SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION doi: 0.038/nPHYS734 Supplementary Information The Uncertainty Principle in the Presence of Quantum Memory Mario Berta,, 2 Matthias Christandl,, 2 Roger Colbeck, 3,, 4 Joseph M. Renes, 5 and Renato Renner Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland. 2 Faculty of Physics, Ludwig-Maximilians-Universität München, Munich, Germany. 3 Perimeter Institute for Theoretical Physics, 3 Caroline Street North, Waterloo, ON N2L 2Y5, Canada. 4 Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland. 5 Institute for Applied Physics, Technische Universität Darmstadt, Darmstadt, Germany. Here we present the full proof of our main result, the uncertainty relation given in Equation (3) of the main manuscript (Theorem below). In order to state our result precisely, we introduce a few definitions. Consider two measurements described by orthonormal bases { ψ j } and { φ k } on a d-dimensional Hilbert space H A (note that they are not necessarily complementary). The measurement processes are then described by the completely positive maps R : ρ j S : ρ k ψ j ρ ψ j ψ j ψ j and φ k ρ φ k φ k φ k respectively. We denote the square of the overlap of these measurements by c, i.e. c := max j,k ψ j φ k 2. () Furthermore, we assume that H B is an arbitrary finite-dimensional Hilbert space. The von Neumann entropy of A given B is denoted H(A B) and is defined via H(A B) := H(AB) H(B), where for a state ρ on H A we have H(A) := tr(ρ log ρ). The statement we prove is then Theorem. For any density operator ρ AB on H A H B, H(R B)+H(S B) log 2 + H(A B), (2) c where H(R B), H(S B), and H(A B) denote the conditional von Neumann entropies of the states (R I)(ρ AB ), (S I)(ρ AB ), and ρ AB, respectively. In the next section, we introduce the smooth min- and max- entropies and give some properties that will be needed in the proof. Before that, we show that the statement of our main theorem is equivalent to a relation conjectured by Boileau and Renes []. Corollary 2. For any density operator ρ ABE on H A H B H E, H(R E)+H(S B) log 2 c. (3) Proof. To show that our result implies (3), we first rewrite (2) as H(RB)+H(SB) log 2 c + H(AB)+H(B). In the case that ρ ABE is pure, we have H(RB) =H(RE) and H(AB) =H(E). This yields the expression H(RE)+H(SB) log 2 c + H(E)+H(B), which is equivalent to (3). The result for arbitrary states ρ ABE follows by the concavity of the conditional entropy (see e.g. [2]). That (3) implies (2) can be seen by taking ρ ABE as the state which purifies ρ AB in (3) and reversing the argument above. I. (SMOOTH) MIN- AND MAX-ENTROPIES DEFINITIONS As described above, we prove a generalized version of (2), which is formulated in terms of smooth min- and maxentropies. This section contains the basic definitions, while Section V B summarizes the properties of smooth entropies needed for this work. For a more detailed discussion of the smooth entropy calculus, we refer to [3 6]. nature physics

2 doi: 0.038/nphys734 We use U = (H) := {ρ : ρ 0, trρ =} to denote the set of normalized states on a finite-dimensional Hilbert space H and U (H) := {ρ : ρ 0, trρ } to denote the set of subnormalized states on H. The definitions below apply to subnormalized states. The conditional min-entropy of A given B for a state ρ U (H AB ) is defined as H min (A B) ρ := sup H min (A B) ρ σ, σ where the supremum is over all normalized density operators σ U = (H B ) and where H min (A B) ρ σ := log 2 inf{λ : ρ AB λ A σ B }. In the special case where the B system is trivial, we write H min (A) ρ instead of H min (A B) ρ. It is easy to see that H min (A) ρ = log 2 ρ A and that for ρ τ, H min (A B) ρ H min (A B) τ. Furthermore, for ρ U (H A ), we define H max (A) ρ := 2 log 2 tr ρ. It follows that for ρ τ, H max (A) ρ H max (A) τ (since the square root is operator monotone). In our proof, we also make use of an intermediate quantity, denoted H. It is defined by H (A) ρ := log 2 sup{λ : ρ A λ Π supp(ρa)}, where Π supp(ρa) denotes the projector onto the support of ρ A. In other words, H (A) ρ is equal to the negative logarithm of the smallest non-zero eigenvalue of ρ A. This quantity will not appear in our final statements but will instead be replaced by a smooth version of H max (see below and Section V B). The smooth min- and max-entropies are defined by extremizing the non-smooth entropies over a set of nearby states, where our notion of nearby is expressed in terms of the purified distance. It is defined as (see [6]) P (ρ, σ) := F (ρ, σ) 2, (4) where F (, ) denotes the generalized fidelity (which equals the standard fidelity if at least one of the states is normalized), F (ρ, σ) := ρ ( trρ) σ ( trσ). (5) (Note that we use F (ρ, σ) := ρ σ to denote the standard fidelity.) The purified distance is a distance measure; in particular, it satisfies the triangle inequality P (ρ, σ) P (ρ, τ) + P (τ,σ). As its name indicates, P (ρ, σ) corresponds to the minimum trace distance 2 between purifications of ρ and σ. Further properties are stated in Section V A. We use the purified distance to specify a ball of subnormalized density operators around ρ: B ε (ρ) := {ρ : ρ U (H),P(ρ, ρ ) ε}. Then, for any ε 0, the ε-smooth min- and max-entropies are defined by Hmin(A B) ε ρ := sup H min (A B) ρ ρ B ε (ρ) H ε max(a) ρ := inf ρ B ε (ρ) H max(a) ρ. In the following, we will sometimes omit the subscript ρ when it is obvious from context which state is implied. In the case of finite dimensional Hilbert spaces (as in this work), the infima and suprema used in our definitions can be replaced by minima and maxima. 2 The trace distance between two states τ and κ is defined by 2 τ κ where Γ = tr ΓΓ. 2 nature physics

3 doi: 0.038/nphys734 II. OVERVIEW OF THE PROOF The proof of our main result, Theorem, is divided into two main parts, each individually proven in the next sections. In the first part, given in Section III, we prove the following uncertainty relation, which is similar to the main result but formulated in terms of the quantum entropies H min and H. Theorem 3. For any ρ AB U (H AB ) we have H min (R B) (R I)(ρ) + H (SB) (S I)(ρ) log 2 c + H min(ab) ρ. The second part of the proof involves smoothing the above relation and yields the following theorem (see Section IV) 3. Theorem 4. For any ρ U = (H AB ) and ε>0, H 5 ε min (R B) (R I)(ρ) + H ε max(sb) (S I)(ρ) log 2 c + Hε min(ab) ρ 2 log 2 ε. From Theorem 4, the von Neumann version of the uncertainty relation (Theorem ) can be obtained as an asymptotic special case for i.i.d. states. More precisely, for any σ U = (H AB ) and for any n N, we evaluate the inequality for ρ = σ n where R I and S I are replaced by (R I) n and (S I) n, respectively. Note that the corresponding overlap is then given by c (n) = max ψ j φ k...ψ jn φ kn 2 = max j...j n,k...k n j,k ψ n j φ n k 2 = c n. The assertion of the theorem can thus be rewritten as ε n H5 min (Rn B n ) ((R I)(σ)) n + n Hε max(s n B n ) ((S I)(σ)) n log 2 c + n Hε min(a n B n ) σ n 2 n log 2 Taking the limit n and then ε 0 and using the asymptotic equipartition property (Lemma 9), we obtain H(R B)+H(SB) log 2 c + H(AB), from which Theorem follows by subtracting H(B) from both sides. ε. III. PROOF OF THEOREM 3 In this section we prove a version of Theorem, formulated in terms of the quantum entropies H min and H. We introduce D R = j e 2πij d ψ j ψ j and D S = k e 2πik d φ k φ k (D R and D S are d-dimensional generalizations of Pauli operators). The maps R and S describing the two measurements can then be rewritten as R : ρ d S : ρ d d DRρD a a R a=0 d DSρD b b S. b=0 We use the two chain rules proved in Section V B (Lemmas and 2), together with the strong subadditivity of the min-entropy (Lemma 0), to obtain, for an arbitrary density operator Ω A B AB, H min (A B AB) Ω H (A AB) Ω H min (B A AB) Ω Ω H min (B AB) Ω Ω H min (B A B) Ω H min (A B) Ω. (6) 3 We note that a related relation follows from the work of Maassen and Uffink [7] who derived a relation involving Rényi entropies (the order α Rényi entropy [8] is denoted H α) and the overlap c (defined in ()). They showed that H α(r) ρ + H β (S) ρ log 2 c, where α + β = 2. The case α, β 2 yields H min(r) ρ + H max(s) ρ log 2 c. nature physics 3

4 doi: 0.038/nphys734 We now apply this relation to the state Ω A B AB defined as follows 4 : Ω A B AB := d 2 a,b aa A bb B (DRD a S b )ρ AB (D b S D a R ), where { a A } a and { b B } b are orthonormal bases on d-dimensional Hilbert spaces H A and H B. This state satisfies the following relations: H min (A B AB) Ω = 2 log 2 d + H min (AB) ρ (7) H (A AB) Ω = log 2 d + H (SB) (S I)(ρ) (8) H min (B A B) Ω log 2 d + H min (R B) (R I)(ρ) (9) H min (A B) Ω log 2 c. (0) Using these in (6) establishes Theorem 3. We proceed by showing (7) (0). Relation (7) follows because Ω A B AB is unitarily related to d a,b aa 2 A bb B ρ AB, and the fact that the unconditional min-entropy is invariant under unitary operations. To see (8), note that Ω A AB is unitarily related to d a aa 2 A b (Sb )ρ AB (S b ) and that d )ρ AB (S b ) = (S I)(ρ AB ). To show inequality (9), note that Ω B AB = d 2 bb B a b (DRD a S b )ρ AB (D b S D a R ). b (Sb To evaluate the min-entropy, define λ such that H min (B A B) Ω = log 2 λ. It follows that there exists a (normalized) density operator σ B such that λ B A σ B d 2 bb B a b (DRD a S b )ρ AB (D b S D a R ). Thus, for all b, λ A σ B d 2 (DRD a S b )ρ AB (D b a S D a R ), and in particular, for b = 0, we have λ A σ B d 2 (DR a )ρ AB (D a R ) = d (R I)(ρ AB). We conclude that 2 Hmin(R B) (R I)(ρ) λd, from which (9) follows. To show (0), we observe that Ω AB = d 2 (DRD a S b )ρ AB (D b ab a S D a R ) = ((R S) I)(ρ AB). 4 The idea behind the use of this state first appeared in [9]. 4 nature physics

5 doi: 0.038/nphys734 5 Then, ((R S) I)(ρ AB ) = (R I) φ k φ k tr A (( φ k φ k )ρ AB ) = jk max lm = max lm = max lm k φ k ψ j 2 ψ j ψ j tr A (( φ k φ k )ρ AB ) φl ψ m 2 ψ j ψ j tr A (( φ k φ k )ρ AB ) jk φl ψ m 2 A tr A (( φ k φ k )ρ AB ) φl ψ m 2 A ρ B. k It follows that 2 Hmin(A B) ((R S) I)(ρ) maxlm φ l ψ m 2 = c, which concludes the proof. IV. PROOF OF THEOREM 4 The uncertainty relation proved in the previous section (Theorem 3) is formulated in terms of the entropies H min and H. In this section, we transform these quantities into the smooth entropies H ε min and Hε max, respectively, for some ε>0. This will complete the proof of Theorem 4. Let σ AB U (H AB ). Lemma 5 applied to σ SB := (S I)(σ AB ) implies that there exists a nonnegative operator Π such that tr(( Π 2 )σ SB ) 3ε and H ε max(sb) (S I)(σ) H (SB) Π(S I)(σ)Π 2 log 2 ε. () We can assume without loss of generality that Π commutes with the action of S I because it can be chosen to be diagonal in any eigenbasis of σ SB. Hence, Π(S I)(σ AB )Π=(S I)(Πσ AB Π), and tr(( Π 2 )σ AB ) = tr((s I)(( Π 2 )σ AB )) = tr(( Π 2 )σ SB ) 3ε. (2) Applying Theorem 3 to the operator Πσ AB Π yields Note that ΠσΠ σ and so H min (R B) (R I)(ΠσΠ) + H R (SB) (S I)(ΠσΠ) log 2 c + H min(ab) ΠσΠ. (3) Using () and (4) to bound the terms in (3), we find H min (AB) ΠσΠ H min (AB) σ. (4) H min (R B) (R I)(ΠσΠ) + H ε max(sb) (S I)(σ) log 2 c + H min(ab) σ 2 log 2 ε. (5) Now we apply Lemma 8 to ρ AB. Hence there exists a nonnegative operator Π which is diagonal in an eigenbasis of ρ AB such that and H min (AB) Πρ Π H ε min (AB) ρ. Evaluating (5) for σ AB := Πρ AB Π thus gives tr(( Π 2 )ρ AB ) 2ε (6) H min (R B) (R I)(Π Πρ ΠΠ) + H ε max(sb) (S I)( Πρ Π) log 2 c + Hε min(ab) ρ 2 log 2 ε, (7) where Π is diagonal in any eigenbasis of (S I)( Πρ AB Π) and satisfies tr(( Π 2 ) Πρ AB Π) 3ε. (8) nature physics 5

6 doi: 0.038/nphys734 Since ρ AB Πρ AB Π, we can apply Lemma 7 to (S I)(ρAB )and(s I)( Πρ AB Π), which gives The relation (7) then reduces to H ε max(sb) (S I)(ρ) H ε max(sb) (S I)( Πρ Π). (9) H min (R B) (R I)(Π Πρ ΠΠ) + H ε max(sb) (S I)(ρ) log 2 c + Hε min(ab) ρ 2 log 2 ε. (20) Finally, we apply Lemma 7 to (6) and (8), which gives Hence, by the triangle inequality P (ρ AB, Πρ AB Π) 4ε P ( Πρ AB Π, Π ΠρAB ΠΠ) 6ε. P (ρ AB, Π Πρ AB ΠΠ) ( 4+ 6) ε<5 ε. Consequently, (R I)(Π Πρ AB ΠΠ) has at most distance 5 ε from (R I)(ρAB ). This implies Inserting this in (20) gives H 5 ε min (R B) (R I)(ρ) H min (R B) (R I)(Π Πρ ΠΠ). H 5 ε min (R B) (R I)(ρ) + H ε max(sb) (S I)(ρ) log 2 c + Hε min(ab) ρ 2 log 2 ε, which completes the proof of Theorem 4. V. TECHNICAL PROPERTIES A. Properties of the purified distance The purified distance between ρ and σ corresponds to the minimum trace distance between purifications of ρ and σ, respectively [6]. Because the trace distance can only decrease under the action of a partial trace (see, e.g., [2]), we obtain the following bound. Lemma 5. For any ρ U (H) and σ U (H), ρ σ 2P (ρ, σ). The following lemma states that the purified distance is non-increasing under certain mappings. Lemma 6. For any ρ U (H) and σ U (H), and for any nonnegative operator Π, P (ΠρΠ, ΠσΠ) P (ρ, σ). (2) Proof. We use the fact that the purified distance is non-increasing under any trace-preserving completely positive map (TPCPM) [6] and consider the TPCPM E : ρ ΠρΠ tr( Π 2 ρ Π 2 ). We have P (ρ, σ) P (E(ρ), E(σ)), which implies F (ρ, σ) F (E(ρ), E(σ)). Then, F (ρ, σ) F (E(ρ), E(σ)) = F (ΠρΠ, ΠσΠ) + (trρ tr(π 2 ρ))(trσ tr(π 2 σ)) + ( trρ)( trσ) F (ΠρΠ, ΠσΠ) + ( tr(π 2 ρ))( tr(π 2 σ)) = F (ΠρΠ, ΠσΠ), which is equivalent to the statement of the Lemma. 6 nature physics

7 doi: 0.038/nphys734 The second inequality is the relation (trρ tr(π2 ρ))(trσ tr(π 2 σ)) + ( trρ)( trσ) ( tr(π 2 ρ))( tr(π 2 σ)), which we proceed to show. For brevity, we write trρ tr(π 2 ρ)=r, trσ tr(π 2 σ)=s, trρ = t and trσ = u. We hence seek to show rs + tu (r + t)(s + u). For r, s, t and u nonnegative, we have rs + tu (r + t)(s + u) rs +2 rstu + tu (r + t)(s + u) 4rstu (ru + st) 2 0 (ru st) 2. Furthermore, the purified distance between a state ρ and its image ΠρΠ is upper bounded as follows. Lemma 7. For any ρ U (H), and for any nonnegative operator, Π, P (ρ, ΠρΠ) trρ (trρ)2 (tr(π 2 ρ)) 2. Proof. Note that ρ ΠρΠ = tr ( ρπ ρ)( ρπ ρ) = tr(πρ), so we can write the generalized fidelity (see (5)) as F (ρ, ΠρΠ) = tr(πρ)+ ( trρ)( tr(π 2 ρ)). For brevity, we now write trρ = r, tr(πρ) =s and tr(π 2 ρ)=t. Note that 0 t s r. Thus, F (ρ, ΠρΠ) 2 = r + t rt s 2 2s ( r)( t). We proceed to show that r( F (ρ, ΠρΠ) 2 ) r 2 + t 2 0: r( F (ρ, ΠρΠ) 2 ) r 2 + t 2 = r r + t rt s 2 2s ( r)( t) r 2 + t 2 This completes the proof. r r + t rt s 2 2s( r) r 2 + t 2 = rt r 2 t + t 2 2rs +2r 2 s rs 2 rt r 2 t + t 2 2rs +2r 2 s rt 2 = ( r)(t 2 + rt 2rs) ( r)(s 2 + rs 2rs) = ( r)s(s r) 0. Lemma 8. Let ρ U (H) and σ U (H) have eigenvalues r i and s i ordered non-increasingly (r i+ s i+ s i ). Choose a basis i such that σ = i s i ii and define ρ = i r i ii, then P (ρ, σ) P ( ρ, σ). Proof. By the definition of the purified distance P (, ), it suffices to show that F (ρ, σ) F ( ρ, σ). r i and F (ρ, σ) ( trρ)( trσ) = ρ σ = max Re tr(u ρ σ) U max Re tr(u ρv σ) U,V = ri si = F ( ρ, σ) ( tr ρ)( trσ). i The maximizations are taken over the set of unitary matrices. The second and third equality are Theorem and Equation (7.4.4) (on page 436) in [0]. Since tr ρ = trρ, the result follows. nature physics 7

8 doi: 0.038/nphys734 B. Basic properties of (smooth) min- and max-entropies Smooth min- and max-entropies can be seen as generalizations of the von Neumann entropy, in the following sense [5]. Lemma 9. For any σ U = (H AB ), lim lim ε 0 n lim lim ε 0 n n Hε min(a n B n ) σ n = H(A B) σ n Hε max(a n ) σ n = H(A) σ. The von Neumann entropy satisfies the strong subadditivity relation, H(A BC) H(A B). That is, discarding information encoded in a system, C, can only increase the uncertainty about the state of another system, A. This inequality directly generalizes to (smooth) min- and max-entropies [3]. In this work, we only need the statement for H min. Lemma 0 (Strong subadditivity for H min [3]). For any ρ S (H ABC ), Proof. By definition, we have H min (A BC) ρ ρ H min (A B) ρ ρ. (22) 2 Hmin(A BC) ρ ρ A ρ BC ρ ABC 0. Because the partial trace maps nonnegative operators to nonnegative operators, this implies 2 Hmin(A BC) ρ ρ A ρ B ρ AB 0. This implies that 2 Hmin(A B) ρ ρ 2 H min(a BC) ρ ρ, which is equivalent to the assertion of the lemma. The chain rule for von Neumann entropy states that H(A BC) =H(AB C) H(B C). This equality generalizes to a family of inequalities for (smooth) min- and max-entropies. In particular, we will use the following two lemmas. Lemma (Chain rule I). For any ρ S (H ABC ) and σ C S (H C ), H min (A BC) ρ ρ H min (AB C) ρ H min (B C) ρ. Proof. Let σ C S (H C ) be arbitrary. Then, from the definition of the min-entropy we have ρ ABC 2 Hmin(A BC) ρ ρ A ρ BC 2 Hmin(A BC) ρ ρ 2 Hmin(B C) ρ σ AB σ C. This implies that 2 Hmin(AB C) ρ σ 2 H min(a BC) ρ ρ2 H min(b C) ρ σ and, hence Hmin (A BC) ρ ρ H min (AB C) ρ σ H min (B C) ρ σ. Choosing σ such that H min (B C) ρ σ is maximized, we obtain H min (A BC) ρ ρ H min (AB C) ρ σ H min (B C) ρ. The desired statement then follows because H min (AB C) ρ σ H min (AB C) ρ. Lemma 2 (Chain rule II). For any ρ S (H AB ), H min (AB) ρ H (B) ρ H min (A B) ρ ρ. Note that the inequality can be extended by conditioning all entropies on an additional system C, similarly to Lemma. However, in this work, we only need the version stated here. Proof. From the definitions, ρ AB 2 Hmin(AB) A Π supp(ρb) 2 Hmin(AB) 2 H (B) A ρ B. It follows that 2 Hmin(A B) ρ ρ 2 H min(ab) 2 H (B), which is equivalent to the desired statement. The remaining lemmas stated in this appendix are used to transform statements that hold for entropies H min and H R into statements for smooth entropies H ε min and Hε max. We start with an upper bound on H in terms of H max. 8 nature physics

9 doi: 0.038/nphys734 Lemma 3. For any ε>0 and for any σ S (H A ) there exists a projector Π which is diagonal in any eigenbasis of σ such that tr(( Π)σ) ε and H max (A) σ >H (A) ΠσΠ 2 log 2 ε. Proof. Let σ = i r i ii be a spectral decomposition of σ where the eigenvalues r i are ordered non-increasingly (r i+ r i ). Define the projector Π k := i k ii. Let j be the smallest index such that tr(π jσ) ε and define Π := Π j. Hence, tr(πσ) tr(σ) ε. Furthermore, tr σ tr(π j σ) tr(πj σ)π j σπ j 2. We now use tr(π j σπ j ) > ε and the fact that Π j σπ j cannot be larger than the smallest non-zero eigenvalue of ΠσΠ, 5 which equals 2 H (A)ΠσΠ. This implies tr σ>ε 2 H (A)ΠσΠ. Taking the logarithm of the square of both sides concludes the proof. Lemma 4. For any ε>0 and for any σ S (H A ) there exists a nonnegative operator Π which is diagonal in any eigenbasis of σ such that tr(( Π 2 )σ) 2ε and H ε max(a) σ H max (A) ΠσΠ. Proof. By definition of H ε max(a) σ, there is a ρ B ε (σ) such that H ε max(a) σ = H max (A) ρ. It follows from Lemma 8 that we can take ρ to be diagonal in any eigenbasis of σ. Define ρ := ρ {ρ σ} + = σ {σ ρ} + where { } + denotes the positive part of an operator. We then have ρ ρ, which immediately implies that H max (A) ρ H max (A) ρ. Furthermore, because ρ σ and because ρ and σ have the same eigenbasis, there exists a nonnegative operator Π diagonal in the eigenbasis of σ such that ρ =ΠσΠ. The assertion then follows because tr(( Π 2 )σ) = tr(σ) tr(ρ ) = tr({σ ρ} + ) ρ σ 2ε, where the last inequality follows from Lemma 5 and P (ρ, σ) ε. Lemma 5. For any ε>0 and for any σ S (H A ) there exists a nonnegative operator Π which is diagonal in any eigenbasis of σ such that tr(( Π 2 )σ) 3ε and H ε max(a) σ H (A) ΠσΠ 2 log 2 ε. Proof. By Lemma 4, there exists a nonnegative operator Π such that H ε max(a) σ H max (A) Πσ Π and tr(( Π 2 )σ) 2ε. By Lemma 3 applied to Πσ Π, there exists a projector Π such that H max (A) Πσ Π H (A) ΠσΠ 2 log 2 ε and tr(( Π) Πσ Π) ε, where we defined Π := Π Π. Furthermore, Π, Π and, hence, Π, can be chosen to be diagonal in any eigenbasis of σ. The claim then follows because tr(( Π 2 )σ) = tr(( Π Π2 )σ) = tr(( Π 2 )σ) + tr(( Π) Πσ Π) 3ε. 5 If ΠσΠ has no non-zero eigenvalue then H (A) ΠσΠ = and the statement is trivial. nature physics 9

10 doi: 0.038/nphys734 Lemma 6. Let ε 0, let σ S (H A ) and let M : σ i φ iφ i φ i σ φ i be a measurement with respect to an orthonormal basis { φ i } i. Then H ε max(a) σ H ε max(a) M(σ). Proof. The max-entropy can be written in terms of the (standard) fidelity (see also [4]) as H max (A) σ = 2 log 2 F (σ A, A ). Using the fact that the fidelity can only increase when applying a trace-preserving completely positive map (see, e.g., [2]), we have Combining this with the above yields F (σ A, A ) F (M(σ A ), M( A )) = F (M(σ A ), A ). H max (A) σ H max (A) M(σ), (23) which proves the claim in the special case where ε = 0. To prove the general claim, let H S and H S be isomorphic to H A and let U be the isometry from H A to span{ φ i S φ i S } i H S H S defined by φ i A φ i S φ i S. The action of M can then equivalently be seen as that of U followed by the partial trace over H S. In particular, defining σ SS := Uσ AU,wehaveM(σ A )=σ S. Let ρ S(H SS ) be a density operator such that and H max (S) ρ = H ε max(s) σ (24) P (ρ SS,σ SS) ε. (25) (Note that, by definition, there exists a state ρ S that satisfies (24) with P (ρ S,σ S ) ε. It follows from Uhlmann s theorem (see e.g. [2]) and the fact that the purified distance is non-increasing under partial trace that there exists an extension of ρ S such that (25) also holds.) Since σss has support in the subspace span{ φ i S φ i S } i, we can assume that the same is true for ρ SS. To see this, define Π as the projector onto this subspace and observe that tr S (Πρ SSΠ) cannot be a worse candidate for the optimization in Hmax(S) ε σ : From Lemma 8, we can take ρ S to be diagonal in the { φ i} basis, i.e. we can write ρ S = i λ i φ i φ i, where λ i 0. We also write ρ SS = c ijkl φ i φ j φ k φ l, ijkl for some coefficients c ijkl. To ensure ρ S = tr S ρ SS, we require k c ijkk = λ i δ ij. Consider then tr S (Πρ SS Π) = tr S c ijij φ i φ j φ i φ j ij = i c iiii φ i φ i. It follows that tr S (Πρ SS Π) ρ S (since k c iikk = λ i and c iikk 0) and hence we have Furthermore, from Lemma 6, we have H max (S) trs (Πρ SS Π) H ε max(s) ρ. P (Πρ SS Π,σ SS )=P (Πρ SS Π, Πσ SS Π) P (ρ SS,σ SS) ε, 0 nature physics

11 doi: 0.038/nphys734 from which it follows that tr S (Πρ SS Π) Bε (σ S). We have hence shown that there exists a state ρ SS satisfying (24) and (25) whose support is in span{ φ i S φ i S } i. We can thus define ρ A := U ρ SS U so that ρ S = M(ρ A) and hence (24) can be rewritten as and (25) as Using this and (23), we conclude that H max (A) M(ρ) = H ε max(a) M(σ), P (ρ A,σ A ) ε. H ε max(a) M(σ) = H max (A) M(ρ) H max (A) ρ H ε max(a) σ. Lemma 7. Let ε 0, and let σ S (H A ) and σ S (H A ). If σ σ then H ε max(a) σ H ε max(a) σ. Proof. By Lemma 6, applied to an orthonormal measurement M with respect to the eigenbasis of σ, wehave H ε max(a) σ H ε max(a) M(σ ). Using this and the fact that M(σ ) M(σ) =σ, we conclude that it suffices to prove the claim for the case where σ and σ are diagonal in the same basis. By definition, there exists ρ such that P (ρ, σ) ε and H max (A) ρ = H ε max(a) σ. Because of Lemma 8, ρ can be assumed to be diagonal in an eigenbasis of σ. Hence, there exists an operator Γ which is diagonal in the same eigenbasis such that ρ =ΓσΓ. We define ρ := Γσ Γ for which ρ 0 and tr(ρ ) tr(ρ). Furthermore, since ρ ρ, wehave H max (A) ρ H max (A) ρ = H ε max(a) σ. Because σ and σ can be assumed to be diagonal in the same basis, there exists a nonnegative operator Π which is diagonal in the eigenbasis of σ (and, hence, of Γ and ρ) such that σ =ΠσΠ. We then have ρ =Γσ Γ = ΓΠσΠΓ = ΠΓσΓΠ = ΠρΠ. Using the fact that the purified distance can only decrease under the action of Π (see Lemma 6), we have P (ρ,σ )=P (ΠρΠ, ΠσΠ) P (ρ, σ) ε. This implies H ε max(a) σ H max (A) ρ and thus concludes the proof. Lemma 8. For any ε 0 and for any (normalized) σ S = (H A ), there exists a nonnegative operator Π which is diagonal in any eigenbasis of σ such that tr(( Π 2 )σ) 2ε and H ε min(a) σ H min (A) ΠσΠ. Proof. Let ρ B ε (σ) be such that H min (A) ρ = H ε min (A) σ. It follows from Lemma 8 that we can take ρ to be diagonal in an eigenbasis i of σ. Let r i (s i ) be the list of eigenvalues of ρ (σ) and define σ A = i min(r i,s i ) ii. It is easy to see that there exists a nonnegative operator Π such that σ =ΠσΠ. Since σ ρ, wehave H min (A) ΠσΠ = H min (A) σ H min (A) ρ = H ε min(a) σ. Furthermore, tr(( Π 2 )σ) = tr(σ σ )= i: s i r i (s i r i ) σ ρ. The assertion then follows because, by Lemma 5, the term on the right hand side is bounded by 2P (σ, ρ) 2ε. nature physics

12 doi: 0.038/nphys734 [] Renes, J. M. & Boileau, J.-C. Conjectured strong complementary information tradeoff. Physical Review Letters 03, (2009). [2] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2000). [3] Renner, R. Security of Quantum Key Distribution. Ph.D. thesis, ETH Zürich (2005). URL quant-ph/ [4] König, R., Renner, R. & Schaffner, C. The operational meaning of min- and max-entropy. IEEE Transactions on Information Theory 55, (2009). [5] Tomamichel, M., Colbeck, R. & Renner, R. A fully quantum asymptotic equipartition property. IEEE Transactions on information theory 55, (2009). [6] Tomamichel, M., Colbeck, R. & Renner, R. Duality between smooth min- and max-entropies (2009). URL org/abs/ [7] Maassen, H. & Uffink, J. B. Generalized entropic uncertainty relations. Physical Review Letters 60, (988). [8] Rényi, A. On measures of information and entropy. In Proceedings 4th Berkeley Symposium on Mathematical Statistics and Probability, (96). [9] Christandl, M. & Winter, A. Uncertainty, monogamy and locking of quantum correlations. IEEE Transactions on Information Theory 5, (2005). [0] Horn, R. A. & Johnson, C. R. Matrix Analysis (Cambridge University Press, 985). 2 nature physics