The structure of preserved information in quantum processes
Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola
aa r X i v : . [ qu a n t - ph ] O c t The structure of preserved information in quantum processes
Robin Blume-Kohout , Hui Khoon Ng , David Poulin , and Lorenza Viola Institute for Quantum Information, Center for the Physics of Information, Caltech, Pasadena, CA 91125, USA Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA (Dated: October 29, 2018)We introduce a general operational characterization of information-preserving structures (IPS) –encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correctingcodes – by demonstrating that they are isometric to fixed points of unital quantum processes. Usingthis, we show that every IPS is a matrix algebra. We further establish a structure theorem for thefixed states and observables of an arbitrary process, which unifies the Schr¨odinger and Heisenbergpictures, places restrictions on physically allowed kinds of information, and provides an efficientalgorithm for finding all noiseless and unitarily noiseless subsystems of the process.
PACS numbers: 03.67Lx, 03.67.Pp, 03.65Yz, 89.70.+c
Quantum processes – a.k.a. quantum channels, quan-tum operations, or completely positive (CP) maps – [1, 2]– are central to the theory and practice of quantum in-formation processing (QIP). They describe how quan-tum states evolve over a period of time in the pres-ence of noise, or how a device’s output depends on itsinput. They are also complex and unwieldy: to fullyspecify a quantum process on a d -dimensional systemrequires d real numbers. Most of this data is irrele-vant to what one really wants to know: What informa-tion can pass unharmed through the process? Besidebeing central to QIP, a general answer is broadly rele-vant to both fundamental physics and quantum technolo-gies, for the information-preserving degrees of freedomare precisely those that may be reliably characterizedand exploited. Information-preserving structures ( IPS )in quantum processes – what they are and how to findthem – are the subject of this Letter.The quest for such structures has a long history inquantum physics.
Pointer states (PS), defined in thecontext of quantum measurement theory, are “most clas-sical” states that resist decoherence [3]. QIP science hasspurred interest in the preservation of quantum informa-tion, leading to the notion of noiseless subsystems (NS)[4] as passive IPS that emerge from the existence of sym-metries in the noise, and recover both decoherence-freesubspaces (DFS) [5], and PS in special limits. Processesadmitting no NS may still preserve information, whichcan be actively protected using quantum error correction (QEC) [6, 7] to create an effective NS. Rapid experimen-tal progress in implementing DFS [8], NS [9], and QEC[10] heightens the need for a complete and constructivecharacterization of preserved information.In this Letter, we formulate a general operational the-ory of IPS. The key insight is to identify preserved in-formation with sets of states (or codes ) whose mutualdistinguishability is left unchanged. We prove that everypreserved code can, through error correction, be made noiseless , then show that every optimal noiseless code isisometric [11] to the fixed-point set of the dynamics. This set, in turn, is isometric to a matrix algebra, thus we con-clude that every IPS is an algebra. Finally, we provide anexplicit structure for the fixed points of an arbitrary pro-cess, and an efficient algorithm to determine its noiselessand unitarily noiseless IPS.Our results fill several gaps in existing work. Startingfrom basic operational definitions, our approach encom-passes everything that could represent information per-fectly preserved by a quantum process, and shows an ex-plicit connection to fixed points. Our structure theoremfor fixed points is general, whereas previous results ap-plied only to unital [12, 13] maps, or ones with a full-rankfixed state [14]. While information preservation has beenaddressed in both the Schr¨odinger and Heisenberg [15]dynamical pictures, we consistently unify them. Avail-able algorithms to find IPS are either inefficient (e.g.,Zurek’s “predictability sieve” for PS [16], or Choi andKribs’s method for NS [17]), or restricted to purely noise-less information [18] or unital channels [19]. By explicitlyshifting focus from the noise commutant to the fixed-point set (recent work, e.g. [15], has also moved in thisdirection) our approach paves the way to analyzing “ap-proximate” IPS, beyond existing results on the stabilityof DFS/NS under symmetry-breaking perturbations [20].
Quantum states and processes:
We consideran open quantum system with a [finite] d -dimensionalHilbert space H . Its state is described by a non-negative,trace-1, d × d density matrix ρ , which is also a vectorin the system’s Hilbert-Schmidt space B ( H ) (the spaceof bounded operators on H ). The system’s dynamicalevolution over time t is described by a quantum process E : B ( H ) → B ( H ). E is linear, trace-preserving (TP),and CP, which ensures that E does not produce negativeprobabilities operating on arbitrary states. E is CP iff E ( ρ ) = P i K i ρK † i for some set of Kraus operators { K i } ,and TP iff P i K † i K i = 1l. E is unital iff, in addition, E (1l) = P i K i K † i = 1l (see [1, 2] for further details). Preserved information and distinguishability:
To encode information, we prepare the system in a state ρ , chosen from a set C of possible states. We denote anysuch C a code , without a priori assuming any structurefor C . The code defines the kind of information encoded;in particular, our definition includes all the familiar ex-amples: e.g., a QEC code contains all the states in asubspace P ⊆ H ; a classical code comprises a discreteset of orthogonal states. Many other kinds of codes arepossible, and our first goal is to classify them.To access the information, we must distinguish be-tween states ρ, ρ ′ ∈ C . If we assign prior probabilities { q, − q } to ρ and ρ ′ , and make the optimal measure-ment to distinguish them, we guess correctly with proba-bility p = (1 + k qρ − (1 − q ) ρ ′ k ) (see Ref. [21], IV.2).Clearly, if E makes the states in C less distinguishable,then information was not perfectly preserved. We there-fore propose the following operational criterion: A code C is preserved by a process E iff each pair of states ρ, ρ ′ ∈ C is just as distinguishable after E as before it. The distin-guishability result cited above implies then a technicaldefinition: C is preserved by E iff, for every ρ, ρ ′ ∈ C and x ∈ R + , kE ( ρ − xρ ′ ) k = k ρ − xρ ′ k . A useful consequence is that preserved codes can al-ways be closed under (real) linear combination, so wecan assume that C comprises all the states in an oper-ator subspace of B ( H ). C is preserved if it is isometric to E ( C ), that is, E acts as a 1:1 trace-distance-preservingmap on C . Several operational notions of “preserved”will be relevant. From strongest to weakest:1. C is noiseless for E iff it is preserved by any convexmixture P n p n E n , with p n ≥ P n p n = 1.2. C is unitarily noiseless [22] for E iff it is preservedby E n , for every fixed n ∈ N .3. C is correctable for E iff there exists a correctionprocess R such that C is noiseless for R ◦ E .Thus, while both noiseless and unitarily noiseless codespreserve information indefinitely without any interven-tion, they differ in how the preserved information “movesaround”. The optimal measurement to distinguish twostates in a noiseless code is independent of the number n of applications of E (and can be derived from E ∞ , seeproof of Theorem 2). In contrast, for a unitarily noiselesscode (e.g., a system evolving unitarily, as E ( ρ ) = U ρU † )this measurement may depend on n , so we must keeptrack of how many times E has occurred. Correctablecodes, on the other hand, are not inherently stable – butthey can be stabilized indefinitely by applying R . Wecan collapse the lowest levels of the above hierarchy: Theorem 1
A code C is preserved by the process E iff itis correctable for E . While we defer a full proof to Ref. [23], the central ideais simple: If C is preserved, then we can correct it withthe transpose channel [24], b E P ( ρ ) = X i (cid:16) P K † i P E ( P ) − / (cid:17) ρ (cid:16) E ( P ) − / P K i P (cid:17) , where P is the joint support of every ρ ∈ C ,and P projects onto it. Notice that b E P ◦ E ( P ) = E † ( E ( P ) − / E ( P ) E ( P ) − / ) = P, thus the corrected mapis not only TP but also unital on the code’s support.Because a correctable code for E is a noiseless codefor some other channel R ◦ E , we can characterize allpreserved codes by characterizing noiseless codes. Thefirst step is to relate E ’s noiseless codes to its fixed points: Theorem 2 If C is a noiseless code for E , then C is iso-metric to a subset of the fixed states of E . Proof : C is preserved by any channel of the form P n p n E n ( P p n = 1), including E N = N +1 P Nn =0 E n ,and therefore also by E ∞ = lim N →∞ E N [25] (the limitis well-defined for finite-dimensional H ). Thus, C isisometric to E ∞ ( C ). But E ◦ E ∞ = E ∞ , so if σ = E ∞ ( ρ ),then E ( σ ) = σ . Therefore, E ∞ projects onto the fixedpoints of E , so E ∞ ( C ) is a subset of E ’s fixed states. ✷ Theorem 2 has two important consequences for opti-mal codes – ones that encode as many states as possible.First, every optimal noiseless code for E is isometric tothe set of all fixed states of E . The fixed states are them-selves a noiseless code C , so if C is not isometric to C ,then it is isometric to a proper subset, and cannot beoptimal. Next, every optimal preserved code for E is iso-metric to the set of all fixed states of a unital, TP map .This follows from Theorem 1. If C is preserved, then b E P corrects it, so C is noiseless for b E P ◦ E , and (by Theo-rem 2), isometric to its fixed points. Optimal preservedcodes come in equivalence classes characterized by fixedgeometries (the pairwise distances between elements of aset define its geometry): C and C ′ are equivalent iff theyare isometric. Equivalent codes use different states toencode the same information – they are manifestationsof the same IPS: Definition 1
An IPS of a process E is the geometricstructure common to an equivalence class of optimal pre-served codes. An optimal preserved code is isometric to the fixed-point set of b E P ◦ E . Because this set (and its geometry)depend on P , E may have several distinct IPS. However,all its optimal noiseless codes belong to a single class asthey all share the geometry of E ’s fixed-point set. Theyare manifestations of a unique noiseless IPS: Definition 2
The noiseless IPS of a process E is theunique geometric structure common to all of its optimalnoiseless codes. The structure of codes:
The next step toward char-acterizing the possible IPS is to determine the structureof fixed states for arbitrary E . Because E is linear, itsfixed points are closed under linear combination, henceform an operator subspace of B ( H ). For the special casewhere E is unital , several authors have shown [12, 13]that: (a) The fixed points of E form a complex matrixalgebra A ; (b) A is the commutant of E ’s Kraus opera-tors; (c) E and E † have the same fixed points.This is a powerful result because finite-dimensionalmatrix algebras share an elegant structure: Every suchmatrix algebra is a direct sum of the form, A = M k M d k ⊗ n k , n k , d k ∈ N , (1)where M d k is the algebra of all d k × d k matrices, and1l n k is the trivial algebra containing the n k -dimensionalidentity [26]. Thanks to this result, we have all the ingre-dients to describe the structure of preserved informationfor an arbitrary (not necessarily unital) E : Every opti-mal preserved code is isometric to a matrix algebra.
Thisfollows from Theorem 1 (preserved codes are correctable,with
R◦E unital) and Theorem 2 (optimal noiseless codesare isometric to fixed point sets), together with the struc-ture theorem cited above. We conclude:
Any IPS of aprocess on a d -dimensional system is a subalgebra of M d . Fixed points of arbitrary maps:
While theabove IPS characterization is fully general, it is non-constructive as long as the projector P required to con-struct the transpose map is unknown. However, on onehand noiseless codes are isometric to the fixed states of E itself (rather than b E P ◦ E ). On the other hand, theset of all fixed states is an optimal noiseless code, whoseunique isometric algebra A fully specifies E ’s noiselessIPS. To obtain a constructive characterization of thisIPS, we need (1) a general description of the fixed statesof E ; and (2) a way to extract the algebra to which theyare isometric. Unfortunately, the structure theorem forunital maps does not extend to arbitrary processes. Thefollowing example violates every point listed earlier: E and E † have different fixed-point sets, which do not formalgebras, and do not commute with the Kraus operators! Example:
Let A be a qutrit and B a qubit, and E = E A ⊗ E B be a process on H A ⊗ H B , with Kraus operators E ∼ n | ih | + | ih | , √ | ih | , √ | ih | o A ⊗ n | ih | , | ih | , √ | ih | , √ | ih | o B . E does nothing to the {| i , | i} subspace of A , but maps | i A into an equal mixture of | ih | A , | ih | A . At the sametime, it forces B into τ B = | ih | B + | ih | B . E ’s fixedstates are σ A ⊗ τ B (for any 2 × σ A ), and thefixed observables of E † are (cid:0) σ A + Tr( σ A ) | ih | A (cid:1) ⊗ B .The commutant of the Kraus operators is nothing but 1l.Still, we can characterize fixed states and observables: Theorem 3
Let E be a quantum process on B ( H ) , Σ thefixed points of E , and B the fixed points of E † . Then: (i) Σ and B are each isometric to a matrix algebra A ⊆ B ( P ) , where P is a subspace of H . (ii) Σ is supported on P , and contains all operators σ = L k M d k ⊗ τ n k , where M d k is an arbitrary d k × d k operator, and τ n k is a fixed n k × n k state. (iii) B contains all operators of the form X = A P ⊕F P→P ( A P ) , where A P ∈ A , P is the complement of P in H , and F P→P is a fixed linear map from B ( P ) to B ( P ) . (iv) Projecting B onto the support P of Σ yields a rep-resentation of A . The proof is deferred to [23]. The central result – thatthe fixed states are isometric to a matrix algebra – isalready implied by the fact they form a preserved code.Notice that if E is unital, Σ coincides with A – the non-negative, trace-1 operators in Σ directly determine theprocess’ optimal noiseless codes, hence its noiseless IPS.Familiar examples of noiseless IPS correspond to spe-cific ways in which information is encoded in one ormore blocks of M d via Eq. (1). The simplest IPS cor-responds to encoding purely classical information by achoice among multiple blocks. For a pointer basis, inparticular, all blocks are one-dimensional. Quantum in-formation is preserved within a single higher-dimensionalblock. A DFS is represented by a single block with a triv-ial co-factor, and a NS by a single block tensored with anidentity (“noise-full”) subsystem. The most general IPS,a hybrid quantum memory [27], has n blocks of (possi-bly) different sizes d k . It can be concisely described byits shape , the vector { d , d , . . . d n } .In each of the examples above, E must (by Theorem2) have a set of fixed points. For a pointer basis, theprojectors onto each PS are fixed. For a DFS, every stateon the subspace is fixed. The fixed points associatedwith a NS are less obvious. If E has a NS, H may bedecomposed as H = H A ⊗ H B ⊕ H C , and for all ρ A and ρ B , E ( ρ A ⊗ ρ B ) = ρ A ⊗ σ B [28]. That is, E acts on H A ⊗ H B as E = 1l A ⊗ E B , and by Schauder’s fixed pointtheorem [29], E B must have a fixed point τ B . Thus, forany ρ A , ρ A ⊗ τ B is in Σ. Note how, for each σ B , there isa distinct noiseless code C σ = { ρ A ⊗ σ B ∀ ρ A } , which isisometric to the unique fixed code C = { ρ A ⊗ τ B ∀ ρ A } .In general, the explicit form of the fixed states given inTheorem 3(ii) illustrates what it means to be “isometricto a matrix algebra”: The “noise-full” subsystems arerepresented, not by 1l n k , but by a fixed state τ n k . Fixedobservables have a different structure, also derived fromthat of A . Their restriction to P coincides with A , buteach has an “echo” of itself on P . E † extends observableson P to P , so that they detect states initially outside of P . This is the Heisenberg-picture manifestation of thefact that E maps states on P to P . Finding the Noiseless IPS:
By construction, E ’snoiseless IPS contains all of E ’s NS. To find this IPS:1. Write E as a d × d matrix.2. Diagonalize it, and extract the λ = 1 right and lefteigenspaces (Σ and B , respectively).3. Compute P , the joint support of all ρ ∈ Σ, andproject B onto P to obtain a basis for A .4. Find the shape of A , using (e.g.) tools in [30].Our algorithm runs in time O ( d ) (matrix diagonaliza-tion is O (( d ) )), and uses standard numerical tools. Assuch, it is more efficient than algorithms (e.g., [16, 17])that require exhaustive search over states or subspaces in H – for these sets grow exponentially in volume with d .The above algorithm may be easily generalized to uni-tarily noiseless IPS, provided that we shift our focus from E ’s fixed points to its rotating points , defined as follows: The rotating points of E comprise the span of its unit-modulus eigenoperators. We then have:
Theorem 4
Every optimal unitarily noiseless code for E is isometric to the [positive trace-1 states in the] rotatingpoints of E . The key observation for the proof (deferred to [23]) isthat there exist high powers of E that project onto itsrotating points. Thus, E has a unique unitarily noiselessIPS, which can be found using the algorithm above pro-vided that “the λ = 1 eigenspace” is replaced with “thespan of all the unit-modulus ( λ = e iφ ) eigenoperators”. Discussion:
Our IPS framework can be used in mul-tiple ways. An experimentalist who has characterized asystem using quantum process tomography can apply ouralgorithm to find noiseless and unitarily NS – then usethe IPS shape as a concise language to report the results.On a theoretical front, we have classified all optimal pre-served codes. This rules out certain kinds of information,as unphysical – e.g., no process acting on a single qubitcan perfectly preserve only 1l, σ x , and σ y (a “rebit”).Physically, the IPS shape distills the invariant proper-ties of a process ( what kind of information is preserved),discarding the details ( which states are preserved) thatare needed to design quantum hardware, but not to un-derstand what it can do. It is closely related to E ’seigenvalues, but is both more concise and more informa-tive [31]. One might hope to generalize our algorithm tofind all correctable codes, not just noiseless ones. How-ever, a constructive algorithm seems difficult, and findingthe best codes for even a classical process is NP-hard.Thus, while we now know what every code must looklike, finding one may be intractable. Acknowledgments:
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