Coherence generating power of quantum unitary maps and beyond
CCoherence generating power of quantum unitary maps and beyond
Paolo Zanardi, Georgios Styliaris, and Lorenzo Campos Venuti
Department of Physics and Astronomy, and Center for Quantum Information Science & Technology,University of Southern California, Los Angeles, CA 90089-0484
Given a preferred orthonormal basis B in the Hilbert space of a quantum system we define a measure of thecoherence generating power of a unitary operation with respect to B . This measure is the average coherencegenerated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical formin any dimension and provide an operational protocol to directly detect it. We characterize the set of unitarieswith maximal coherence generating power and study the properties of our measure when the unitary is drawn atrandom from the Haar distribution. For large state-space dimension a random unitary has, with overwhelmingprobability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to generalunital quantum operations and the relation to the concept of asymmetry are discussed. Introduction:–
One of the most fundamental attributes ofquantum dynamical systems is their ability to exist in linearsuperpositions of different physical states. In fact any purequantum state can be regarded, in infinitely many differentways, as a linear superposition of a basis of distinguishablequantum states. The experimental signature of such a super-position structure (in the given basis) is known as quantumcoherence [1]. The latter is also known as one the basic ingre-dients for quantum information processing [2] and its protec-tion e.g., by decoherence-free subspaces [3–5], is one of thefundamental challenges in the field.Over the last few years we have witnessed a strong renewalof interest in the quantitative theory of coherence [6, 7]. Thisis partly practically motivated by the role that quantum co-herence plays in quantum metrological protocols (see e.g.,discussion in [8]) and, on a more conceptual ground, by itsrelation to the general resource theory of asymmetry [9–11].Quantum coherence is also believed to play a role in somefundamental biological process [12–14] as well as in quan-tum thermodynamics [15, 16]. The general idea is that onecan quantify quantum coherence by introducing a real-valuedfunction over the quantum state-space, a coherence measure ,such that it vanishes for all the states that are deemed to be incoherent and cannot increase under some class of opera-tions that preserve incoherence [17]. Even if a preferred basisis chosen the choice of the coherence measure is not uniqueand different options have been discussed in the literature[6, 8, 18, 19].In this paper we address a closely related problem, whichwas first tackled in [20]: the quantification of the power of aquantum operation to generate coherence. Again, even whenan underlying coherence measure is assumed, the definitionof the coherence generating power (CGP) of a CompletelyPositive (CP)-map is not unique and different lines of attackare possible [20–22] (see Sect IV C of [7] for a comprehensivelist of references). All of these approaches, however, are castin terms of an optimization problem that is extremely hard tohandle for generic channels in arbitrary dimensions.Following the spirit of Ref. [23] in entanglement theory, weshall here pursue a different strategy based on probabilisticaverages. We define the CGP of a map as the average coher-ence that is generated when the corresponding quantum opera- D B S S | ϕ + ⟩ ⟨ ϕ + || ϕ + ⟩ ⟨ ϕ + || ϕ + ⟩ ⟨ ϕ + | D B UU ⟨ ⟩ SD B D B Figure 1. Protocol for the direct detection of the Coherence Gen-erating Power (CGP) Eq. (2) of the unitary CP map U based onEq. (7). Here D B is the dephasing super-operator for the preferredbasis B , the measurement of the swap operator is denoted by S and | Φ + (cid:105) := d − / (cid:80) di =1 | i (cid:105) ⊗ . tion is performed over a suitable input ensemble of incoherentstates. We shall here firstly focus on unitary maps and intro-duce a definition of CGP based on a uniform ensemble (seebelow for a precise definition) of incoherent states.Our measure of CGP is analytically computable for arbi-trary unitary map in any dimension. It also enjoys several nat-ural and desirable properties e.g., invariance under pre- andpost-processing by incoherent unitaries. We shall present asimple operational protocol for the direct detection of the CGPof a given map which does not involve the ensemble gener-ation or quantum process tomography [24, 25]. The set ofunitary operations with maximal CGP is easily characterizedand some universal statistical properties of our measure overthe group of unitaries can be established rigorously. We willalso provide some numerical study of the distribution of CGPin various dimensions d (for d = 2 analytical form is avail-able). Finally, extensions of CGP to arbitrary unital operationsare discussed as well as the connection to the broader conceptof asymmetry generating power of a map. The proofs of thePropositions can be found in [26]. Preliminaries:–
Let B = {| i (cid:105)} di =1 be an orthonormal basisin the Hilbert space H ∼ = C d . Given B one has the associ-ated B -dephasing map over L ( H ) given by X (cid:55)→ D B ( X ) = (cid:80) di =1 | i (cid:105)(cid:104) i | (cid:104) i | X | i (cid:105) [27]. The dephasing map D B can be real-ized physically as the measurement CP map associated to anynon degenerate observable H diagonal in the basis B . For any B, the dephasing map is an orthogonal projection over L ( H ) a r X i v : . [ qu a n t - ph ] N ov Figure 2. Probability distribution densities (PDD) of the normalized ˜ C B ( U ) for d = 2 , . . . , . An ensemble of Haar-distributed U ’s hasbeen generated numerically. For d = 2 the analytical form of thePDD is P CGP ( c ) = (1 − c ) − / (see [26]). equipped with the standard Hilbert-Schmidt scalar product (cid:104) X, Y (cid:105) := tr( X † Y ) . We will denote by Q B := 1I − D B the complementary projection of D B . Naturally, one defines B -incoherent operators (states) as operators (states) that arediagonal in the preferred basis B . Definition 1.–
The set of B -incoherent operators is therange of the B -dephasing map, i.e., Im D B . We will denotethe set of B -incoherent states ρ ( ρ ≥ , tr ρ = 1 ) by I B .From the point of view of this definition one can say that D B ( Q B ) projects an operator onto its incoherent (coherent)component. The set I B is clearly isomorphic to a ( d − -dimensional simplex spanned by convex combinations of the | i (cid:105)(cid:104) i | , i = 1 , . . . , d . CP maps (all such maps are assumedto be trace preserving in this paper) T mapping I B into itselfwill play a distinguished role in this paper. A necessary andsufficient condition for the invariance of I B under T is givenby T D B = D B T D B [8]. However, in this paper we willadopt a slightly stronger invariance condition. Definition 2.–
A CP map T on L ( H ) will be called B -incoherent iff [ T , D B ] = 0 . We will write T ∈ CP B .Note that B -incoherent maps leave both the subspace of B -incoherent operators and its orthogonal complement ( ∼ =Ker D B = Im Q B ) invariant. Let us first establish the fol-lowing, almost obvious, fact. Proposition 1.–
A unitary CP map U ( X ) = U XU † (with U unitary) is B -incoherent iff U | i (cid:105) = η i | σ U ( i ) (cid:105) where σ U is a ( U -dependent) permutation of { , . . . , d } and the η i ’s are U (1) -phases. B -incoherent unitary maps form a subgroup of CP B . Measures of coherence generating power:–
Loosely speak-ing a coherence measure is a way to quantify how far a givenstate is from being incoherent, moreover this quantification isrequested to fulfill some natural properties. More precisely,let us consider the function ˜ c B ( ρ ) := (cid:107) ρ − D B ( ρ ) (cid:107) = (cid:107)Q B ( ρ ) (cid:107) ( (cid:107) X (cid:107) denotes the 1-norm of X , i.e. the sum of thesingular values of X ); this is vanishing iff ρ is B -incoherent.Moreover if T ∈ CP B then ˜ c B ( T ( ρ )) = (cid:107)Q B T ( ρ ) (cid:107) = (cid:107)T Q B ( ρ ) (cid:107) ≤ (cid:107)Q B ( ρ ) (cid:107) = ˜ c B ( ρ ) , where we have usedDefinition 2 and the monotonicity of the -norm under generalCP maps. These remarks show that c B is a good coherencemeasure with respect to B -incoherent operations [8]. Unfor- tunately the -norm is hard to handle therefore in this paper wewill adopt the Hilbert-Schmidt -norm (cid:107) X (cid:107) = (cid:112) (cid:104) X, X (cid:105) .We define the function c B ( ρ ) := (cid:107)Q B ( ρ ) (cid:107) . (1)Again, it is immediate to see that c B vanishes iff ρ ∈ I B and ˜ c B ( ρ ) ≤ (cid:112) d c B ( ρ ) . On the other hand it is now nottrue that c B is necessarily non-increasing under general B -incoherent CP maps (as the -norm does not have that prop-erty either). However if T is unital i.e., T (1I) = 1I , then (cid:107)T ( X ) (cid:107) ≤ (cid:107) X (cid:107) [28]. Thereby the desired monotonicityproperty is recovered if one restricts to the set of unital B -incoherent CP maps.Let us now introduce the main novel concept of this paper Definition 3.–
The coherence generating power (CGP) C B ( U ) of a unitary CP map X (cid:55)→ U ( X ) := U XU † , ( U ∈ U ( H )) with respect the basis B is defined as C B ( U ) := (cid:104) c B ( U off ( | ψ (cid:105)(cid:104) ψ | )) (cid:105) ψ (2)where U off := Q B UD B and the average over ψ is taken ac-cording to the Haar measure.The operational idea behind our definition (2) of CGP issimple: the power of a unitary U to generate coherence (in apreferred basis B ) is given by the average coherence, as mea-sured by the function (1), obtained by U acting over an ensem-ble of incoherent states. The latter is prepared by a stochas-tic process that involves first the generation of (Haar) randomquantum states, and then their B -dephasing e.g., by perform-ing a non-selective measurement of any non-degenerate B -diagonal observable. Note that the ensemble so generated co-incides with the uniform one over the simplex I B (see [26]).Of course other definitions are possible. For example, besidesthe freedom of choosing a coherence measure different from(1), one might have resorted to a different ensemble of B -diagonal states or even replace the average by a supremumover the ensemble [20–22]. However, our choice, thanks tothe high symmetry of the Haar measure, will allow us to es-tablish properties of CGP on general grounds as well as tocompute it in an explicit analytic fashion. The most basicproperties of the CGP can be derived directly from Eq. (2). Proposition 2.– a) C B ( U ) ≥ and C B ( U ) = 0 iff U ∈ CP B . b) If W is a unitary such that
W ∈ CP B then C B ( U W ) = C B ( W U ) = C B ( U ) . c) Let {| ˜ i (cid:105) := V | i (cid:105)} di =1 be a new basis ˜ B := BV obtained from B by the (right) ac-tion of the unitary V then: C BV ( U ) = C B ( V † U V ) .Part b) shows that CGP does not change if the ensembleis pre- or post-processed by incoherent unitaries. Moreover,Part c) shows that computing the CGP for a single given ba-sis B is in principle sufficient for obtaining it for any basis B (for, given any pair of bases, there is always a unitary connect-ing them). It also implies, as we will see, that the statisticalproperties of the CGP over the unitary group are universal inthe sense of being basis independent: just the Hilbert spacedimension d matters.It is important to stress that Prop. 2 holds for a more gen-eral choice of c B than Eq. (1) e.g., for ˜ c B [29]. The choice c P C G P ( c ) d = Figure 3. Probability distribution density (PDD) of the normalized ˜ C B ( U ) for d = 40 . A Gaussian fit is superimposed on the numeri-cally generated PDD to highlight the central-limit type behavior. of the Hilbert-Schmidt norm in the definition of CGP, onthe other hand, while imposing the somewhat severe unital-ity constraint, has the great advantage of allowing one for an explicit computation of C B ( U ) . Proposition 3.–
Let | Φ + (cid:105) = 1 / √ d (cid:80) di =1 | i (cid:105) ⊗ be the max-imally entangled d × d singlet, then: a) C B ( U ) = 1 d + 1 [1 − tr ( Sω B ( U ))] , (3)where ω B ( U ) := ( D B UD B ) ⊗ ( | Φ + (cid:105)(cid:104) Φ + | ) and S = (cid:80) di,j =1 | ij (cid:105)(cid:104) ji | is the swap operator over H ⊗ ; b) tr ( Sω B ( U )) = 1 /d (cid:80) di,j =1 |(cid:104) i | U | j (cid:105)| ; c) C B ( U ) ≤ − /dd +1 =: C d . The upper bound is saturated iff |(cid:104) i | U | j (cid:105)| =1 /d ( ∀ i, j ) .Part c) of Prop. 3 above shows the fact that for U tobe a unitary with maximal CGP the base B and the base BU := { U | i (cid:105)} di =1 have to be mutually unbiased [30,31]. For example the unitary U such that (cid:104) h | U | m (cid:105) =1 / √ d exp( i πd hm ) , ( h, m = 1 , . . . , d ) has maximal CGP.We also remark that from a) and b) above it follows easilythat C B ( U ) = C B ( U † ) .Eq. (3) naturally leads to an operational protocol for thedetection of the CGP of a unitary U which does not require thegeneration of a Haar distributed ensemble of states or quantumprocess tomography [24, 25]. Protocol for CGP detection: 1)
Prepare | Φ + (cid:105) ; B -dephase both subsystems; ) Apply U to both subsystems; B -dephase again both subsystems; Measure the expecta-tion value of the observable S ; Plug the obtained value inEq. (3). This protocol is depicted in Fig. (1). Since D ⊗ B ( | Φ + (cid:105)(cid:104) Φ + | ) = 1 d d (cid:88) i =1 | i (cid:105)(cid:104) i | ⊗ =: ρ B , (4)steps 1) and 2) above can be replaced by 1’) Prepare the max-imally classically B -correlated state ρ B (for which is enoughto B -dephase one subsystem). This shows that entanglementis not really needed in the detection of C B ( U ) . However, in5) one is required to measure S which involves non-trivial in-teractions between the two d -dimensional subsystems. This isthe experimentally more challenging part of the protocol. No-tice, however, that for two-qubits, this amounts to a standardBell’s basis measurement. CGP as a random variable over the unitary group:–
Wenow investigate some of the properties of the CGP of Eq. (3)seen as a random variable over the unitary group U ( H ) equipped with the Haar measure dµ ( U ) . Proposition 4.– a)
The probability distribution density P CGP ( c ) := (cid:82) U ( H ) dµ ( U ) δ ( c − C B ( U )) for the CGP Eq. (3)is independent of B . b) The first moment is given by (cid:104) C B ( U ) (cid:105) U = (cid:90) dcP CGP ( c ) = d − d + 1) . (5) c) Let us define the normalized CGP ˜ C B ( U ) := C B ( U ) /C d ≤ then (cid:104) ˜ C B ( U ) (cid:105) U = (1 + 1 /d ) − . UsingLevy’s lemma for unitaries [32] one obtains Prob (cid:16) ˜ C B ( U ) ≥ − /d / (cid:17) ≥ − exp (cid:16) − d / / (cid:17) . (6)Eq. (6) shows that in high-dimension a random unitary willhave, with overwhelming probability, nearly maximal CGP.In Fig. 2 are reported numerical simulations of the probabil-ity distribution function of ˜ C B ( U ) for Haar distributed U indifferent dimensions. In particular numerics shows that thevariance of ˜ C B ( U ) is O (1 /d ) (see [26]). Moreover, Fig. 3shows that for large Hilbert space dimension d a central-limittype behavior emerges and the P CGP can be well approxi-mated by a normal distribution.
Beyond Unitarity and finite dimensions:–
In this sectionwe will briefly discuss how our approach extends to CP mapsthat are not necessarily unitary and how one might extend ourformalism to infinite dimensions e.g., optical modes. Sincewe still would like to employ the Hilbert-Schmidt norm wewill focus here on unital maps. We can still adopt Eq. (1) forthe definition of a coherence measure. Moreover, incoherent(according to Def. 2) E will not increase it. We can now definethe CGP of E by the same Eq. (2) (with E replacing U ). It isstill true that C B ( E ) = 0 ⇔ E off := Q B ED B = 0 but this is ( a ) ( b ) ( c ) U U U U U U U U U Figure 4. Coherence generating power for convex combinationsof unitaries of the form E ( · ) = (cid:80) k =1 p k U k · U k † . One alwayshas the convexity inequality ˜ C B ( E ) ≤ (cid:80) k =1 p k ˜ C B ( U k ) as notedin the main text. (a) Here d = 3 and we fixed U k such that ˜ C B ( U ) = 1 , ˜ C B ( U ) = 1 / and ˜ C B ( U ) = 0 . (b) For thisexample ( d = 10 ) U is the Discrete Fourier Transform matrix (cid:104) l | U | m (cid:105) = d − / exp ( ilm π/d ) while U ( U ) is obtained byinterchanging the first (last) 2 rows of U . All U k have maximalCGP (simplex vertices) but the CGP of mixtures can drop signifi-cantly. (c) This is a typical case for randomly chosen unitaries U k oflarge dimension (here d = 40 ). One observes that ˜ C B ( U k ) is nearlymaximal consistently with the concentration phenomenon. now a weaker property than incoherence as it does not imply [ E , D B ] = 0 . The corresponding measure of CGP is thereforenot faithful i.e., C B ( E ) = 0 ⇒ E ∈ CP B doesn’t hold (justthe converse does) [33]. The following proposition shows howProp. 3 generalizes to unital maps more general than unitaries. Proposition 5–
Let E ( · ) = (cid:80) k A k · A † k , ( (cid:80) k A † k A k = 1I) be a unital CP-map over L( H ) . If we define its CGP by Eq. (2)(replacing U with E ) then it follows that a) C B ( E ) ≥ andit vanishes if E is B -incoherent. b) If T is B -incoherent then C B ( T E ) ≤ C B ( E ) . c) C B ( E ) = 1 d + 1 [tr ( S ˜ ω B ( E )) − tr ( Sω B ( E ))] ≤ C d , (7)where ω B ( E ) := ( D B E ) ⊗ ( ρ B ) and ˜ ω B ( E ) := E ⊗ ( ρ B ) . d) C B ( E ) = [ d ( d + 1)] − (cid:80) di,l (cid:54) = m =1 | (cid:80) k ( A k ) li ( A k ) ∗ mi | Property b) in Prop. 5 is the analog of Eq. (3) and can besimilarly interpreted by an operational protocol involving themeasurement of S over the states ω B ( T ) and ˜ ω B ( T ) . Pointd) above gives the CGP explicitly as a function of the matrixelements of the Kraus operators of E ; it corresponds to b) inProp. 3. We also note that the function E (cid:55)→ C B ( E ) is convex (since it is a convex combination of the convex functions E (cid:55)→ c B ( E off ( | ψ (cid:105)(cid:104) ψ | )) ∀| ψ (cid:105) ). It follows that the maximum CGPof a convex set of maps will be achieved over extremal points.This phenomenon can be seen in the in Fig. 4.Remarkably, Eq. (7) seems to suggest a natural way inwhich our results can be extended to infinite dimensions. Letus consider, for simplicity, the unitary case and normalizeEq. (3) by dividing by C d . Now sending d → ∞ the d -dependent pre-factor of CGP disappears and one is led to con-sider the expression ˜ C ( ∞ ) B ( U ) = 1 − tr ( Sω ( ∞ ) B ( U )) with ω ( ∞ ) B ( U ) = ( D B U ) ⊗ ( ρ ( ∞ ) B ) where ρ ( ∞ ) B is some infinite-dimensional generalization of the maximally classically B -correlated state Eq. (4). For example, for any λ ∈ (0 , , one could choose ρ ( ∞ ) B := (1 − λ ) (cid:80) ∞ i =0 λ i | i (cid:105)(cid:104) i | ⊗ [34].With this choice it is immediate to check that ˜ C ( ∞ ) B ( U ) = 0 iff U is incoherent and that post-processing with incoherentunitaries leaves the CGP invariant [35]. Developments in theinfinite-dimensional case will be presented elsewhere [36]. Asymmetry:–
Closely related to the theory of coherence isthe notion of asymmetry [9–11]. Given an observable H onesays that a state ρ (CP map E ) is H -symmetric ( H -covariant)iff [ H, ρ ] =: H ( ρ ) = 0 ( [ H , E ] = 0 ). An asymmetry measure is a real valued function a H ( ρ ) that vanishes over symmet-ric states and is non-increasing under covariant CP maps i.e., a H ( E ( ρ )) ≤ a H ( ρ ) [11]. Following the main idea of this pa-per one could define the asymmetry generating power (AGP)of a CP map E by A H ( E ) := (cid:104) a H ( E ( ω )) (cid:105) ω where the averageis performed over a suitable ensemble of H -symmetric states ω . First results in this direction and connection between AGPand the CGP defined in this paper are discussed in [26]. Conclusions:–
In this paper we have discussed a way toquantify the coherence generating power (CGP) of a quan-tum operation. As a coherence measure we have convenientlyadopted the Hilbert-Schmidt norm of the coherent part of a quantum state. Our approach is to look at the average coher-ence produced when the operation is performed over a uni-form ensemble of input incoherent states. The input ensembleis obtained by dephasing, with respect to the chosen basis, anensemble of pure states distributed according the Haar mea-sure and coincides with the uniform measure over the simplexof states spanned by the pure basis states.Under these assumptions one obtains an analytically com-putable measure of CGP for arbitrary unital operations in anydimension. Operational protocols for the direct detection ofCGP have been described. Neither the ability to generate theHaar distributed input ensemble nor quantum process tomog-raphy are required. We focused on unitary maps, character-ized those with maximal CGP, studied the distribution of thismeasure over the unitary group, both analytically and numer-ically. For unitary maps this distribution is universal (basisindependent) and for large Hilbert space dimension a central-limit type phenomenon emerges. A random unitary has, withoverwhelming probability, nearly maximal CGP. Finally, weextended our approach to quantify the power of an operationto generate a more general type of asymmetry.The analytical framework here established is particularlysuited for unital quantum maps. Going beyond unitality, fi-nite dimensionality, and extending to general resource theo-ries represent challenging tasks for future investigations.
Acknowledgements.-
This work was partially supported bythe ARO MURI grants W911NF-11-1-0268 and W911NF-15-1-0582. P.Z. thanks I. Marvian for introducing him to asym-metry measures and F.G.S. L. Brand˜ao for pointing out theright form of the Levy’s Lemma. [1] R. J. Glauber, Phys. Rev. , 2766 (1963).[2] M. A. Nielsen and I. L. Chuang,
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This condition is clearly a sufficient for B -incoherence ascommutativity of U and D B can be explicitly checked in astraightforward fashion. It is also necessary. Indeed if U is B -incoherent then U ( | i (cid:105)(cid:104) i | ) must be B -diagonal for all i ; be- cause of unitarity, it also must be a one-dimensional projectorwhence U ( | i (cid:105)(cid:104) i | ) has necessarily the form | j (cid:105)(cid:104) j | where | j (cid:105) is auniquely defined element j =: σ U ( i ) of B . The only degreesof freedom of U left are then U (1) -phases. The last statementof the proposition is evident. (cid:3) Equivalence of ensembles
Here we show that the ensemble constructed in the maintext coincides in fact with the uniform distribution over thesimplex spanned by the states | i (cid:105)(cid:104) i | , i = 1 , . . . , d . For any(measurable) function f the expectation value over the en-semble is given by (cid:104) f ( D B ( | ψ (cid:105)(cid:104) ψ | ) (cid:105) ψ . Calling ψ i = (cid:104) i | ψ (cid:105) and p i = |(cid:104) i | ψ (cid:105)| we can write it as (cid:104) f ( D B ( | ψ (cid:105)(cid:104) ψ | ) (cid:105) ψ = M (cid:90) dψ · · · (cid:90) dψ d ×× f ( p , . . . , p d ) δ (1 − d (cid:88) i =1 p i ) (8)where M is a normalization constant and dψ i = d Re( ψ i ) d Im( ψ i ) . Switching to polar coordinates one has dψ i = r i dr i dϑ i = dp i dϑ i / . Performing the integrationover the angles ϑ i we obtain (cid:104) f ( D B ( | ψ (cid:105)(cid:104) ψ | ) (cid:105) ψ = M (cid:48) (cid:90) dp · · · (cid:90) dp d ×× f ( p , . . . , p d ) δ (1 − d (cid:88) i =1 p i ) , (9)that is, the uniform measure over the simplex ( M (cid:48) is anothernormalization constant). Proof of Proposition 2 a) By definition the CGP is non-negative, moreover C B ( U ) = 0 implies U off ( | ψ (cid:105)(cid:104) ψ | ) = 0 , ∀| ψ (cid:105) , which in turnimplies that U off = UD B − D B UD B = 0 . This equa-tion, as remarked in the above, shows that Im D B is invari-ant under U but, since U is normal, also the orthogonal com-plement Ker D B is invariant. It follows that [ U B , D B ] = 0 that is what we wanted to prove. b) C B ( W U ) = C B ( U ) ( C B ( U W ) = C B ( U ) ) follows from the commutativity of W and Q B ( D B ) and the unitary invariance of the Hilbert-Schmidt norm (Haar measure). c) By definition of ˜ B = BV one finds D ˜ B = VD B V † , ( V ( · ) = V · V † ) inserting this rela-tion in Eq. (2) in the main text and using again unitary invari-ance of the Hilbert-Schmidt norm and of the Haar measureone completes the proof. (cid:3) Lemma If S is the swap operator over H ⊗ ( S = (cid:80) di,j =1 | ij (cid:105)(cid:104) ji | , H = span {| i (cid:105)} di =1 ) then: a) (cid:107) X (cid:107) =tr ( SX ⊗ X ) ; b) (cid:104)| ψ (cid:105)(cid:104) ψ | ⊗ (cid:105) ψ = [ d ( d + 1)] − (1I + S ) where the average is taken over Haar distributed ψ in H , (seee.g., [1]). Proof of Proposition 3 a) Using the
Lemma and Definition 3 one can immedi-ately write C B ( U ) = [ d ( d + 1)] − tr (cid:104) S U ⊗ off (1I + S ) (cid:105) .The first term in this expression is vanishing; indeed U ⊗ off (1I) = Q ⊗ B (1I) = 0 (the identity is a diagonaloperator for any B ). Using the fact that ∀ Y one has tr (cid:2) S Q ⊗ B Y (cid:3) = tr (cid:2) S (1I − D ⊗ B ) Y (cid:3) the second term can bewritten as tr (cid:104) S U ⊗ off ( S ) (cid:105) = tr (cid:2) S (1I − D ⊗ B )( UD B ) ⊗ ( S ) (cid:3) Moreover, D ⊗ B ( S ) = (cid:80) di =1 | i (cid:105)(cid:104) i | ⊗ =: dρ B thereforethe first term in the last equation can be now writtenas ( d + 1) − tr (cid:0) S U ⊗ ( ρ B )) (cid:1) = ( d + 1) − . The lastequality follows from the fact that U ⊗ ( ρ B ) is entirelysupported in the eigenvalue one subspace of S (sym-metric subspace). Observing now that is also true that D ⊗ B ( S ) = d D ⊗ B ( | Φ + (cid:105)(cid:104) Φ + | ) completes the proof of part a).Let us now move to part b). One has tr (cid:0) S ( D B U ⊗ )( ρ B ) (cid:1) =1 /d (cid:80) di =1 tr (cid:0) S ( D B U ) ⊗ ( | i (cid:105)(cid:104) i | ⊗ ) (cid:1) =1 /d (cid:80) di =1 (cid:107) ( D B U ( | i (cid:105)(cid:104) i | ) (cid:107) . But D B U ( | i (cid:105)(cid:104) i | ) = (cid:80) dj =1 |(cid:104) i | U | j (cid:105)| | j (cid:105)(cid:104) j | . Bringing together the last twoequations completes the Proof of part b). Now part c). Fromthe above one sees that d tr ( Sω B ( U )) is the sum of d purities (cid:107)D B U ( | i (cid:105)(cid:104) i | ) (cid:107) , ( i = 1 , . . . , d ) . Therefore the minimumof this quantity occurs when they are all their minimum i.e., /d . Adding over i one finds (cid:104) S (cid:105) ω B ( U ) ≥ /d from whichthe desired upper bound c) follows, This bound is achieved iff D B U ( | i (cid:105)(cid:104) i | ) = 1I /d, ( ∀ i ) . This, in turn, from the expressiona few lines above, implies |(cid:104) i | U | j (cid:105)| = 1 /d . Notice that thisconclusion can be also derived directly from the formula c). (cid:3) Proof of Proposition 4 a) Given a fixed basis B and any other base B one has thatthere exists a V ∈ U ( H ) such that B = B V (see com-ment after Prop. 2 in the main text). Therefore P B ( c ) = P B V ( c ) dc = (cid:82) dµ ( U ) δ ( c − C B V ( U )) = (cid:82) dµ ( U ) δ ( c − C B ( V † U V )) = (cid:82) dµ ( U ) δ ( c − C B ( V † U V )) = (cid:82) dµ ( V W V † ) δ ( c − C B ( W )) = P B ( c ) dc . Where wehave used c) of Prop. 2 and the unitary invariance of theHaar measure i.e., dµ ( V W V † ) = dµ ( V ) . b) Let us con-sider the terms |(cid:104) i | U | j (cid:105)| from part b) of Prop. 3 and per-form average with respect a Haar distributed U . Denoting by | ψ (cid:105) = U | j (cid:105) this amounts to average with respect | ψ (cid:105) the fol-lowing quantity ( (cid:104) i | ψ (cid:105)(cid:104) ψ | i (cid:105) ) = tr (cid:0) | i (cid:105)(cid:104) i | ⊗ | ψ (cid:105)(cid:104) ψ | ⊗ (cid:1) . Us-ing now the Lemma one finds (cid:104)|(cid:104) i | U | j (cid:105)| (cid:105) U = (cid:104)|(cid:104) i | ψ (cid:105)| (cid:105) ψ =[ d ( d + 1)] − tr (cid:0) | i (cid:105)(cid:104) i | ⊗ (1I + S ) (cid:1) = 2[ d ( d + 1)] − . Addingover i and j and using Eq. (3) in the main text one obtainsEq. (5). c) Here we need a version of the Levy Lemma for-mulated for Haar distributed d × d unitaries: Prob { X ( U ) −(cid:104) X ( U ) (cid:105) U ≥ (cid:15) } ≤ exp (cid:104) − d(cid:15) K (cid:105) where K is a Lipschitz con-stant of X : U ( d ) (cid:55)→ R i.e., | X ( U ) − X ( V ) | ≤ K (cid:107) U − V (cid:107) [2]. Let us set X ( U ) := 1 − ˜ C B ( U ) then X ( U ) −(cid:104) X ( U ) (cid:105) U =1 − ˜ C B ( U ) − / ( d + 1) from which Prob { ˜ C B ( U ) ≤ − (cid:15) − /d } ≤ exp( − d(cid:15) / (4 K )) . If we now set (cid:15) = d − α with α ∈ (0 , / we get Prob { ˜ C B ( U ) ≤ − /d α } ≤ exp( − d − α / (4 K )) . To complete the proof we have to estimate the Lips-chitz constant K . For this, from Eq. (3), and the def-initions above, is clearly enough to consider the function f ( U ) = 1 /d (cid:80) di =1 tr (cid:0) S ( D ⊗ B ( | i U (cid:105)(cid:104) i U | ⊗ ) (cid:1) =: 1 − d/ ( d −
1) ˜ C B ( U ) where | i U (cid:105) := U | i (cid:105) . Let us consider each ofthe d terms, called f i ( U ) , separately: | f i ( U ) − f i ( V ) | ≤| tr (cid:0) S D ⊗ B ( | i U (cid:105)(cid:104) i U | ⊗ − | i V (cid:105)(cid:104) i V | ⊗ ) (cid:1) | ≤ (cid:107)| i U (cid:105)(cid:104) i U | ⊗ −| i V (cid:105)(cid:104) i V | ⊗ (cid:107) , where we have used tr( AB ) ≤ (cid:107) A (cid:107) ∞ (cid:107) B (cid:107) , (cid:107) S (cid:107) ∞ = 1 and, since B -dephasing is a CP map, (cid:107)D ⊗ B ( X ) (cid:107) ≤ (cid:107) X (cid:107) . Now, the last trace-norm distancecan be upper bounded by twice the Hilbert space distance (cid:107)| i U (cid:105) ⊗ −| i V (cid:105) ⊗ (cid:107) ≤ (cid:107) U ⊗ − V ⊗ (cid:107) ∞ = (cid:107) − ( U † V ) ⊗ (cid:107) ∞ .if ∆ := U − V and K := U † ∆ has U † V = 1 − K and then thelast norm becomes (cid:107) − (1 − K ) ⊗ (cid:107) ∞ = (cid:107) K ⊗
1I + 1I ⊗ K + K ⊗ K (cid:107) ∞ ≤ (cid:107) K (cid:107) ∞ (2 + (cid:107) K (cid:107) ∞ ) ≤ (cid:107) K (cid:107) ≤ (cid:107) U − V (cid:107) ∞ ≤ (cid:107) U − V (cid:107) where we have used standard operator norm in-equalities. Bringing all together (cid:107) f ( U ) − f ( V ) (cid:107) ≤ (cid:107) U − V (cid:107) showing that one can take K = 8 . Setting α = 1 / and con-sidering he complementary inequality one obtains Eq. (6) inthe main text. (cid:3) Scaling of the Variance
See Fig. (5).
Proof of Proposition 5
Proceed exactly as in the unitary case. The only differenceis that, for general E the state E ⊗ ( ρ B ) = ˜ ω B is not entirelysupported in the eigenvalue one eigenspace of S . (cid:3) PDD for CGP in d = 2 Using Eq. (3) for a SU (2) matrix one finds C B ( U ) = (1 −| a | − | b | ) , where a = (cid:104) | U | (cid:105) = (cid:104) | U | (cid:105) ∗ , b = (cid:104) | U | (cid:105) = −(cid:104) | U | (cid:105) ∗ . Since | a | + | b | = 1 one can use the Bloch sphereparametrization | a | = cos( θ/ , | b | = sin( θ/ from which it - - d V a r C G P Figure 5. This log-log plot shows the numerically computed vari-ance of the random variable ˜ C B ( U ) (where U is distributed accord-ing to the Haar measure) for different values of the dimension d ofthe Hilbert space. A power-law A/d α least square fitting (taking intoaccount only the points d = 6 , , , ) gives α = 3 . , suggest-ing that the variance of ˜ C B ( U ) is O (1 /d ) . follows ˜ C B ( U ) = C B ( U ) /C d =2 = sin ( θ ) . The distributiondensity of c = ˜ C B ( U ) ∈ [0 , is given by P CGP ( c ) = 14 π (cid:90) π dφ (cid:90) π d (cos θ ) δ ( c − sin θ )= 12 (cid:90) − dx δ ( c − x ) = 12 √ − c , where we have used (cid:82) dx δ ( f ( x )) = (cid:80) x : f ( x )=0 | f (cid:48) ( x ) | − . Asymmetry
In order to directly connect Asymmetry Generating Power(AGP) and and CGP we assume from here on that the Hamil-tonian H is non degenerate and that B = {| i (cid:105)} di =1 is the as-sociated basis of eigenvectors. In this case the notion of H -symmetric state and the one of B -incoherent collapse. It is in-deed immediate to see that H ( ρ ) =: [ H, ρ ] = 0 ⇔ D B ( ρ ) = ρ (in the degenerate case incoherence implies symmetry). Atthe CP map level, however, one has just that H -covarianceimplies B -incoherence but not the converse. For example uni-taries in Prop. 1 realizing a non-trivial permutation of B areincoherent but not covariant. As a consequence the set of co-herence measures is smaller than the set of asymmetry mea-sures [3, 4]. We introduce the following notion of AGP forunital maps E A H ( E ) = (cid:104)(cid:107)HED B ( | ψ (cid:105)(cid:104) ψ | ) (cid:107) (cid:105) ψ (10) where once again the average is taken with respect to the Haarmeasure. As the CGP Eq. (7) in the main text (see commentafter Prop. 5) also the AGP Eq. (10) is a convex function ofits argument. Furthermore, if H = (cid:80) di =1 (cid:15) i | i (cid:105)(cid:104) i | , δ ( H ) :=min l (cid:54) = m | (cid:15) l − (cid:15) m | > (non-degeneracy) and (cid:107)H(cid:107) :=max l (cid:54) = m | (cid:15) l − (cid:15) m | , then the AGP (10) fulfills the followingproperties: Proposition 6.– a) A H ( E ) = 0 for all incoherent maps E .In particular all H -covariant maps have vanishing AGP. b) if T is a unital H -covariant map A H ( T E ) ≤ A H ( E ) . Forunitary H -covariant T the inequality becomes an equality. c) A H ( E ) = [ d ( d + 1)] − (cid:80) i,l (cid:54) = m ( (cid:15) l − (cid:15) m ) |(cid:104) l |E ( | i (cid:105)(cid:104) i | ) | m (cid:105)| . d) δ ( H ) C B ( E ) ≤ A H ( E ) ≤ (cid:107)H(cid:107) C B ( E ) . e) If E ( · ) = U · U † and the unitary U ’s are Haar distributed then the induceddistribution of A H ( U ) depends just on the gap spectrum { (cid:15) l − (cid:15) m } l (cid:54) = m . Proof.– a) Follows from HD B = 0 and ED B = D B ED B which holds for incoherent maps. b) Use [ T , D B ] = 0 for H -covariant maps and the non-increasing property of the Hilbert-Schmidt norm under unital maps. c) Following the samesteps in the proof of a) in Prop. 3 one arrives at A H ( E ) =[ d ( d + 1)] − (cid:80) di =1 (cid:107)HE ( | i (cid:105)(cid:104) i | ) (cid:107) . Expanding the norms inthis equation and using H ( | l (cid:105)(cid:104) m | ) = ( (cid:15) l − (cid:15) m ) | l (cid:105)(cid:104) m | ) onecompletes the proof. d) From c) using δ ( H ) ≤ | (cid:15) l − (cid:15) m | ≤(cid:107)H(cid:107) , ( ∀ l, m ) . e) If the Hamiltonian eigenbasis is changed by | i (cid:105) (cid:55)→ W | i (cid:105) ( W unitary) then from the result in c) one seesthat E (cid:55)→ W † EW ( W ( · ) = W · W † ). If E ( · ) = U · U † the last equation implies U (cid:55)→ W † U W . The proof canbe now completed following the same reasoning of point c)in the proof of Prop. 2 and observing that H enters now,having modded the basis away, just through the differences (cid:15) l − (cid:15) m ( l (cid:54) = m = 1 , . . . d ) . (cid:3) [1] M. Keyl, Phys. Rep. , 431-548 (2002)[2] Corollary 4.4.28 in G. W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices . Cambridge UniversityPress, (2009).[3] I. Marvian, R. Spekkens and P. Zanardi, Phys. Rev. A93