theory of Computing, 212 (1996)(arXiv:quant-ph/9605043)[8] A. Ekert and R. Jozsa, Rev. Mod. Phys. , 733 (1996).[9] P. W. Shor, SIAM Journal on Scientific and Statistical Com-puting , 1484 (1997).[10] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Proc. R.Soc. Lond. A , 339 (1998).[11] A. M. Childs and W. van Dam, Rev. Mod. Phys. , 1 (2008).[12] M. A. Nielsen and I. L. Chuang, Quantum Computation andQuantum Information , (Cambridge University Press, Cam-bridge, United Kingdom, 2000).[13] P.W. Shor, Phys. Rev. A , R2493 (1995).[14] A. M. Steane, Phys. Rev. Lett. , 793 (1996).[15] D. Gottesman, Stabilizer Codes and Quantum Error Correc-tion , 2 (2003).[18] H. Bombin and M. A. Martin-Delgado, Phys. Rev. Lett. ,180501 (2006).[19] H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber andM. A. Martin-Delgado6, Phys. Rev. X , 021004 (2012).[20] D. Gottesman, Physics , 50 (2012).[21] I. Bengtsson and K. ˙Zyczkowski, Geometry of QuantumStates: An Introduction to Quantum Entanglement , (Cam-bridge University Press, Cambridge, 2006).[22] A. Uhlmann, Rep. Math. Phys. , 273 (1976).[23] R. Jozsa, J. Mod. Opt , 2315 (1994).[24] D. J. C. Bures, Trans. Am. Math. Soc. , 199 (1969).[25] V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and A. Wun-sche, J. Mod. Opt. 47, 633 (2000).[26] S. Luo and Q. Zhang, Phys. Rev. A , 032106 (2004).[27] A. P. Majtey, P. W. Lamberti, and D. P. Prato, Phys. Rev.A , 052310 (2005). [28] E. B. Davies, The Theory of Open Quantum Systems (Aca-demic Press, London, 1976).[29] H.-P. Breuer and F. Petruccione,
The Theory of Open Quan-tum Systems (Oxford University Press, Oxford, 2002).[30] P. Pechukas, Phys. Rev. Lett. , 1060 (1994).[31] P. Pechukas, Phys. Rev. Lett. , 3021 (1995).[32] P. Stelmachovic and V. Buzek, Phys. Rev. A , 062106(2001).[33] N. Boulant, J. Emerson, T. F. Havel, D. G. Cory, and S.Furuta, J. Chem. Phys. , 2955 (2004).[34] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev.A , 052110 (2004).[35] K. M. F. Romero, P. Talkner, and P. H¨anggi, Phys. Rev. A , 052109 (2004).[36] A. Smirne, H.-P. Breuer, J. Piilo, and B. Vacchini, Phys.Rev. A , 062114 (2010).[37] H.-T. Tan and W.-M. Zhang, Phys. Rev. A , 032102(2011).[38] M. Ban, S. Kitajima, and F. Shibata, Phys. Lett. A ,2283 (2011).[39] V. G. Morozov, S. Mathey, and G. R¨opke, e-printarXiv:1106.5654.[40] C.-F. Li, J.-S. Tang, Y.-L. Li, and G.-C. Guo, Phys. Rev. A , 064102 (2011).[41] A. Smirne, D. Brivio, S. Cialdi, B. Vacchini, and M. G. A.Paris, Phys. Rev. A , 032112 (2011).[42] J. Dajka and J. Luczka, Phys. Rev. A , 012341 (2010).[43] E.-M. Laine, J. Piilo, and H.-P. Breuer, Europhys. Lett. ,60010 (2010).[44] J. Dajka, J. Luczka, and P. H¨anggi, Phys. Rev. A , 032120(2011).[45] P. W. Lamberti, M. Portesi, and J. Sparacino, Int. J. Quan-tum Inf. , 1009 (2009).[46] Notation remark: In order to keep consistency with othernotations found in literature, the notation for the metric D N introduced in reference [45] was changed to D E in this work.[47] A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys. Rev.A , 062310 (2005).[48] In literature is jargon to use the term quantum operation inphysically motivated contexts, whereas the terms, completelypositive trace-preserving map (CPTP-map) and superopera-tor are normally used in more mathematical contexts.[49] Gilchrist, Langford and Nielsen [47] also propose two morecriteria for a good measure of distance between quantum pro-cesses: (i) Easy to calculate and (ii)
Easy to measure . How-ever, as we are interested in as broad applications as possibleof the metric ∆ E analyzed in this work (both theoretical andexperimental), for example, as a bound of what can be phys-ically feasible for a given process, or as a diagnostic measureto evaluate a quantum protocol experimentally implemented,we will regard these two extra criteria as not exclusive. [50] D. Aharonov, A. Kitaev, and N. Nisan, Proceedings of thethirtieth annual ACM Symposium on Theory of Computing(STOC), pp20-30 (1998), arXiv:quant-ph/9806029.[51] A. Galindo and M. A. Mart´ın-Delgado, Rev. Mod. Phys. ,347 (2002)[52] G. Jaeger, Quantum Information: An Overview , (Springer,2007).[53] K. Kraus, Ann. Phys. , 311 (1971)[54] K. Kraus, States, Effects, and Operations: Fundamental No-tions of Quantum Theory , (Springer-Verlag, Berlin, Heidel-berg 1983).[55] Post-selection is usually associated with obtainig some spe-cific outcome after a measurement procedure is performed.[56] A. Jamio lkowski, Rep. Math. Phys. , 275 (1972)[57] V. Vedral, Introduction to Quantum Information Science ,(Oxford University Press, Oxford, 2006).[58] An important remark, however, is that not all density matri-ces may be realized as states ρ E . It turns out that the class E of quantum states that may arise in this way from a trace-preserving operation is simply all those ρ E whose reduceddensity matrix on the copy of the original system is the com-pletely mixed state, I/d [69]. This fact does not constitutes a drawback for the purpose of this paper since we are inter-ested on mapping a quantum process E into a density matrix ρ E and not the converse.[59] P. W. Lamberti, A. P. Majtey, A. Borras, M. Casas, and A.Plastino, Phys. Rev. A , 052311 (2008).[60] J. Bri¨et and P. Harremo¨es, Phys. Rev. A , 052311 (2009).[61] W. Roga, M. Fannes, and K. ˙Zyczkowski, Phys. Rev. Lett. , 040505 (2010).[62] M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiur´aˇsek,and C. Macchiavello, Phys. Rev. Lett. , 170501 (2004).[63] J. I. Cirac, A. K. Ekert, and C. Macchiavello, Phys. Rev.Lett. , 4344 (1999).[64] A. Uhlmann, Rep. Math. Phys. , 407 (2000).[65] P. E. M. F. Mendon¸ca, R. d. J. Napolitano, M. A. Marchiolli,C. J. Foster, and Y. C. Liang, Phys. Rev. A , 052330(2008).[66] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010)[67] X-L. Qi and S-C. Zhang, Rev. Mod. Phys. , 1057 (2011)[68] O. Viyuela, A. Rivas and M. A. Martin-Delgado and M. A.Martin-Delgado6, Phys. Rev. B , 155140 (2012).[69] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.A60