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we consider pure bipartite states shared by alice and bob . let @xmath4 and @xmath5 be the labels of their respective quantum systems , each with @xmath6 dimensions . suppose alice and bob want to transform a state @xmath1 to another state @xmath2 . they each have complete classical descriptions of the states , but they can only perform local quantum operations and classical communication ( locc )  @xcite . the transformation can be performed under many scenarios . the final state can be exactly or approximately the desired output , and the transformation can work deterministically or with certain probability . it is also possible to consider converting between an asymptotically large numbers of two different states . we consider here _ only _ the scenario of exact , deterministic , and nonasymptotic transformation using locc . if such transformation is possible , we write @xmath7 , otherwise , we write @xmath8 . a necessary and sufficient condition for @xmath7 was found by nielsen  @xcite , with beautiful connections to the theory of majorization . for a pure state @xmath9 , let @xmath10 denote the list of eigenvalues of the reduced density matrix @xmath11 in descending order , and let @xmath12 be the @xmath13-th element in @xmath10 . nielsen s criterion can be stated as | | _ m d _ i=1^m _ ^(i ) _ i=1^m _ ^(i ) an intriguing consequence of the criterion is that , even when @xmath14 , it may still be possible to transform the state @xmath1 to the state @xmath2 as part of a larger transformation . for example , @xmath15 may be true for some state @xmath16 , which functions as a catalyst  @xcite . it is also possible for @xmath17  @xcite .    in ref . @xcite , interesting properties about multiple - copy transformation between @xmath18 and @xmath19 under various scenarios are investigated . in the scenario of exact , deterministic , asymptotic transformation considered in this paper , they have derived a necessary condition for @xmath20 for some @xmath0 , @xmath21 and @xmath22 . they have also made a very plausible conjecture , |^k |^k _ r k |^r |^r . the conjecture is obviously true when @xmath0 divides @xmath23 . however , we find counterexamples for @xmath24 , @xmath25 , and @xmath26 . one example for @xmath27 is given by _ = \{0.493 , 0.284 , 0.158 , 0.035 , 0.030 } + _ = \{0.598 , 0.145 , 0.129 , 0.125 , 0.003 } and a simple example for @xmath28 is given by _ = \{0.24 , 0.22 , 0.22 , 0.19 , 0.10 , 0.03 } + _ = \{0.27 , 0.25 , 0.16 , 0.16 , 0.15 , 0.01 } the conjecture fails because of the exact and nonasymptotic nature of the transformation . we remark that , for fixed @xmath0 , a counterexample in @xmath29 dimensions can always be turned into one in @xmath30 dimensions . the reason is that the criteria for a counterexample involve only continuous functions of @xmath31 and @xmath32 . therefore , the above counterexamples suffice to disprove the conjecture for @xmath24 and @xmath33 . in the following , we describe numerical results that provide not only counterexamples but also statistics of violations of the conjecture . we performed a numerical search for @xmath34 and @xmath24 . the @xmath35 case is uninteresting , as @xmath8 implies @xmath36 . for each @xmath6 , a large number of random pairs @xmath37 are sampled , and without loss of generality , @xmath21 . we search for @xmath38 corresponding to @xmath39 satisfying + @xmath8 and @xmath40 + @xmath41 + the search parameters and the results can be summarized :    @xmath42 \hline \rule{0pt}{2.4ex }   \begin{array}{l } \mbox{sample size } \end{array } &   4 \times 10 ^ 8 & 2 \times 10 ^ 8 & 10 ^ 7 & 10 ^ 7 & 10 ^ 7 & 10 ^ 7 \\ \hline \begin{array}{l } \mbox{fraction of sample }                   \\ \mbox{satisfying { \bf i } } \end{array } &   8.31 \times 10^{-3 } & 1.56 \times 10^{-2 } &   2.16 \times 10^{-2 } & 2.63 \times 10^{-2 } &   3.01 \times 10^{-2 } & 3.31 \times 10^{-2 } \\ \hline \begin{array}{l } \mbox{fraction of sample }          \\ \mbox{satisfying { \bf ii } }           \mbox { given { \bf i } } \end{array } &   0 & 3.21 \times 10^{-7 } &   3.94 \times 10^{-4 } & 3.12 \times 10^{-4 } &   1.60 \times 10^{-4 } & 1.63 \times 10^{-4 } \\ \end{array } $ ]    the fractions of sample satisfying * i * have negligible statistical uncertainties , while the fractions satisfying * ii * conditioned on * i * have about 11 - 14% uncertainties and @xmath43 for which few counterexamples are found . ] . for each @xmath44 , tens of counterexamples for the conjecture are found . for @xmath27 , exactly one counterexample is found in @xmath45 random samples . for @xmath46 no counterexample is found in @xmath47 random samples , so that the conjecture may still be true for this value of @xmath6 . the conjecture investigated in this paper fails because of the exact , deterministic , and nonasymptotic nature of the transformation . the conclusion drawn from our results , along with refs . @xcite , is that even if one knows for all @xmath48 , whether @xmath49 , one still can not infer whether @xmath50 without additional information on the states . this is in sharp contrast with the much simpler conditions for asymptotic manipulations . in particular , asymptotically @xmath51 and @xmath52 are interconvertible with high fidelity and probability if @xmath53 where @xmath54 is the entropy function . nielsen s criterion on transformation of pure bipartite states has led to many important results in the various possible scenarios , some of which are counterintuitive . the results in the exact , deterministic , nonasymptotic scenario are mathematically interesting , yet their very oddity suggests that the asymptotic or probabilistic scenarios are perhaps more physically relevant . we thank charles  h.  bennett for interesting discussions and the ibm watson cafeteria for enlightening coffee .
in this brief comment , we consider the exact , deterministic , and nonasymptotic transformation of multiple copies of pure states under locc . 
 it was conjectured in quant - ph/0103131 that , if @xmath0 copies of @xmath1 can be transformed to @xmath0 copies of @xmath2 , the same holds for all @xmath3 . 
 we present counterexamples to the above conjecture .