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in quantum information processing , information is stored and processed with a quantum system . a quantum system is always in contact with its surrounding environment , which leads to decoherence in the quantum system . decoherence must be suppressed for quantum information stored in qubits to be intact . there are several proposals to fight against decoherence . quantum error correction , abriviated as qec hereafter , is one of the most promising candidate to suppress environmental noise , which leads to decoherence @xcite . by adding extra ancillary qubits , in analogy with classical error correction , it is possible to encode a data qubit to an @xmath4-qubit codeword in such a way that an error which acted in the error quantum channel is identified by measuring another set of ancillary qubits added for error syndrome readout . then the correct codeword is recovered from a codeword suffering from a possible error by applying a recovery operation , whose explicit form is determined by the error syndrome readout . in contrast with the conventional scheme outlined in the previous paragraph , there is a scheme in which neither syndrome readouts nor syndrome readout ancilla qubits are required @xcite . in particular , in @xcite , a general efficient scheme was proposed . a data qubit is encoded with encoding ancilla qubits by the same encoding circuit as the conventional one , after which a noisy channel is applied on the codeword . subsequently , the inverse of the encoding circuit is applied on a codeword , which possibly suffers from an error . the resulting state is a tensor product of the data qubit state with a possible error and the ancilla qubit state . it is possible to correct erroneous data qubit state by applying correction gates with the ancilla qubits as control qubits and the data qubit as a target qubit . this paper presents two examples of error correcting codes falling in the second category . the noisy quantum channel is assumed to be fully correlated @xcite , which means all the qubits constituting the codeword are subject to the same error operators . in most physical realizations of a quantum computer , the system size is typically on the order of a few micrometers or less , while the environmental noise , such as electromagnetic wave , has a wavelength on the order of a few millimeters or centimeters . then it is natural to assume all the qubits in the register suffer from the same error operator . to demonstrate the advantage of the second category , we restrict ourselves within the noise operators @xmath7 in the following , where @xmath3 is the number of constituent qubits in the codeword . we show that there exists an @xmath4-qubit encoding which accommodates an @xmath5-qubit data state if @xmath4 is odd and an @xmath6-qubit date state if @xmath4 is even . although the channel is somewhat artificial as an error channel , we may apply our error correction scheme in the following situation . suppose alice wants to send qubits to bob . their qubit bases differ by unitary operations @xmath8 or @xmath9 . even when they do not know which basis the other party employs , the can correctly send qubits by adding one extra qubits ( when @xmath4 is odd ) or two extra qubits ( when @xmath4 is even ) .    we state the theorems and prove them in the next section . the last section is devoted to summary and discussions . in the following , @xmath10 denotes the @xmath11th component of the pauli matrices and we take the basis vectors @xmath12 so that @xmath13 is diagonalized . we introduce operators @xmath14 and @xmath15 acting on the @xmath4-qubit space @xmath16 , where @xmath3 as mentioned before . let @xmath17 be @xmath18 complex matrices , and let @xmath19 . denote by @xmath20 the ( joint ) rank-@xmath21 numerical range of @xmath22 , which is the collection of @xmath23 such that @xmath24 for some @xmath18 rank-@xmath21 orthogonal projection @xmath25 @xcite . a quantum channel of the form @xmath26 has a @xmath21-dimensional quantum error correcting code ( qecc ) if and only if @xmath27 . to prove this statement , we need to recall the knill - laflamme correctability condition , which asserts that given a quantum channel @xmath28 with error operators @xmath29 , @xmath30 is a qecc of @xmath31 if and only if @xmath32 , where @xmath33 is the projection operator with the range space @xmath30 @xcite . it should be clear that @xmath34 if and only if there is a qecc with dimension @xmath21 . now it follows from @xmath35 and the relations @xmath36 when @xmath4 is even and @xmath37 when @xmath4 is odd that the channel ( [ eq : corrch ] ) has a @xmath21-dimensional qecc if and only if @xmath38 by noting that @xmath39 irrespective of rank  @xmath25 , we find @xmath40 if and only if @xmath41 . [ thm1 ] suppose @xmath3 is odd . then @xmath42 . our proof is constructive . for @xmath43 , denote @xmath44 . let @xmath45 then @xmath46 , where @xmath47 is the number of @xmath48-combinations from @xmath4 elements . since @xmath49 we have @xmath50 let @xmath25 be the orthogonal projection onto @xmath30 . then the above observation shows that @xmath51 and @xmath52 . therefore , @xmath53 , which shows that @xmath54 and hence @xmath30 is shown to be a @xmath55-dimensional qecc . _    now let us turn to the even @xmath4 case . we first state a lemma which is necessary to prove the theorem . [ lem1 ] let @xmath56 be a normal matrix . then the rank-@xmath21 numerical range of @xmath57 is the intersection of the convex hulls of any @xmath58 eigenvalues of @xmath57 . the proof of the lemma is found in @xcite . [ thm2 ] suppose @xmath3 is even . then @xmath59 but @xmath60 .    _ proof let @xmath61 . by theorem [ thm1 ] , @xmath62 . consider @xmath63 observe that the projection @xmath25 onto @xmath64 satisfies @xmath65 and @xmath52 and hence @xmath66 , which proves @xmath59 . _    since @xmath67 is a commuting family , @xmath8 and @xmath9 can be diagonalized simultaneously . we may assume that @xmath68 since @xmath69 , we have @xmath70    let us show that @xmath71 . we first note the identity @xmath72 for hermitian @xmath73 . let us replace @xmath74 by @xmath75 and @xmath76 by @xmath77 to obtain @xmath78 . since @xmath75 and @xmath77 commute , @xmath79 is normal and lemma [ lem1 ] is applicable . from eqs . ( [ eq1 ] ) and ( [ eq2 ] ) , we find @xmath79 has eigenvalues @xmath80 and each eigenvalue is @xmath81-fold degenerate . by taking @xmath82 and @xmath83 in lemma [ lem1 ] , we find the rank-@xmath55 numerical range of @xmath79 is the intersection of the convex hulls of any @xmath84 eigenvalues . since each eigenvalue has multiplicity @xmath81 , each convex hull involves at least three eigenvalues . by inspecting four eigenvalues plotted in the complex plane , we easily find the intersection of all the convex hulls is a single point @xmath85 , which proves @xmath71 . similarly , we prove @xmath86 . from these equalities we obtain @xmath87 suppose @xmath42 . let @xmath25 be a rank-@xmath55 projection such that @xmath88 . let @xmath89\ ] ] where each @xmath90 has size @xmath91 . from @xmath92 and @xmath93 , we have four independent equations @xmath94 let @xmath95 be the singular value decomposition of @xmath96 , where @xmath97 is a nonnegative diagonal matrix and @xmath98 . then the above equations are solved as @xmath99 by collecting these results , we find the projection operator is decomposed as @xmath100   \[\begin{array}{cc}d&d\\   d&d\end{array}\]\[\begin{array}{cc}u^\dagger&0\\ 0&v^\dagger\end{array}\]\,.\ ] ] since rank  @xmath101 and @xmath102 , it follows from @xmath103 that @xmath104 . let @xmath105 then both @xmath57 and @xmath106 are non - singular . on the other hand , the assumption @xmath107 implies @xmath108 and hence @xmath109 , which is a contradiction . therefore , @xmath60 .    in the following , we give an explicit construction of qecc for @xmath31 in eq . ( [ eq : corrch ] ) with odd @xmath4 . the technique is based on theorem [ thm1 ] and the results in @xcite . let @xmath110 be the @xmath111 matrix with columns in the set @xmath112    define the @xmath113 matrix @xmath114 $ ] . in our qec , an @xmath5-qubit state @xmath115 is encoded with one ancilla qubit @xmath116 as @xmath117 . then a noisy quantum channel @xmath31 is applied on the encoded state and subsequently the recovery operation @xmath118 is applied so that the decoded state automatically appears in the output with no syndrome measurements . our qec is concisely summarized as @xmath119 where @xmath120 . choosing an encoding amounts to assigning each of @xmath81 column vectors in @xmath110 a basis vector of the whole hilbert space without repetition . therefore there are large degrees of freedom in the choice of encoding . in the following examples , we have chosen encoding whose quantum circuit can be implemented with the least number of cnot gates . since our decoding circuit is the inverse of the encoding circuit , it is also implemented with the least number of cnot gates .    when @xmath121 , the unitary operation @xmath122 can be chosen as @xmath123 when @xmath124 , @xmath122 can be chosen as @xmath125    figure [ nodd ] shows quantum circuits of the matrix @xmath122 for @xmath126 and @xmath127 . [ nodd ]    -qubit state @xmath115 with a single ancilla qubit initially in the state @xmath128 . ( a ) is for @xmath126 while ( b ) is for @xmath127 . the quantum channel in the box represents a quantum operation with fully correlated noise given in eq . the output ancilla state is @xmath129 for error operators @xmath130 and @xmath131 ( @xmath132 and @xmath133 ) for @xmath126 and @xmath129 for @xmath134 and @xmath135 ( @xmath136 and @xmath137 ) for @xmath127.,width=529 ]    it follows from eq . ( [ qecc ] ) that the recovery circuit is the inverse of the encoding circuit . it seems , at first sight , that the implementations given in fig . 1 contradict with eq . ( [ qecc ] ) since the controlled not gate in the end of the recovery circuit is missing in the encoding circuit . note , however , that the top qubit is set to @xmath138 initially and the controlled not gate is safely omitted without affecting encoding . we construct a decoherence - free encoding when @xmath4 is even as follows . the codeword in this case is immune to the noise operators , which is an analogue of noiseless subspace / subsystem introduced in @xcite . let @xmath139 then evidently a vector @xmath140 is separately invariant under the action of @xmath141 and @xmath9 . there are @xmath142 orthogonal vectors of such form , e.g. we have four vectors , @xmath143 for @xmath144 . thus we find a decoherence - free encoding for @xmath145 qubits by projecting onto this invariant subspace spanned by these basis . it should be noted that the projection operator @xmath25 to the subspace @xmath146 spanned by the four vectors in eq . ( [ nss ] ) satisfies rank  @xmath147 and @xmath148 , which shows @xmath149 . it is easy to generalize this result to cases with arbitrary @xmath150 . figure [ neven ] ( a ) and ( b ) depict quantum circuits for ( a ) @xmath144 and ( b ) @xmath151 , respectively . [ neven ]    -qubit state @xmath115 with two ancilla qubit initially in the state @xmath152 . ( a ) is for @xmath144 while ( b ) is for @xmath151 . the quantum channel in the box represents a quantum operation with fully correlated noise given in eq . the output ancilla state is always @xmath153 , irrespective of error operators acted in the channel.,width=529 ] we have shown that there is a quantum error correction which suppresses fully correlated errors of the form @xmath154 , in which @xmath4 qubits are required to encode ( i ) @xmath155 data qubit states when @xmath4 is odd and ( ii ) @xmath156 data qubit states when @xmath4 is even . we have proved these statements by using operator theoretical technique . neither syndrome measurements nor ancilla qubits for syndrome measurement are required in our scheme , which makes physical implementation of our scheme highly practical . examples with @xmath126 and @xmath127 are analyzed in detail and explicit quantum circuits implementing our qec with the least number of cnot gate were obtained . since the error operators are closed under matrix multiplication , errors can be corrected even when they act on the codeword many times . a somewhat similar qec has been reported in @xcite . they analyzed a partially correlated noise , where the error operators acts on a fixed number of the codeword qubits simultaneously . they have shown that the quantum packing bound was violated by taking advantage of degeneracy of the codes . justification of such a noise physically , however , seems to be rather difficult . they have also shown that correlated noise acting on an arbitrary number @xmath4 of qubits can encode @xmath157 data qubits . in contrast , we have analyzed a fully correlated noise , which shows the highest degeneracy , and have shown that @xmath158 data qubits can be encoded with an @xmath4-qubit codeword when @xmath4 is odd . clearly , our qec suppressing fully correlated errors is optimal as it is clear that one can not encode @xmath4 qubits as data qubits for odd @xmath4 and we have shown that one can not encode @xmath155 qubits for even @xmath4 . ckl was supported by a usa nsf grant , a hk rgc grant , the 2011 fulbright fellowship , and the 2011 shanxi 100 talent program . he is an honorary professor of university of hong kong , taiyuan university of technology , and shanghai university . mn and ht were supported by `` open research center '' project for private universities : matching fund subsidy from mext ( ministry of education , culture , sports , science and technology ) . ytp was supported by a usa nsf grant . nss was supported by a hk rgc grant .
we investigate an efficient quantum error correction of a fully correlated noise . 
 suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by @xmath0 , @xmath1 and @xmath2 , where @xmath3 is the number of qubits encoding the codeword . 
 it is proved that ( i ) @xmath4 qubits codeword encodes @xmath5 data qubits when @xmath4 is odd and ( ii ) @xmath4 qubits codeword implements an error - free encoding , which encode @xmath6 data qubits when @xmath4 is even . 
 quantum circuits implementing these schemes are constructed . 
 quantum error correction , higher rank numerical range , recovery operator , mixed unitary channel