Approximate reversal of quantum Gaussian dynamics

Transcription

1 Approximate reversal of quantum Gaussian dynamics L. Lami, S. Das, and M.M. Wilde arxiv: or how to compute Petz maps associated with Gaussian states and channels. Motivation: data processing inequality, recoverability, and Petz map Gaussian states in a nutshell Why computing the action of the Gaussian Petz is possible Explicit form of the Gaussian Petz map

2 Motivation. Quantum data processing inequality Von Neumann entropy: Umegaki relative entropy: S( ).= Tr [ log ] D( k ).= Tr[ (log log )] Data processing inequality (DPI): D( k ) D(N ( )kn ( )) quantum channel = CPTP map Conditional mutual information: Strong subadditivity (SSA): I(A; B C).= S(AC) + S(BC) S(ABC) S(C) I(A; B C) 0 SSA can be deduced from (in fact, is equivalent to) DPI: 7! ABC, 7! AC 1 B, N 7! Tr A I(A : B C) = D( ABC k AC 1 B ) D( BC k C 1 B ) 0

3 Equality case in DPI was solved in the 1980s by Petz: D( k )= D(N ( )kn ( )) () 9 channel P :(P N)( ) =, (P N)( )=. P = P,N :! 7! 1/2 N N ( ) 1/2!N ( ) 1/2 1/2 Petz map The definition looks a bit complicated let s start from the classical version: X q N y x! Y q 0 The Petz map P goes Y X and reconstructs q from q, so its matrix elements satisfy P x y q 0 (y) =N y x q(x) =) P x y = q(x) q 0 (y) N y x Sanity check: also the quantum Petz map reconstructs σ from N(σ). How to find the correct quantum expression?

4 Underlying intuition (wrong to some extent): a purification of a quantum state ω consists of two copies of ω, one per party.! Alice has Bob has =) (I N)( )! Alice has Bob has N ( ) N ( )! Alice has N ( ) Bob has N ( ) =) (P I)( N ( ) )! Alice has Bob has N ( ) Idea (Andreas). The Petz map attempts to reverse the action of N on σ. We can derive its action by equating the expressions on the right (up to some transpositions): (I N)( T ) T =(P I)( N ( ) ) Some calculations show that the expression one finds from here coincides with our original definition of Petz map.

5 Equality case in DPI: D( k )= D(N ( )kn ( )) () 9 channel P :(P N)( ) =, (P N)( )=. P = P,N :! 7! 1/2 N N ( ) 1/2!N ( ) 1/2 1/2 What about approximate equality in DPI? For classical probability distributions (and classical channels) one can show that [Li & Winter, ] D(pkq) D(N(p)kN(q)) D (pk(p q,n N)(p)). Unfortunately, the same is false for quantum The following possible generalization was put forward by Winter & Li in 2012: D( k ) D(N ( )kn ( ))? D( k(r,n N)( )), where the recovery map R has to depend only on σ and N. This holds for some interesting special cases, but not in general [Fawzi & Fawzi, ].

6 Motivation. Improvements of DPI Some important progress was made in 2014 by Fawzi & Renner, who showed that I(A; B C) log F ( ABC, R C!BC ( AC )) F (!, ).= k p! p k 2 1 R C!BC ( ) =V BC P U C ( )U C V BC Quantum fidelity Rotated Petz recovery map P( ) =P AC 1 B,Tr A ( ) = 1/2 BC 1/2 C ( ) 1/2 C 1/2 BC Petz map for this problem

7 Several related improvements: [Wilde, ]: D( k ) D(N ( )kn ( )) inf t2r log F, (R t,n N)( ) [Sutter et al., ]: D( k ) D(N ( )kn ( )) D M ( k(r N)( )) [Junge et al., ]: D( k ) D(N ( )kn ( )) Z +1 1 dt 0(t) log F, (R t,n N)( ) [Sutter et al., ]: D( k ) D(N ( )kn ( )) D M Z +1 1 dt 0(t)(R t,n N)( )

8 Our contribution A relevant case is when both the state σ and the channel N are (bosonic) Gaussian. How does the resulting Petz map act? We showed that if σ and N are a Gaussian state and channel then the associated Petz map is another Gaussian channel, and we computed its action explicitly. The techniques we use in the proof are very general, and can be applied to related problems. In another paper [Seshadreesan et al., ], we use them to compute the Rényi relative entropies between Gaussian states (fidelity had already been computed previously). As an immediate application, we found examples of Gaussian states and channels for which the following inequality is violated: D( k ) D(N ( )kn ( )) D( k(p,n N)( )),

9 Gaussian states in a nutshell Quantum optics is quantum mechanics applied to a number of harmonic oscillators: ˆr (ˆx 1,...,ˆx n, ˆp 1,...,ˆp n ) T! [ˆx i, ˆp j ]=i ij n [ˆr, ˆr T 0 1 ]=i =i 1 0 n This is an -dim Hilbert space with a lot of room for weird things to happen. However, physics tends to be more restrictive Nature has a special preference for quadratic Hamiltonians, like the free field one: Ĥ free = 1 2 P n i=1 (ˆx2 i +ˆp2 i )= 1 2 ˆrT ˆr Not surprisingly, thermal states of quadratic Hamiltonians play a major role. In fact, they are so ubiquitous that they deserve a name: Gaussian states. Ok, Gaussian states pop up all the time in the lab. How to study them? Can we find an efficient parametrisation?

10 In a sense, we already have a good candidate for a parametrisation: Quadratic Hamiltonian: Ĥ = 1 2 ˆrT H ˆr s T H ˆr Thermal state: ˆ G / exp h i Ĥ However, there is a more convenient one. Exactly like a Gaussian (multivariate) random variable, a Gaussian state is uniquely identified by its mean, or displacement vector, and by its covariance matrix. displacement vector: s =Trˆ ˆr 2 R 2n covariance matrix: i H V =Tr[{ˆr s, (ˆr s) T } ˆ ] = coth i 2M 2n (R) 2 ˆ G (V,s) = Z d 2n u (2 ) n e 1 4 u T Vu is T u ˆD u unitary displacement operators classical Gaussian weight Unlike that of a classical random variable, the covariance matrix of a Gaussian state must be bounded from below, by virtue of Heisenberg s uncertainty principle: a symmetric V that satisfies this is called Quantum Covariance Matrix (QCM) V i = QCMs are in particular strictly positive definite

11 Quantum Gaussian channels A classical Gaussian channel acts on a random variable z as Z 0 = XZ + Y on mean and covariance matrix: How to define quantum Gaussian channels? s 0 = Xs + V 0 = XV X T + Y Definition. A quantum Gaussian channel transforms input density matrices by: (1) adding an ancillary system in a fixed Gaussian state; (2) performing a joint Gaussian unitary; (3) discarding the ancillary system. time evolution driven by a quadratic Hamiltonian A quantum Gaussian channel acts on the canonical operators as ˆr 0 = Xˆr + Y id on displacement vector and QCM: Complete positivity: Y + i ix X T 0. s 0 = Xs + V 0 = XV X T + Y Gaussian channels send Gaussian states to Gaussian states. Under certain conditions, the converse is also true: a completely positive and trace-preserving map that preserves Gaussianity is a Gaussian channel [De Palma et al., ].

12 Computations with Gaussian states and channels Remember the form of the Petz map? P,N ( ) = 1/2 N N ( ) 1/2 ( )N ( ) 1/2 1/2 If we want to compute its action for σ Gaussian state and N Gaussian channel we have to be able to: (a) take square roots of Gaussian states; (b) multiply Gaussian states and square roots thereof; (c) determine the action of the adjoint of a Gaussian channel. (a)-(b): note that quadratic Hamiltonians form a Lie algebra [Balian & Brezin, 1969]: Ĥ Q [K, s, a] = i 2 ˆrT Kˆr + is T ˆr + i 2 a hĥq [K 1,s 1,a 1 ], Ĥ Q [K 2,s 2,a 2 ]i [K1,K 2 ],K 1 s 2 K 2 s 1, s T 1 s 2 = ĤQ

13 Campbell-Baker-Hausdorff formula tells us that any product of exponentials of quadratic Hamiltonians can itself be written as an exponential of another quadratic Hamiltonian [Balian & Brezin, 1969]. But there is more: the product depends only on the algebraic structure of the commutation relations! Moral of the story: if we find it more convenient, we can choose a matrix representation of our Lie algebra, compute everything there, and bring back the result via the isomorphism. The following isomorphic matrix representation of the above Lie algebra is useful: Ĥ Q [K, s, a]! 0 1 st T a 0 K sa 0 0 0

14 As for (c), it is easy to show that the adjoint of a channel, generally defined by Tr [N (A) B].= Tr[A N (B)], in the Gaussian case acts on displacement operators as [Genoni et al., ] N ˆD u = e 1 4 ut Yu+iu T ˆD XT u. For invertible X, it sends Gaussian states to (unnormalised) Gaussian states: N : 7 7 V! XV X T + Y s! Xs + =) N : V 7! X 1 (V + Y ) X T s 7! X 1 (s ) This is already enough to establish that the Petz map is a Gaussian channel, since: (i) it s completely positive and trace-preserving; and (ii) preserves exponentials of quadratic operators, in particular maps Gaussian states to Gaussian states.

15 Gaussian Petz map: sketch of the computation Quadratic Hamiltonians form a Lie algebra Campbell-Baker- Hausdorff formula Action of the adjoint of a Gaussian channel The Petz map P σ, N is Gaussian if so are and σ and N. P,N (!) = 1/2 N N ( ) 1/2! N ( ) 1/2 1/2,! V,s,!! V!,s!, N!X, Y,. Step 1. We can assume that s σ = 0 = δ. We will carry out the computation for s ω = 0 and invertible X.

16 Step 2. Using the matrix Lie algebra isomorphism, one shows that [Banchi et al., ] q V 1/2! = V 1/2 q1 +(V ) 2 V (V! + V ) 1 V 1 +( V ) 2 ( ) First, we have to apply ( ) with τ = N (σ) -1 : P,N (!) = 1/2 N N ( ) 1/2! N ( ) 1/2 1/2 How to interpret Vτ? Remember that for zero-mean Gaussian states apple 1 i H / exp 2 ˆrT H ˆr, V = coth i. 2 This seems to suggest the ansatz apple 1 i ( 1 H ) / exp 2 ˆrT ( H )ˆr! V 1 = coth 2 i = V. Plugging this into ( ) yields V N ( ) 1/2! N ( ) 1/2 = V N ( ) q 1 + V N ( ) 2 V N ( ) V! V N ( ) 1 VN ( ) q1 + V N ( ) 2.

17 Step 3. Apply the adjoint of the channel N : P,N (!) = 1/2 N N ( ) 1/2! N ( ) 1/2 1/2 V N (N ( ) 1/2! N ( ) 1/2 ) = X 1 V N ( ) 1/2!N ( ) + Y X T apple q 1/2 q = X V N ( ) V N ( ) V! V N ( ) VN ( ) 1 + V N ( ) + Y X T V N ( ) = V X 1 q 1 + V N ( ) 2 V N ( ) V! V N ( ) 1 VN ( ) q 1 + V N ( ) 2 X T V N ( ) = XV X T + Y

18 Step 4. Use once more the sandwiching formula ( ) with τ = σ : V P,N (!) = V 1/2 N (N ( ) 1/2!N ( ) 1/2 ) q 1/2 = V 1 +(V ) 2 V q V X V N ( ) V N ( ) V! V N ( ) VN ( ) q1 + V N ( ) X T + V q V 1 +( V ) 2 q = V + qi +(V ) 2 V X T 2 1 I + V N ( ) V 1 N ( ) V! V N ( ) q V 1 N ( ) 1 + V N ( ) 2 1 q XV 1 +(V ) 2 = X P V! X T P + Y P, 1 X P.= q1 +(V ) 2 V X T q 1 + V N ( ) 2 1 V 1 N ( ), Y P.= V X P V N ( ) X T P. Gaussian Petz map:

19 Step 5. The above derivation was largely based on ansatzes and cavalier manipulation of unbounded operators. A crucial part of the proof is the verification that the formula we found reproduces the true Petz map. Following Petz, we know that P σ, N is the unique channel satisfying the Petz equation h Tr A 1/2 N (B) 1/2i h =Tr P,N (A)N ( )1/2 BN ( ) 1/2i, 8 bounded A, B. It suffices to show the above equation when A,B are Hilbert-Schmidt operators. In fact, Hilbert-Schmidt operators are weakly dense in the set of bounded operators, and the bilinear function (A, B) 7! Tr ha 1/2 N (B) 1/2i is continuous in the product weak topology for trace-class τ and channels N. For Hilbert-Schmidt operators, the whole machinery of characteristic functions is available, thus checking the Petz equation boils down to a lengthy but fully rigorous computation. This establishes the proposed Gaussian Petz map to be the correct one.

20 Summary and conclusions The Petz map is a central object in modern quantum information theory. Recent progress has shown that it plays a major role in establishing operational lower bounds on the quantum conditional mutual information (recoverability theory). We have computed explicitly the action of the Petz map constructed out of a Gaussian state and a Gaussian channel, and shown this is again a Gaussian channel. The proof leverages the (previously known) fact that the exponentials of quadratic Hamiltonians form a group given that quadratic Hamiltonians form a Lie algebra. We developed a matrix isomorphism that enabled us to carry out multiplications in a systematic way. Take home message: computations can be cumbersome, but the formalism is elegant and versatile. Recently, we also showed how to use it to compute all sort of Rényi relative entropies between Gaussian states.

21 Summary and conclusions The Petz map is a central object in modern quantum information theory. Recent progress has shown that it plays a major role in establishing operational lower bounds on the quantum conditional mutual information (recoverability theory). We have computed explicitly the action of the Petz map constructed out of a Gaussian state and a Gaussian channel, and shown this is again a Gaussian channel. The proof leverages the (previously known) fact that the exponentials of quadratic Hamiltonians form a group given that quadratic Hamiltonians form a Lie algebra. We developed a matrix isomorphism that enabled us to carry out multiplications in a systematic way. Take home message: computations can be cumbersome, but the formalism is elegant and versatile. Recently, we also showed how to use it to compute all sort of Rényi relative entropies between Gaussian states. Thank you!